ISSN 2590-9770
The Art of Discrete and Applied Mathematics 4 (2021) #P1.01
https://doi.org/10.26493/2590-9770.1312.5b4
(Also available at http://adam-journal.eu)
Palindromic products
⇤
Richard H. Hammack† , Jamie L. Shive
Virginia Commonwealth University, Dept. of Mathematics, Richmond, VA 23284, USA
Received 28 July 2019, accepted 20 January 2020, published online 8 September 2020
Abstract
A graph G on n vertices is said to be palindromic if there is a vertex-labeling bijection
f : V (G) ! {1, 2, . . . , n} with the property that for any edge vw 2 E(G) there is an edge
xy 2 E(G) for which f(x) = n  f(v) + 1 and f(y) = n  f(w) + 1.
This notion was defined and explored in a recent paper [R. Beeler, Palindromic graphs,
Bulletin of the ICA, 85 (2019) 85–100]. The paper gives sufficient conditions on the factors
of a Cartesian product of graphs that ensure the product is palindromic, but states that it
is unknown whether the conditions are necessary. We prove that the conditions are indeed
necessary. Further, we prove a parallel result for the strong product of graphs.
Keywords: Palindromic graphs, cartesian product of graphs, strong product of graphs.
Math. Subj. Class. (2020): 05C76, 05C78
1 Introduction
A recent article by R. Beeler [1] introduced a new concept. A graph G on n vertices is
palindromic provided that there is a vertex-labeling bijection f : V (G) ! {1, 2, . . . , n}
with the property that to each vw 2 E(G) there corresponds an xy 2 E(G) for which
f(x) = n+ 1  f(v) and f(y) = n+ 1  f(w).
Palindromic graphs, like palindromic words, have a certain symmetry. The mapping
V (G) ! V (G) whose effect on labels is k 7! n+1 k is an involution (an automorphism
of order 2). View it as a mirror symmetry, where the vertices are ordered on a line by their
labels, as in Figure 1.
This induced involution has no fixed vertex if n is even, and exactly one fixed vertex
if n is odd. Indeed, we have the following characterization of palindromic graphs as those
graphs admitting an involution that fixes at most one vertex. (The order of a graph is its
number of vertices. For other standard terms and notations not defined here see West [5].)
⇤We thank the referees.
†Supported by Simons Foundation Collaboration Grant for Mathematicians 523748.
E-mail addresses: rhammack@vcu.edu (Richard H. Hammack), shivejl@vcu.edu (Jamie L. Shive)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Art Discrete Appl. Math. 4 (2021) #P1.01
v w y x
1 2 3 4 n
v w y x
1 2 3 4 n
Figure 1: Palindromic graphs of even order admit an involution with no fixed points. Palin-
dromic graphs of odd order admit an involution with exactly one fixed point.
Theorem 1.1 (Beeler [1]). A graph of even order is palindromic if and only if it admits
an involution with no fixed vertices. A graph of odd order is palindromic if and only if it
admits an involution with exactly one fixed vertex.
Guided by this theorem, we define a graph to be even palindromic if it is palindromic
and of even order; it is odd palindromic if it is palindromic and of odd order. An invo-
lution that fixes at most one vertex is called a palindromic involution; one that fixes no
vertex is an even palindromic involution, and one that fixes exactly one vertex is an odd
palindromic involution. Thus a graph is even palindromic if and only if admits an even
palindromic involution; it is odd palindromic if and only if it admits an odd palindromic
involution. A fixed point is a fixed vertex.
Beeler [1] characterizes several classes of palindromic graphs, including hypercubes
(see Figure 2). More generally he addresses the Cartesian product of graphs, and we will
expand upon this in the next section.
Figure 2: Every hypercube is palindromic. Here is the 4-cube.
2 Cartesian Products
The Cartesian product of graphs G and H is the graph G⇤H with vertices V (G)⇥V (H)
and edges
E(G⇤H) =
 
(x, y)(x0, y0) | xx0 2 E(G) and y = y0, or x = x0 and yy0 2 E(H)
 
.
(See Figure 3.) This product is commutative and associative in the sense that the maps
(x, y) 7! (y, x) and ((x, y), z) 7! (x, (y, z)) are isomorphisms G⇤H ! H⇤G and
(G⇤H)⇤K ! G⇤(H⇤K).
Given automorphisms ↵ : G ! G and   : H ! H , it is straightforward from the
definitions that (x, y) 7! (↵(x), (y)) is an automorphism of G⇤H . For example, in
Figure 3, let ↵ : G ! G be the even palindromic involution of G reflecting G across
R. H. Hammack and J. L. Shive: Palindromic products 3
G
H G⇤H
Figure 3: Cartesian product of graphs.
a vertical axis. Let   : H ! H be the identity. Then (x, y) 7! (↵(x), y) is an even
palindromic involution of G⇤H reflecting it across a vertical axis. This suggests that if
one factor of a product is even palindromic, then the product will be even palindromic.
Indeed, we have the following result [1, Theorem 4.4].
Lemma 2.1. If G or H is even palindromic, then G⇤H is even palindromic. If G and H
are odd palindromic, then G⇤H is odd palindromic.
Proof. Let one of G or H (say G) be even palindromic. Theorem 1.1 yields an even
palindromic involution ↵ : G ! G. Form the even palindromic involution (x, y) 7!
(↵(x), y) of G⇤H . Thus the product is even palindromic. For the second statement, say
both G and H are odd palindromic. By Theorem 1.1, G has an involution ↵ with exactly
one fixed point x0. (That is, ↵(x0) = x0.) For the same reason, H has an involution   with
exactly one fixed point y0. Then (x, y) 7! (↵(x), (y)) is an involution of G⇤H that has
exactly one fixed point (x0, y0). Therefore G⇤H is odd palindromic.
Lemma 2.1 spells out conditions on the factors that are sufficient for a palindromic
product. Beeler [1] states that it is unknown whether these conditions are also necessary.
We will shortly prove that in fact they are, but we first need to review prime factorizations
over the Cartesian product.
Observe that K1⇤G ⇠= G for any graph G, so K1 is the unit for the Cartesian product.
A nontrivial graph G is prime over ⇤ if for any factoring G ⇠= A⇤B, one of A or B is K1
and the other is isomorphic to G. Certainly every graph can be factored into prime factors.
Sabidussi and Vizing [3, 4] proved that each connected graph has a unique prime factoring
up to order and isomorphism of the factors. More precisely, we have the following.
Theorem 2.2 ([2, Theorem 6.8]). Let G and H be isomorphic connected graphs G =
G1⇤ · · ·⇤Gk and H = H1⇤ · · ·⇤H`, where each factor Gi and Hi is prime. Then k = `,
and for any isomorphism ' : G ! H , there is a permutation ⇡ of {1, 2, . . . , k} and
isomorphisms 'i : G⇡(i) ! Hi for which
'(x1, x2, . . . , xk) =
 
'1(x⇡(1)),'2(x⇡(2)), . . . ,'k(x⇡(k))
 
.
Now we can prove our main result about palindromic Cartesian products.
Theorem 2.3. Suppose G and H are connected graphs. Then:
(1) G or H is even palindromic if and only if G⇤H is even palindromic.
4 Art Discrete Appl. Math. 4 (2021) #P1.01
(2) G and H are odd palindromic if and only if G⇤H is odd palindromic.
Proof. One direction is Lemma 2.1. Conversely, suppose G⇤H is palindromic and let '
be a palindromic involution of it. Take prime factorings G = G1⇤ · · ·⇤Gj and H =
Gj+1⇤ · · ·⇤Gk, so ' is an involution of G⇤H = (G1⇤ · · ·⇤Gj)⇤(Gj+1⇤ · · ·⇤Gk).
The involution ' permutes the prime factors of this product in the sense of Theorem 2.2,
where the permutation ⇡ satisfies ⇡2 = id. Using commutativity of ⇤, group together the
prime factors Gi of G for which 1 < ⇡(i)  j, and call their product A. (By convention,
A = K1 if there are no such factors Gi. The same applies for the graphs B and D defined
below.) Let B be the product of the remaining factors Gi of G. Also group together the
prime factors Gi of H for which j+1 < ⇡(i)  k, and call their product D. The Cartesian
product of the remaining factors of H is then a graph isomorphic to B. The structure
of ' under this scheme is as indicated below, where the arrows represent isomorphisms
'i : G⇡(i) ! Gi between factors.
'
G
G
H
H
=
=
(
(
(
(
)
)
)
)
⇤
⇤
⇤
⇤
⇤
⇤
⇤
⇤
⇤
⇤
⇤
⇤
⇤
⇤
⇤
⇤
⇤
⇤
⇤
⇤
⇤
⇤
G1
G1
G2
G2
G3
G3
G4
G4
G5
G5
G6
G6
G7
G7
G8
G8
G9
G9
G10
G10
G11
G11
Gz }| { Hz }| {
Az }| { Bz }| { Bz }| { Dz }| {
| {z }
A
| {z }
B
| {z }
B
| {z }
D
We have coordinatized G and H as G = A⇤B and H = B⇤D, and ' is an involution
of G⇤H = (A⇤B)⇤(B⇤D) for which '
 
(a, b), (b0, d)
 
=
 
(↵(a), (b0)), ( (b),  (d))
 
,
for automorphisms ↵ : A ! A,  ,   : B ! B and   : D ! D. But because '2 is the
identity, it must be that ↵2 = id,   =   1 and  2 = id. Thus we have involutions ↵ and  
of A and D, respectively, and
'
 
(a, b), (b0, d)
 
=
 
(↵(a), (b0)), (  1(b),  (d))
 
, (2.1)
From (2.1) it is evident that the fixed points of ' (if any) are precisely
 
(a0, (b)), (b, d0)
 
with ↵(a0) = a0,  (d0) = d0, and b 2 V (B). (2.2)
Thus ' has a fixed point if and only if both ↵ and   have fixed points. Further, if ' has a
fixed point, then it has exactly |V (B)| of them.
Now suppose G⇤H is even palindromic. Let ' be an even palindromic involution of
G⇤H (having no fixed point). From (2.2), at least one of ↵ or   has no fixed point; say it
is ↵. Then ↵ is an even palindromic involution of A, so A is even palindromic. By the first
part of the theorem, G = A⇤B is even palindromic. Similarly H is even palindromic if  
has no fixed points.
Suppose G⇤H is odd palindromic. Let ' be an odd palindromic involution whose
sole fixed point is
 
(a0, (b0)), (b0, d0)
 
. The remark following (2.2) implies ' has at
least |V (B)| fixed points, so B = K1. Thus we can drop B from our discussion, so
G = A, H = D and '(a, d) =
 
↵(a),  (d)
 
. We now have involutions ↵ : G ! G and
  : H ! H with fixed points a0 and d0, respectively. Also (a0, d0) is a fixed point of '. If
R. H. Hammack and J. L. Shive: Palindromic products 5
the involution ↵ of G had a second fixed point a1, then (a0, d0) and (a1, d0) would be two
distinct fixed points of '. Thus a0 is the only fixed point of ↵, so ↵ (hence also G) is odd
palindromic. By the same reasoning H is odd palindromic.
3 Strong Products
The strong product of graphs G and H is the graph G⇥H with vertex set V (G)⇥V (H),
where distinct vertices (x, y) and (x0, y0) are adjacent whenever
 
xx
0 2 E(G) or y = y0
 
and
 
x = x0 or yy0 2 E(H)
 
.
See Figure 3. We quickly review this product’s properties; Chapter 7 of [2] proves all
assertions made here. The strong product is commutative and associative. If NG[x] :=
N(x) [ {x} is the closed neighborhood of a vertex x 2 V (G), then
NG⇥H [(x, y)] = NG[x]⇥NH [y]. (3.1)
Also K1 ⇥ G ⇠= G for all graphs G. A graph G is prime over ⇥ if for any factoring
G = A⇥B, one of A or B is K1 and the other is isomorphic to G.
G
H G⇥H
Figure 4: Strong product of graphs.
Given automorphisms ↵ : G ! G and   : H ! H , it is straightforward from the
definitions that (x, y) 7!
 
↵(x), (y)
 
is an automorphism of G ⇥ H . For instance, in
Figure 4, let ↵ : G ! G be the even palindromic involution of G reflecting G across
a vertical axis. Say   : H ! H is the identity. Then (x, y) 7!
 
↵(x), y
 
is an even
palindromic involution of G⇥H reflecting it across a vertical axis (relative to the drawing).
This suggests that we might expect a result for the strong product that is parallel to
Theorem 2.3 for the Cartesian product. Indeed, this is exactly the case, but the proof is
more involved. The complication is that in general the strong product has no result parallel
to Theorem 2.2, unless we impose an additional restriction. A graph is called S-thin if
no two distinct vertices have the same closed neighborhood. We will need the following
analogue of Theorem 2.2 for S-thin graphs.
Theorem 3.1 ([2, Theorem 7.16]). Let ' be an automorphism of an S-thin connected
graph G with prime factorization G = G1 ⇥G2 ⇥ · · ·⇥Gk. Then there is a permutation
⇡ of {1, 2, . . . , k} and isomorphisms 'i : G⇡(i) ! Gi for which '(x1, x2, . . . , xk) = 
'1(x⇡(1)),'2(x⇡(2)), . . . ,'k(x⇡(k))
 
.
6 Art Discrete Appl. Math. 4 (2021) #P1.01
We say that vertices x and y of a graph are in relation S, written xSy, provided that
each has the same closed neighborhood, that is, N [x] = N [y]. It is easy to check that S is
an equivalence relation of the graph’s vertex set. We call an S-equivalence class of V (G)
an S-class of G. (Note that a graph is S-thin if and only if each S-class consists of a single
vertex.) In general, if X is an S-class of graph G, then the subgraph of G induced on X is
the complete graph K|X|. Also, for any distinct S-classes X and Y , either each vertex of
X is adjacent to all vertices of Y , or no vertex of X is adjacent to any vertex of Y .
Given a graph G, we define the quotient G/S to be the graph whose vertices are the
S-classes of G, and for which XY 2 E(G/S) provided that X 6= Y and G has an edge
joining X to Y . Check that G/S is always S-thin.
Because S is defined in terms of the adjaceny structure of a graph, any isomorphism
' : G ! H sends S-classes of G bijectively onto S-classes of H . From the discussion
above it should be clear that any isomorphism ' : G ! H induces an isomorphism e' :
G/S ! H/S where e'(X) = '(X), that is, e'(X) is the image of the S-class X under '.
But the existence of an isomorphism e' : G/S ! H/S does not necessarily mean that
there is an isomorphism ' : G ! H . However, if |X| = |e'(X)| for each X 2 V (G/S),
then we can lift e' to an isomorphism ' : G ! H simply by declaring ' to restrict to a
bijection X ! e'(X) for each X .
Using Equation (3.1), one can show that the S-classes of G⇥H are precisely the (set)
Cartesian products X ⇥ Y , where X is an S-class of G and Y is an S-class of H . In other
words, the vertices of (G ⇥ H)/S are X ⇥ Y , where X 2 V (G/S) and Y 2 V (H/S).
Further, there is a natural isomorphism
(G⇥H)/S  ! G/S ⇥H/S
X ⇥ Y 7 ! (X,Y ). (3.2)
In the proof our main theorem we will switch between X⇥Y and (X,Y ) when expedient.
The proof also uses all ideas discussed so far in this section.
Theorem 3.2. Suppose G and H are connected graphs. Then:
(1) G or H is even palindromic if and only if G⇥H is even palindromic.
(2) G and H are odd palindromic if and only if G⇥H is odd palindromic.
Proof. If G or H (say G) is even palindromic, then there exists an even palindromic invo-
lution ↵ of G, so (x, y) 7! (↵(x), y) is an even palindromic involution of G ⇥ H . Next
suppose G and H are odd palindromic. Then G has an odd palindromic involution ↵ with
fixed point x0, and H has an odd palindromic involution   with fixed point y0. Then
(x, y) 7! (↵(x), (y)) is an odd palindromic involution of G ⇥H whose sole fixed point
is (x0, y0).
It remains to prove the converses of the two statements. We will do this in three parts.
The first part codifies the structure of involutions of G⇥H .
Part I (Involution structure) Let ' : G⇥H ! G⇥H be an involution. By the remarks
preceding this theorem, ' induces an automorphism e' of the S-thin graph (G ⇥H)/S ⇠=
G/S ⇥ H/S. Because ' is an involution, we have e'2 = id. (Note that e' could be the
identity even if ' is not. This is the case if ' fixes each S-class, i.e., it restricts to a
permutation on each S-class.)
R. H. Hammack and J. L. Shive: Palindromic products 7
Take prime factorings G/S = G1 ⇥ · · · ⇥Gj and H/S = Gj+1 ⇥ · · · ⇥Gk. Then e'
is an automorphism (of order 1 or 2) of the graph
G/S ⇥H/S = (G1 ⇥ · · ·⇥Gj)⇥ (Gj+1 ⇥ · · ·⇥Gk).
Now, e' permutes the prime factors of this product in the sense of Theorem 3.1, where the
permutation ⇡ satisfies ⇡2 = id. As in the proof of Theorem 2.3, group together the prime
factors Gi of G/S for which 1 < ⇡(i)  j, and call their product A. Let B be the product
of the remaining factors of G/S. Also group together the prime factors Gi of H/S for
which j + 1 < ⇡(i)  k, and call their product D. The product of the remaining factors of
H/S is then a graph isomorphic to B. Now we have G/S = A ⇥ B and H/S = B ⇥D,
and e' is an automorphism of
G/S ⇥H/S = (A⇥B)⇥ (B ⇥D)
satisfying e'2 = id, and for which (as in the proof of Theorem 2.3) we have
e'
 
(a, b), (b0, d)
 
=
 
(↵(a), (b0)), (  1(b),  (d))
 
(3.3)
for automorphisms ↵ : A ! A,   : B ! B and   : D ! D, with ↵2 = id and  2 = id.
In (3.3), the ordered pairs (a, b) and (↵(a), (b0)) are vertices of G/S, which are S-
classes of G (subsets of V (G)), and hence they have cardinalities |(a, b)| and |(↵(a), (b0))|.
Similarly, (b0, d) and (  1(b),  (d)) are S-classes of H/S.
By the remarks preceding this theorem, the involution ' of G ⇥ H sends the S-class
(a, b)⇥ (b0, d) bijectively to S-class (↵(a), (b0))⇥ (  1(b),  (d)), so
|(a, b)| · |(b0, d)| =
   ↵(a), (b0)
    ·
     1(b),  (d)
    (3.4)
for all a 2 V (A), b, b0 2 V (B) and d 2 V (D). Putting b0 =   1(b) yields
|(a, b)| · |(  1(b), d)| =
   ↵(a), b)
    ·
     1(b),  (d)
   . (3.5)
In (3.5) replace d with  (d) (and use  2 = id) to get
|(a, b)| · |(  1(b),  (d))| =
   ↵(a), b)
    ·
     1(b), d
   . (3.6)
Equations (3.5) and (3.6) imply |(a, b)| = |(↵(a), b)|. Form an automorphism e↵ :
A⇥B ! A⇥B as e↵(a, b) =
 
↵(a), b)
 
. Then e↵2 = id, so we have an involution (if it is
not the identity map) e↵ : G/S ! G/S that maps each vertex (S-class) (a, b) to the vertex
(S-class) (↵(a), b) of the same cardinality.
Also (3.5) and (3.6) yield
     1(b),  (d)
    =
     1(b), d
   , so |(b,  (d))| = |(b, d)|
for all b 2 V (B) and d 2 V (D). Form the automorphism e  : B ⇥ D ! B ⇥ D where
e (b, d) =
 
b,  (d)
 
. Then e 2 = id, so we have an involution (if not the identity map) e  :
H/S ! H/S mapping each S-class (b, d) to the S-class (b,  (d)) of the same cardinality.
In summary, for any involution ' of G⇥H , we have constructed automorphisms e↵ and
e  of G/S and H/S, respectively, for which e↵2 = id and e 2 = id. And |e↵((a, b))| = |(a, b)|
for any S-class (a, b) of G. Thus we can lift e↵ to an automorphism   : G ! G by declaring
that   restricts to a bijection (a, b) !
 
↵(a), b
 
, for each S-class (a, b) of G. Similarly,
|e ((b, d))| = |(b, d)| for any S-class (b, d) of H , so we can lift e  to an automorphism
µ : H ! H . In parts II and III of the proof these lifts will be palindromic involutions.
8 Art Discrete Appl. Math. 4 (2021) #P1.01
To carry out this plan we will need to consider S-classes of G⇥H that are fixed by '
(i.e., the S-classes whose vertices are permuted by '.) By Equation (3.3), the fixed points
of e' (respectively, the fixed S-classes of ') are
 
(a0, (b)), (b, d0)
 
where ↵(a0) = a0,  (d0) = d0 and b 2 V (B) (3.7)
(a0, (b))⇥ (b, d0) where ↵(a0) = a0,  (d0) = d0 and b 2 V (B). (3.8)
We call an S-class even (odd) if it has even (odd) cardinality.
Part II (Converse of Statement (1)) Suppose G⇥H is even palindromic. Then there is an
even palindromic involution ' of G ⇥H . We retain the development and notation of Part
I of the proof.
Our strategy is to show that one of e↵ : G/S ! G/S or e  : H/S ! H/S has no
odd fixed point (S-class). For if this is the case for (say) e↵, then e↵ can be lifted to an
automorphism   : G ! G sending any S-class (a, b) bijectively to (↵(a), b). Whenever
e↵ fixes an S-class (a, b), we can arrange for   to restrict to an order-2 fixedpoint-free
permutation of the even set (a, b). Then   will be an even palindromic involution of G, so
G is even palindromic.
Suppose to the contrary that e↵ had an odd fixed point (a, b) and e  had an odd fixed
point (b0, d). (So ↵(a) = a and  (d) = d.) By (3.4),
   a, b
   
| {z }
odd
·
   b0, d
   
| {z }
odd
=
   a, (b0)
    ·
     1(b), d
   .
Then (a, (b0)) is odd, so (a, (b0))⇥(b0, d) is an odd S-class of G⇥H . But the involution
' fixes this odd S-class, by (3.8). Thus ' fixes some point of this S-class, contradicting
the fact that ' is even palindromic.
Part III (Converse of Statement (2)) Suppose G⇥H is odd palindromic. Then there is an
odd palindromic involution ' of G⇥H with fixed point (x0, y0). Then ' fixes the S-class
X that contains (x0, y0), which necessarily has form X = (a0, (b0)) ⇥ (b0, d0), where
↵(a0) = a0 and  (d0) = d0. (See (3.8) in Part I.) As the involution ' fixes exactly one
vertex, which is in X , we know X has odd cardinality. Thus (a0, (b0)) is an odd S-class
of G/S, and (b0, d0) is an odd S-class of H/S. Note that (a0, (b0)) is a fixed point of
e↵ and (b0, d0) is a fixed point of e . Suppose e  had another odd fixed point (b1, d1). Then
 (d1) = d1 and by Equation (3.4),
   a0, (b0)
   
| {z }
odd
·
   b1, d1
   
| {z }
odd
=
   a0, (b1)
    ·
   b0, d1
   .
Therefore
   a0, (b1)
    and |(b0, d1)| are odd. Then (a0, (b1))⇥(b1, d1) and (a0, (b0))⇥
(b0, d1) are odd S-classes of G⇥H that are fixed by '. But X = (a0, (b0))⇥ (b0, d0) is
the only such S-class, hence  (b1) =  (b0) and d1 = d0. This means (b1, d1) = (b0, d0).
Conclusion: (b0, d0) is the only odd S-class of H/S that is fixed by e . Therefore we can
lift e  : H/S ! H/S to an odd palindromic involution µ : H ! H sending each S-class
(b, d) bijectively to (b,  (d)), having only one fixed vertex on the odd fixed class (b0, d0)
and no fixed points on any other fixed (even) S-class. Thus H is odd palindromic.
By a symmetric argument, G is also odd palindromic.
R. H. Hammack and J. L. Shive: Palindromic products 9
4 Conclusion and Open Questions
Our Theorems 2.3 and 3.2 characterize palindromic Cartesian and strong products in terms
of the palindromic properties of their factors. There are four standard associative graph
products, the Cartesian, strong, direct and lexicographic products. (See [2].) Here we
have only addressed two of these four products. A natural unexplored problem, then, is
to establish analogous results for palindromic direct and lexicographic products. However,
because the automorphism structure of these products is not as rigid as for the Cartesian
and strong products (cf. Theorems 2.2 and 3.1 above), the results and proofs are likely to
be substantially different from those presented here.
ORCID iDs
Richard H. Hammack https://orcid.org/0000-0002-6384-9330
Jamie L. Shive https://orcid.org/0000-0002-8294-1423
References
[1] R. A. Beeler, Palindromic graphs, Bull. Inst. Combin. Appl. 86 (2019), 85–100.
[2] R. Hammack, W. Imrich and S. Klavžar, Handbook of product graphs, Discrete Mathematics and
its Applications (Boca Raton), CRC Press, Boca Raton, FL, 2nd edition, 2011, with a foreword
by Peter Winkler.
[3] G. Sabidussi, Graph multiplication, Math. Z. 72 (1959/60), 446–457, doi:10.1007/bf01162967.
[4] V. G. Vizing, The cartesian product of graphs, Vyčisl. Sistemy No. 9 (1963), 30–43.
[5] D. B. West, Introduction to graph theory, Prentice Hall, Inc., Upper Saddle River, NJ, 1996.
ISSN 2590-9770
The Art of Discrete and Applied Mathematics 4 (2021) #P1.02
https://doi.org/10.26493/2590-9770.1304.2cf
(Also available at http://adam-journal.eu)
A note on a candy sharing game
⇤
Deepak Bal
Department of Mathematics, Montclair State University, Montclair, NJ, USA
Joseph DeGaetani
Department of Mathematical Sciences, University of Delaware, Newark, DE, USA
Received 9 July 2019, accepted 3 February 2020, published online 10 September 2020
Abstract
Suppose k students sit in a circle and are each distributed some initial amount of candy.
Each student begins with an even amount of candy, but their individual amounts may vary.
Upon the teacher’s signal, each student passes half of their candy to their left and keeps half.
After this step, any student with an odd amount of candy receives an extra piece. The game
ends if all the students are holding the same amount of candy. We prove, in a generalized
setting, that for any initial distribution of n pieces of candy, the game terminates after
O(log n) many iterations and each student ends with nk +O(log n) many pieces. Moreover,
there exist initial distributions for which the O(log n) term cannot be improved.
Keywords: Games on graphs, Markov chains.
Math. Subj. Class. (2020): 05C20, 60J10
1 Introduction
In this note, we analyze a game referred to as the candy sharing game. Suppose a fixed
number k many students sit in a circle and are each distributed some initial amount of
candy. Each student begins with an even amount of candy, but their individual amounts
may vary. Upon the teacher’s signal, each student passes half of their candy to their left
and keeps half of their candy. After this step, any student with an odd amount of candy
receives an extra piece. The game ends if all the students are holding the same amount of
candy. Does the game end after finitely many steps for any initial distribution?
This question originated as a problem in the Beijing Math Olympiad [2]. A web search
shows it remains popular as a fun activity for budding mathematicians as well as a challenge
⇤The authors would like to thank the referees for their careful reading and numerous suggestions which helped
improve the presentation of the paper. Also, for pointing out an error in an earlier version.
E-mail addresses: deepak.bal@montclair.edu (Deepak Bal), degaetanij2@montclair.edu (Joseph DeGaetani)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Art Discrete Appl. Math. 4 (2021) #P1.02
in computer coding competitions. The answer to the question is “yes” and it boils down to
the following three observations: (1) The maximum amount of candy held by any player
can never increase. (2) The number of players holding the currently minimum amount of
candy will decrease by at least one each step. Thus the minimum amount will increase after
at most k steps. (3) Once the maximum and minimum are equal, the game terminates.
In [4], Iba and Tanton consider a generalized version of the game in which each player
fixes an integer and at each step shares portions of their candy to some subset of the other
players. After receipt of candy from the other players, they round up to the nearest multi-
ple of their chosen integer. They prove that under certain conditions, such games are also
bounded, i.e., reach a stable state after some finite number of moves. In [1], Cairns consid-
ers a version of the game where at each step, each player with more than one piece of candy
passes one piece to their left and one to their right. The game is played until it settles into
a fixed state or an oscillatory pattern. He completely characterizes the long term behavior
given any initial distribution in the case then the number of students and candies are both
k.
For the original candy sharing game, given an initial distribution of candy, predicting
the length of the game and the final stabilizing amount is an intriguing open question. The
main purpose of this note is to provide an upper bound on each of the above parameters
which is tight infinitely often (at least in the case when the number of players k = 3).
We will consider the candy sharing game played in a generalized setting. We say a
directed graph G is d-regular if every vertex has in- and out- degree equal to d. The candy
sharing game can be played on any d-regular graph G as follows. Each vertex (player) is
distributed some amount of candy which is divisible by d. At each step, each player hands
1/d proportion of their candy to each of their out-neighbors and is handed candy from each
of their in-neighbors. After this, each student is handed 0, 1, . . . , or d  1 pieces of candy
to ensure they are holding a multiple of d. Thus the original candy sharing game is one
played on the directed cycle of length k with a directed loop at each vertex. A directed
graph G is strongly connected if between any two vertices u and v, there exists a directed
u, v-path. G is aperiodic if the greatest common divisor of the lengths of its cycles is 1.
Our main theorem is as follows.
Theorem 1.1. Let k   d   2 be fixed. For any d = (a1, . . . , ak) such that
Pk
i=1 ai = n
the candy sharing game, played with initial candy distribution d, on a strongly connected,
aperiodic, d-regular directed graph G ends in O(log n) turns, and every player will be
holding
n
k +O(log n) pieces of candy.
The next proposition shows that, at least in some specific cases, the order of the log n
term in the final distribution cannot be improved.
Proposition 1.2. There exist infinitely many values of n such that the candy sharing game
with initial distribution d = (n, 0, 0) played on a directed cycle of length 3 with a directed
loop on each vertex terminates with after ⌦(log n) turns with every player holding n3 +
⌦(log n) pieces of candy.
In the next section we introduce the definitions and results from the theory of Markov
chains necessary for the proof. In Section 3 we prove Theorem 1.1.
D. Bal and J. DeGaetani: A note on a candy sharing game 3
2 Notation and background
Throughout the paper, we consider k and d to be fixed, so O(·) and ⌦(·) notation is sup-
pressing constants which may depend on k and d. All logarithms are natural unless oth-
erwise stated. Vectors will be denoted by boldface characters and for a vector v, we use
the notation v(i) to denote the ith entry of v. Suppose the game is played on a directed
graph G with vertex set V (G) = {1, . . . , k} and edge set E(G). For t = 0, 1, 2, . . ., let
dt = (a1,t, a2,t, . . . , ak,t) where ai,t represents the amount of candy held by player i after
t steps of the game. One can check that for any t   1, if dt 1 = (b1, . . . , bk), then we have
the ith entry of dt is given by
dt(i) = d ·
2
666
1
d
X
j:ji2E(G)
bj
d
3
777
.
At each step of the game, if a player has an amount of candy not divisible by d, then they
receive extra pieces. Each piece of candy introduced in this way is referred to as a draw.
Note that if the game terminates with each player holding s pieces of candy, then the total
amount of candy at termination is sk. So the total number of draws in a candy game with
n pieces initially distributed is sk   n. The number of draws up to and including turn t is
denoted  t.
To continue we need some definitions and results regarding Markov chains. All notation
and definitions follow those in [5]. A Markov chain with state space ⌦ and transition
matrix P is a sequence of random variables (X0, X1, . . .) on ⌦ such that for all i   0,
if Xi has distribution µ, then Xi+1 has distribution µP . We represent distributions on ⌦
as row vectors and P as an |⌦| ⇥ |⌦| matrix where entry Pij represents the probability of
transitioning from state i to state j.
We say a chain is irreducible if, for any two states, there is a finite number of steps in
which it is possible to transition from one state to the other with positive probability. This
number of steps may be dependent on the chosen states. A chain is aperiodic if, for each
state, the greatest common divisor of the set of times that it is possible to transition from
the state back to itself is 1.
Further, a distribution ⇡ is stationary for P if ⇡P = ⇡. Any irreducible chain has
a unique stationary distribution, and each starting state will converge to the stationary.
The total variation distance between two probability distributions, µ and ⌫ is given by
||µ  ⌫||TV = maxA⇢⌦ |µ(A)  ⌫(A)|. There is a convenient formula for calculating the
total variation distance, given by
||µ  ⌫||TV =
1
2
X
x2⌦
|µ(x)  ⌫(x)| . (2.1)
As we are concerned with how quickly the game stabilizes, we will make use of the fol-
lowing theorem (Thm 4.9 in [5]) which gives a bound on the rate of convergence of an
irreducible aperiodic Markov chain.
Theorem 2.1 (Convergence Theorem). If P is the transition matrix of an irreducible and
aperiodic chain, with stationary distribution ⇡, then for any initial probability distribution
x on state space ⌦, there exist constants ↵ 2 (0, 1) and C > 0 such that for all t   0,
||xP t   ⇡||TV  C↵t.
4 Art Discrete Appl. Math. 4 (2021) #P1.02
3 Proof of Theorem 1.1
Proof of Theorem 1.1. Given a d-regular directed graph G on vertex set {1, 2, . . . , k}, we
may form a k ⇥ k transition matrix P whose ij entry is 1/d if ij is an edge, and 0 other-
wise. We can view the Markov chain with transition matrix P as a randomized candy shar-
ing game where at each stage, vertices no longer draw, but instead individually distribute
each piece of candy uniformly at random to one of its out-neighbors. As an example, the
transition matrix for the original candy sharing game is given by
P =
2
666664
1/2 1/2 0 0 . . . 0
0 1/2 1/2 0 . . . 0
0 0 1/2 1/2 . . . 0
...
...
...
...
. . .
...
1/2 0 0 0 . . . 1/2
| {z }
k
3
777775
9
>>>>>=
>>>>>;
k.
Let d0 = (a1, . . . , ak) with n =
P
ai, c̃0 = d0, and for t   1, let c̃t = c̃t 1P = c̃0P t.
Then c̃t(i) represents the expected amount of candy held by vertex i after t turns of the
randomized candy sharing game. Let ct = 1n c̃t.
Note that since the randomized candy sharing game has no draws, we have the follow-
ing inequalities:
min(dt)   min(c̃t)
max(dt)  max(c̃t) + t.
(3.1)
Lemma 3.1. If a candy sharing game is played on a strongly connected, aperiodic, d-
regular directed graph G with k vertices, then in at most k
2  2k+2 turns either the game
will terminate or the minimum amount of candy held by any player will increase.
Proof. Let t represent the current turn of the game, with t = 0 being the initial distribution.
Let c̃t = dt = (a1, a2, . . . , ak) with ai 2 Z+ for i = 1, . . . k, and further assume that not
all entries of this vector are equal (meaning the game has not terminated). In particular,
some entry of c̃t is larger than min(c̃t). Given that G is d-regular the transition matrix, P ,
for this chain is doubly stochastic, that is, all of its row and column sums are 1. It is easily
verifiable that the product of doubly stochastic matrices is also doubly stochastic.
A theorem of Wielandt (see [6]) says that for a k ⇥ k, irreducible, non-negative matrix
P , there exists an integer r  k2   2k+2 such that P rij > 0 for all i, j. (In fact, a theorem
of Dulmage and Mendelsohn [3] says that if P additionally has a non-zero diagonal entry,
we have P r has all positive entries for some r  2k   2.)
Advancing the randomized game, c̃, by r turns and letting A1, . . . Ak represent the
columns of matrix P r, we have
c̃t+r = c̃tP
r = (c̃t ·A1, c̃t ·A2, . . . , c̃t ·Ak) .
Since P r is doubly stochastic, each entry of c̃t+r is a weighted average of the entries of c̃t.
Since there exist entries of c̃t that are larger than min(c̃t), each weighted average is larger
than min(c̃t). Thus, using (3.1), we have
min(dt+r)   min(c̃t+r) > min(c̃t) = min(dt).
D. Bal and J. DeGaetani: A note on a candy sharing game 5
We can clearly see that if max(dt)  min(dt) < 1 for some t, then the discrete game
has ended. Let ⇡ =
 
1
k ,
1
k , . . . ,
1
k
 
. Then one can check that ⇡P = ⇡, that is, ⇡ is the
stationary distribution for P . Let ↵ and C be given by Theorem 2.1 for matrix P and initial
distribution c0. Then after t steps we have ||ct ⇡||TV < C↵t. Let t0 =
l
log(2Cn)
log( 1↵ )
m
. Then
||ct0   ⇡||TV <
1
2n
. (3.2)
Using the inequalities from (3.1), followed by normalizing the randomized game and
utilizing the triangle inequality we have
|max(dt) min(dt)|  |max(c̃t) min(c̃t)|+ t
 n |max (ct) min (ct)|+ t
 n
✓    max (ct) 
1
k
    +
    min (ct) 
1
k
    
◆
+ t.
In the above sum, we compare the distance between two entries of ct and two entries
of ⇡. Including the remaining distances leads to the next inequality as k   2.
n
✓    max (ct) 
1
k
    +
    min (ct) 
1
k
    
◆
+ t  n
kX
i=1
    ct(i) 
1
k
    + t.
By (2.1), we have
n
kX
i=1
    ct(i) 
1
k
    + t = 2n ||ct   ⇡||TV + t.
Now,  t is bounded above by (d   1)kt, since at most every player will have to draw
(d  1) pieces every turn. Therefore, after t0 turns, we have
|max(dt0) min(dt0)|  2n ||ct0   ⇡||TV + t0
 2n
✓
1
2n
◆
+ (d  1)kt0
= 1 + (d  1)k · log(2Cn)
log( 1↵ )
where in the second inequality we have used (3.2). Thus we have that there exists a constant
C
0 = C 0(k, d) such that after t0 turns, |max(dt0) min(dt0)| < C 0 log n.
Recall that max(dt) cannot increase and by Lemma 3.1, min(dt) is guaranteed to in-
crease every k2   2k + 2  k2 turns. Therefore, after at most k2C 0 log n more turns,
|max(dt) min(dt)| will be less than 1, and thus the game will have ended. From this
we know the total number of turns the game took is at most t0 + k2C 0 log n =
log(2Cn)
log( 1↵ )
+
k
2
C
0 log n < C 00 log n for some constant C 00 = C 00(k, d). At worst, each player draws
(d   1) pieces of candy every turn. So the total amount of candy at the end of the game
is at most n + (d   1)kC 00 log n implying that each player has nk plus at most O(log n)
pieces.
6 Art Discrete Appl. Math. 4 (2021) #P1.02
4 Proof of Proposition 1.2
Proof of Proposition 1.2. Consider the sequence (ri)1i=1 defined recursively by
r` = 4r` 1 + 2 and r1 = 2. We examine the game played on a directed cycle of length 3
with loops at each vertex. With initial distribution d = (r`, 0, 0), the sequence of states is
as follows. An arrow indicates advancing a turn, and a number over the arrow represents
how many pieces were drawn that turn. For `   2,
(r`, 0, 0) = (4r` 1 + 2, 0, 0)
+2 ! (2r` 1 + 2, 2r` 1 + 2, 0)
+2 ! (r` 1 + 2, 2r` 1 + 2, r` 1 + 2).
Notice that each player is holding at least r` 1 + 2 pieces of candy. We can imagine
that they each divide their current piles into two: an inner pile consisting of the r` 1 + 2
pieces, and an outer pile containing the remainder. The players would then continue by
playing two concurrent games, following the game procedure on the inner pile and outer
pile simultaneously. Since the inner pile is the same amount for each player, that game has
already terminated and will no longer draw extra pieces. The only draws will then come
from the outer game which, after invoking symmetry, is equivalent to the game played with
initial distribution d0 = (r` 1, 0, 0).
Further, note that the r` sequence has binary representation
10, 1010, 101010, 10101010, . . .
where the `th element is the digits (10) repeated ` times. Each time the above recursion is
applied to a game of the form (r`, 0, 0), two turns elapse, four pieces of candy are drawn
and two digits are removed from the binary representation of the initial candy amount. We
finally see that we need to draw 4 times the length of a base 2 expansion of r`, which is log-
arithmic. Therefore, letting n = r`, the sequence of games played with initial distribution
d = (n, 0, 0) will terminate after ⇥(log n) turns with each player holding n3 + ⇥(log n)
many pieces of candy.
5 Conclusion
In this note, we have proved a relationship between the candy sharing game with rounding
and a Markov chain without rounding and used the convergence theorem to find a bound on
the length of the candy sharing game. The main open problem in candy sharing is to find
a closed form expression for the number of rounds and ending amount of candy in terms
of the initial candy distribution. It seems this may be quite difficult. Another interesting
problem may be to try and prove a result similar to ours for the general games considered
in [4].
ORCID iDs
Deepak Bal https://orcid.org/0000-0003-1441-1823
References
[1] G. Cairns, Equitable candy sharing, Amer. Math. Monthly 124 (2017), 518–526, doi:10.4169/
amer.math.monthly.124.6.518.
D. Bal and J. DeGaetani: A note on a candy sharing game 7
[2] G. Chang and T. W. Sederberg, Over and over again, volume 39 of New Mathematical Library,
Mathematical Association of America, Washington, DC, 1997.
[3] A. L. Dulmage and N. S. Mendelsohn, Gaps in the exponent set of primitive matrices, Illinois J.
Math. 8 (1964), 642–656, http://projecteuclid.org/euclid.ijm/1256059464.
[4] G. Iba and J. Tanton, Candy sharing, Amer. Math. Monthly 110 (2003), 25–35, doi:10.2307/
3072341.
[5] D. A. Levin, Y. Peres and E. L. Wilmer, Markov chains and mixing times, American Mathemat-
ical Society, Providence, RI, 2009, with a chapter by James G. Propp and David B. Wilson.
[6] H. Schneider, Wielandt’s proof of the exponent inequality for primitive nonnegative matrices,
Linear Algebra Appl. 353 (2002), 5–10, doi:10.1016/S0024-3795(02)00414-7.
ISSN 2590-9770
The Art of Discrete and Applied Mathematics 4 (2021) #P1.03
https://doi.org/10.26493/2590-9770.1360.465
(Also available at http://adam-journal.eu)
Maps and  -matroids revisited
Rémi Cocou Avohou⇤
Einstein Institute of Mathematics, The Hebrew University of Jerusalem,
Giv’at Ram, Jerusalem, 91904, Israel, &
ICMPA-UNESCO Chair, 072BP50, Cotonou, Rep. of Benin, &
Ecole Normale Superieure, B.P 72, Natitingou, Benin
Brigitte Servatius , Herman Servatius
Mathematical Sciences, Worcester Polytechnic Institute, Worcester MA 01609-2280, USA
Received 8 September 2019, accepted 23 February 2020, published online 22 September 2020
Abstract
Using Tutte’s combinatorial definition of a map we define a  -matroid purely combi-
natorially and show that it is identical to Bouchet’s topological definition.
Keywords: Matroid,  -matroid, cellular map, topological surface.
Math. Subj. Class.: 05B35, 05C10, 52B40
1 Matroids and  -matroids
A matroid M is a finite set E and a non-empty collection B of subsets of E satisfying the
condition that if
(MB) If B1 and B2 are in B and x 2 B1 \ B2 then there exists y 2 B2 \ B1 such that
(B1 [ {y}) \ {x} = B1 4 {x, y} 2 B.
Axiom (MB) is called the basis exchange axiom. Sets in B are called bases of M .
Replacing the set difference in Axiom (MB) by the symmetric difference we obtain the
symmetric exchange axiom ( F) used by Bouchet [1] to define  -matroids.
A  -matroid D is a finite set E and a collection F of subsets of E satisfying the
condition that if
( F) If F1 and F2 are in F and x 2 F1 4 F2 then there exists a y 2 F2 4 F1 such that
F1 4 {x, y} 2 F .
⇤The author was supported by ISF Grant 1050/16.
E-mail address: avohou.r.cocou@mail.huji.ac.il (Rémi Cocou Avohou), bservat@wpi.edu (Brigitte
Servatius), hservat@wpi.edu (Herman Servatius)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Art Discrete Appl. Math. 4 (2021) #P1.03
Axiom ( F) is called the symmetric exchange axiom and the sets in F are called the fea-
sible sets of D. It is important to note that y may equal x, so |F1 4 {x, y}|   |F1| 2
{0,±1,±2}.
There are two obvious matroids associated with every  -matroid; Mu, the upper ma-
troid, whose bases are the feasible sets with largest cardinality, and Ml, the lower matroid,
whose bases are the feasible sets with least cardinality, [2].
 -matroids and maps on surfaces
In [2], Bouchet associates a  -matroid to any map. A map is a cellular embedding of a
graph G into a compact surface, and, for the  -matroid he defined, the lower matroid is the
cycle matroid of G, and the upper matroid is the dual of the cycle matroid of the geometric
dual, G⇤, of G in the surface. For more information about maps see [3, 4, 5]. In this section
we would like to reformulate the connection between maps and  -matroids in such a way
as to clarify both the geometry and the combinatorics.
Bouchet defined a base B of a map as a selection of edges from the cellularly embedded
graph, B ✓ E, such that, after deleting all the edges of B and all the dual edges of E \B,
together with their endpoints, the resulting non-compact surface is connected. To perform
this operation, it is convenient to use the barycentric subdivision of the map, whose one-
skeleton contains both the graph and the dual-graph, with the edges of each subdivided
in two, see Figure 1(a) and (b). The map graph is the geometric dual of the barycentric
(a) (b) (c) (d)
Figure 1: (a) A cell of a map, (b) its barycentric subdivision, (c) the map graph, (d) deleting
an edge/dual-edge selection.
subdivision, Figure 1(c), where the edges are colored green, red, and black depending on
whether they are parallel to one of the original edges, cross one, or neither. Suppose,
as Bouchet did, we delete, for each edge, either the edge or its dual, together with their
endpoints, as realized in the barycentric subdivision. If it should happen that some vertex
or dual vertex of the map is not deleted, then it is an interior point of the the deleted surface,
and we may puncture the surface there without affecting the connectivity. Then, expanding
the holes at the vertices and dual vertices, there is a deformation of the punctured surface
which respects all the edges and dual edges, so, in particular, respecting the deleted edges.
This deformation can continue, expanding the holes until all that is left is the set of black
edges of the map graph and the green-red quadrilaterals, each of which has been cut in
half, either leaving the green edge pair intact, or the red edge pair. Each of these cut
quadrilaterals can be deformed, expanding the cut, onto the surviving color pair, leaving
the map graph with one color pair deleted from each quadrilateral, green for those in B, and
R. C. Avohou et al.: Maps and  -matroids revisited 3
(a) (b) (c)
Figure 2: Deforming away from the vertex and dual vertex holes.
red for the others. This is a 2-regular subgraph of the map graph, and contains all the black
edges. By the deformation, the surface with the edges and dual edges deleted is connected
if and only if the corresponding 2-regular subgraph of the map graph is connected as a
topological space, which is true if and only if that 2-regular subgraph is a Hamiltonian
cycle.
Bouchet went on to show that the sets B formed the feasible sets of a  -matroid on E,
using Eulerian splitters. Using the map graph, we may establish this simply and directly.
2 Combinatorial maps and  -matroids
Tutte, in the introduction to his paper What is a map? [5] remarks
Maps are usually presented as cellular dissections of topologically defined sur-
faces. But some combinatorialists, holding that maps are combinatorial in
nature, have suggested purely combinatorial axioms for map theory, so that
that branch of combinatorics can be developed without appealing to point-set
topology.
Tutte’s idea is that each edge of a map is associated with four flags, corresponding to the
triangles in the barycentric subdivision. Each flag has three vertices: one corresponding to
a vertex of the embedded graph (an endpoint of the embedded edge e), one corresponding
to an edge (the mid-point of e), and one corresponding to a face (the bary-center of a
face incident with e) of the map. The map can be uniquely described in terms of three
perfect matchings. Two flags are matched if they share a vertex of the same kind. Faces,
Euler characteristic, and orientability can be treated combinatorially without appealing to
topology. We now recall Tutte’s axiomatic approach as presented in [3, 4].
Let   be a connected graph whose edges are partitioned into three classes R, G, and B
which we color respectively red, green, and black.   is called map graph or a combinatorial
map if the following conditions are satisfied:
1. Each color class is a perfect matching;
2. R [G is a union of 4-cycles;
3.   is connected.
The graph   is 3-regular and edge 2-connected.   may have parallel edges, although nec-
essarily not red/green.   contains 2-regular subgraphs which use all the black edges of
4 Art Discrete Appl. Math. 4 (2021) #P1.03
 , which we call fully black 2-regular subgraphs; R [ B and G [ B are examples, and
there always exists a fully black Hamiltonian cycle. To see this, first note that a fully black
2-regular subgraph cannot contain any incident green and red edges, so every red/green
quadrilateral intersects a fully black 2-regular subgraph in either two red, or two green
edges. Now consider a fully black 2-regular subgraph of   with the fewest connected com-
ponents. If there is not a single component, then there is a green/red quadrilateral which
intersects the subgraph in, say, two red edges which belong to two different components,
and swapping red and green on that quadrilateral reduces the number of components of the
subgraph, violating minimality.
Theorem 2.1. Given a combinatorial map  (R,G,B), let E be the set of quadrilaterals
of R [ G, and let F be the collection of subsets of E corresponding to the pairs of green
edges in a fully black Hamilton cycle in  . Then (F , E) is a  -matroid.
Proof. We have to show the symmetric exchange property holds. Let FC and FC0 be sets of
quadrilaterals corresponding to fully black Hamiltonian cycles C and C 0. Let q 2 FC4F 0C ,
so the edges of quadrilateral q are differently colored in C and C 0, say red and green. There
are two cases, either replacing in q the red edges in C with the green of C 0 results in two
components or one. See Figure 3. If it results in just one component, then take q0 = q, and
q q
Figure 3
Fc 4 {q, q0} = Fc 4 {q} is the set of green quadrilaterals of a fully black Hamiltonian
cycle, and hence feasible, as required.
Otherwise, if there are two components, the Hamiltonian cycle of C 0 contains a non-
black edge, say green, of a quadrilateral q1, connecting those two components, and neces-
sarily both red edges of q1 are in C and both green edges of q1 connect the components,
and q0 2 C 4 C 0. See Figure 4. Regardless of how the green edges of q1 are placed,
qq
q1 q1
Figure 4
swapping the edges of both q and q0 in C yields a new fully black Hamiltonian cycle, so
the set Q4 {q, q1} is feasible, as required.
Since R, G and B are perfect matchings, the union of any two them induces a set of
disjoint cycles. Let V be the set of cycles of R [ B, E be the set of cycles of R [ G,
R. C. Avohou et al.: Maps and  -matroids revisited 5
and V ⇤ be the set of cycles of G [ B. There is a graph (V,E) where incidence is defined
between a red-black cycle and a red-green cycle if they share an edge, and, similarly, there
is a graph (V ⇤, E) where incidence is defined between a green-black cycle and a red-green
cycle if they share and edge. We say that   encodes the graph (V,E) and its geometric
dual (V ⇤, E).
Theorem 2.2. Let  (R,G,B) be a combinatorial map and D  = (F, E) its associated
 -matroid. Then the lower matroid of D  is the cycle matroid of the graph (V,E) and the
upper matroid of D  is the cocycle matroid of the graph (V ⇤, E).
Proof. Given  (R,G,B), recall that the feasible sets of D consist of RG quadrilaterals
whose R edges are contained in a fully black Hamilton cycle of  . Any fully black Hamil-
ton cycle C of   must contain the red edges corresponding to a spanning tree of (V,E)
as well as the green edges corresponding to a spanning tree of (V ⇤, E). So the minimal
number of red edges in C is 2(|V |   1), while the maximal number is 2(|E|   |V ⇤| + 1).
The edge sets of the spanning trees of (V,E) are the bases of its cycle matroid, while the
complements of edge sets of spanning trees in (V ⇤, E) are the bases of the cocycle matroid
of (V ⇤, E).
Note that the difference in rank of the upper and lower matroid of (F, E) is given by
(|E|  |V ⇤|+1) (|V | 1) = 2  , where   is the Euler characteristic. Notice also, that if
  is bipartite, all feasible sets of D  = (F, E) must have the same parity, since exchanging
a red and green pair of edges always disconnects a Hamilton cycle of a bipartite  .
Examples of combinatorial maps together with the underlying graph and geometric
dual are provided in Figures 5, 6 and 7.
G*G
Figure 5
The  -matroid associated to the map of Figure 5 has feasible sets
F = {{1, 3, 4}, {1, 3, 5}, {1, 3, 6}, {1, 4, 5}, {1, 4, 6}, {2, 3, 4},
{2, 3, 5}, {2, 3, 6}, {2, 4, 5}, {2, 4, 6}, {3, 4, 5}, {3, 4, 6}}.
Note that F is the set of spanning trees of G and at the same time the set of co-trees of G⇤,
so all feasible sets have the same size and upper and lower matroid are identical.
The  -matroid associated to the map of Figure 6 has feasible sets all the sets in F
together with the two additional sets {1, 2, 3, 4, 5} and {1, 2, 3, 4, 6}. The lower matroid is
again the cycle matroid of G, but the upper matroid is the co-cycle matroid of G⇤, which
has rank 5 and contains exactly one cycle, namely {5, 6}, which is a minimal cutset of G⇤
and also a cycle in G.
6 Art Discrete Appl. Math. 4 (2021) #P1.03
G*G
Figure 6
G*G
Figure 7
The  -matroid associated to the map of Figure 7 has, in addition to the feasible sets of
the previous example the feasible set {1, 2, 3, 4}, whose parity is even, while the parity of
all other feasible sets is odd, so this map is not orientable.
As is clear from these examples, the map cannot, in general be recovered from the  -
matroid information, since the upper or lower matroid do not even determine the graph.
Non-isomorphic graphs may have identical cycle-and co-cycle matroids. It is easy to check
that F is also a list of spanning trees for the graph G0, but G is not isomorphic to G0.
However, if both G and G⇤ are 3-connected, then the map is uniquely recoverable from
the  -matroid information.
Theorem 2.3. Let D be the  -matroid of a map M with 3-connected upper- and lower
matroid. Then M is determined by D.
Proof. By Whitney’s theorem [6], upper and lower matroid uniquely determine G and G⇤.
To recover M from D, we need to specify a rotation system for each vertex v of G. To
determine if two edges e and f with endpoint v follow each other in the rotation about v,
it is enough to check if e and f are both incident in G⇤, since the vertex co-cycles of G⇤
correspond to the facial cycles of the embedded G. Now re-construct the map graph.
For example the lower matroid could be the cycle matroid of K5, while the upper
matroid is the co-cycle matroid of K5 as well, so this matroid information gives us the
graphs G and G⇤ depicted in Figure 8. By the method in the proof of Theorem 2.3 the
map M is easily recovered to be as in Figure 9, which represents the torus map as a doubly
periodic tiling. The faces are colored according to the vertex colors in Figure 8.
R. C. Avohou et al.: Maps and  -matroids revisited 7
a b
c
d
e
E B
C D
A
a d
B
e
A c b
C E
D
Figure 8
B
C
D
A
a b
a b
c d
c d e
e
a b cd e
ab c d e
a b c de
E B
C
D
A
EB
C
D
A
EB
C
D
A
EB
C
D
A
E
Figure 9
3 Another  -matroid from a map
If the objective is to define a natural  -matroid from a combinatorial map, the requirement
that the subgraph of the map graph be Hamiltonian can be weakened provided that some
connectivity is required. Again, let   be a map graph with edge set R [ G [ B, with red
edges R, green edges G and black edges B.
Theorem 3.1. Let K be a fully black 2-valent subgraph of   with the property that K [R
and K [G are both connected. Then the set FK of quadrilaterals in which red is selected
in K form the feasible sets of a  -matroid.
Proof. We have to show the symmetric exchange property. Let FK and FK0 be sets of red
quadrilaterals corresponding to fully black 2-valent subgraphs K and K 0, both of which
can be connected by adding edges of one color only. Let q 2 FK 4 F 0K , so the edges of
quadrilateral q are differently colored in K and K 0, say red and green respectively. If the
red edges of q belong to two different cycles of K, then swapping red for green in q merges
the two cycles, then we may take q0 = q and the FK 4 {q, q0} will be connected by the
same collections of red, respectively green edges as Fk.
So we may assume that the red edges of q belong to the same component K. If swap-
ping red for green in q does not split the component of K they belong to, see the right side
of Figure 3, then just as before, take q0 = q. So we may assume that the red edges of q be-
long to the same component of K, and swapping them for green splits that component, see
the left side of Figure 3. Let the red edges of q be denoted by qr and the green edges of q be
denoted by qg . Clearly (K  qr + qg)+R is connected since K+R ✓ (K  qr + qg)+R
and is connected. The issue is that (K   qr + qg) + G = K + G   qr may have two
8 Art Discrete Appl. Math. 4 (2021) #P1.03
components. If it has just one, again, take q0 = q and we are done. We know that K 0 +G
is connected, so K 0 must have a red edge of some quadrilateral q0 that connects the two
components of (K   qr + qg) + G, so q0 62 FK and q0 2 FK0 , that is q0 2 FK 4 FK0 .
(K   qr + q0r) +G is connected and we already know (K   qr + q0r) +R is connected, so
the collections FK are the feasible sets of a  -matroid.
Let D  be the  -matroid as in Theorem 2.3, with feasible sets the pairs of red edges in
a fully black Hamilton cycle, and DK be the  -matroid as in Theorem 3.1, with feasible
sets the pairs of red edges in a fully black 2-valent subgraph K such that K becomes
connected by addition of red edges only as well as by addition of only green edges. D 
and DK are different. For example for the unitary map there are two quadrilaterals, {q, q0}
and the feasible sets in the first sense are {;, {q, q0}}, whereas in the second sense are all
subsets. For unitary maps the connectivity issue here is void since the R + B and G + B
are both connected. The upper and lower matroid for both D  and DK are clearly the
same. However, the Hamiltonian requirement encodes the orientability of the map, by the
fact that all feasible sets have the same parity in the orientable case and are of both even
and odd cardinality if   is not bipartite, while the second approach does not distinguish
between the two.
ORCID iDs
Brigitte Servatius https://orcid.org/0000-0003-3729-3399
References
[1] A. Bouchet, Greedy algorithm and symmetric matroids, Math. Programming 38 (1987), 147–
159, doi:10.1007/BF02604639.
[2] A. Bouchet, Maps and 4-matroids, Discrete Math. 78 (1989), 59–71, doi:10.1016/
0012-365X(89)90161-1.
[3] C. Godsil and G. Royle, Algebraic Graph Theory, Graduate Texts in Mathematics, Springer New
York, 2001, https://books.google.com.au/books?id=pYfJe-ZVUyAC.
[4] T. Pisanski and B. Servatius, Configurations from a graphical viewpoint, Birkhäuser Advanced
Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser/Springer,
New York, 2013, doi:10.1007/978-0-8176-8364-1.
[5] W. T. Tutte, What is a map?, in: New directions in the theory of graphs (Proc. Third Ann Arbor
Conf., Univ. Michigan, Ann Arbor, Mich., 1971), Academic Press, New York, pp. 309–325, 1973.
[6] H. Whitney, Congruent Graphs and the Connectivity of Graphs, Amer. J. Math. 54 (1932), 150–
168, doi:10.2307/2371086.
ISSN 2590-9770
The Art of Discrete and Applied Mathematics 4 (2021) #P1.04
https://doi.org/10.26493/2590-9770.1368.f37
(Also available at http://adam-journal.eu)
C4-face-magic toroidal labelings on Cm ⇥ Cn
Stephen J. Curran
Department of Mathematics, University of Pittsburgh at Johnstown,
450 Schoolhouse Rd, Johnstown, PA 15904 USA
Richard M. Low
Department of Mathematics, San Jose State University,
1 Washington Sq, San Jose, CA 95192 USA
Stephen C. Locke
Department of Mathematical Sciences, Florida Atlantic University,
777 Glades Rd, Boca Raton, FL 33431 USA
Received 19 August 2019, accepted 8 May 2020, published online 22 September 2020
Abstract
For a graph G = (V,E) naturally embedded in the torus, let F(G) denote the set of
faces of G. Then, G is called a Cn-face-magic toroidal graph if there exists a bijection
f : V (G) ! {1, 2, . . . , |V (G)|} such that for every F 2 F(G) with F ⇠= Cn, the sum
of all the vertex labels along Cn is a constant S. Let xv = f(v) for all v 2 V (G). We
call {xv : v 2 V (G)} a Cn-face-magic toroidal labeling on G. We show that, for all
m,n   2, Cm ⇥ Cn admits a C4-face-magic toroidal labeling if and only if either m = 2,
or n = 2, or both m and n are even. We say that a C4-face-magic toroidal labeling {xi,j :
(i, j) 2 V (C2m ⇥ C2n)} on C2m ⇥ C2n is antipodal balanced if xi,j + xi+m,j+n = 12S,
for all (i, j) 2 V (C2m ⇥ C2n). We show that there exists an antipodal balanced C4-face-
magic toroidal labeling on C2m ⇥ C2n if and only if the parity of m and n are the same.
Furthermore, when both m and n are even, an antipodal balanced C4-face-magic toroidal
labeling on C2m ⇥ C2n is both row-sum balanced and column-sum balanced. In addition,
when m = n is even, an antipodal balanced C4-face-magic toroidal labeling on C2n⇥C2n
is diagonal-sum balanced.
Keywords: C4-face-magic graphs, polyomino, toroidal graphs, Cartesian products of cycles.
Math. Subj. Class. (2020): 05C78
E-mail addresses: sjcurran@pitt.edu (Stephen J. Curran), richard.low@sjsu.edu (Richard M. Low),
lockes@fau.edu (Stephen C. Locke)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Art Discrete Appl. Math. 4 (2021) #P1.04
1 Introduction
Kotzig and Rosa [11] formally introduced graph labelings in the 1970s. There are applica-
tions of graph labelings to graph decomposition problems, radar pulse code designs, X-ray
crystallography and communication network models. The interested reader should read
J.A. Gallian’s comprehensive dynamic survey on graph labelings [8] for further investiga-
tion.
We refer the reader to Bondy and Murty [5] for concepts and notation not explicitly
defined in this paper. All graphs in this paper are simple and connected. For a planar graph
G = (V,E) embedded in R2, let F(G) denote the set of faces of G. Then, G is called a
Cn-face-magic graph if there exists a bijection f : V (G) ! {1, 2, . . . , |V (G)|} such that
for every F 2 F(G) with F ⇠= Cn, the sum of all the vertex labels along Cn is a constant S.
Here, the constant S is called a Cn-face-magic value of G. A Cn-face-magic toroidal (or
cylindrical) graph G is defined similarly, where G is embedded in the torus (or cylinder),
respectively. Cn-face-magic graph labelings are a special case of the more general (a, b, c)-
magic labeling introduced by Lih [12]. For assorted values of a, b and c, Baca and others
[1, 2, 3, 4, 9, 10, 12] have analyzed the problem for various classes of graphs. Wang [13]
showed that the toroidal grid graphs Cm⇥Cn are antimagic for all integers m,n   3. Butt
et al. [6] investigated face antimagic labelings on toroidal and Klein bottle grid graphs.
In this paper, we investigate C4-face-magic toroidal labelings on Cm ⇥ Cn with its
natural embedding in the torus. We show that for all m,n   2, there exists a C4-face-magic
toroidal labeling on Cm ⇥ Cn if and only if either m = 2, or n = 2, or both m and n are
even. In the case when m = n, we say that a C4-face-magic toroidal labeling on C2n⇥C2n
is torus symmetric if the labeling is row-sum balanced, column-sum balanced and diagonal-
sum balanced. Curran and Low [7] show that, up to symmetries on the torus, there are only
three torus symmetric C4-face-magic toroidal labelings on C4 ⇥ C4. See Theorem 3.5
in Section 3 for details. In this paper, we search for C4-face-magic toroidal labelings on
C2m ⇥C2n that are row-sum balanced and column-sum balanced. This investigation leads
naturally to the concept of an antipodal balanced labeling. We say that a C4-face-magic
toroidal labeling {xi,j : (i, j) 2 V (C2m ⇥ C2n)} on C2m ⇥ C2n is antipodal balanced
if xi,j + xi+m,j+n = 12S, for all (i, j) 2 V (C2m ⇥ C2n). We show that there exists an
antipodal balanced C4-face-magic toroidal labeling on C2m ⇥C2n if and only if the parity
of m and n are the same. Furthermore, when m = n is even, we show that any antipodal
balanced C4-face-magic toroidal labeling on C2n ⇥ C2n is torus symmetric.
2 Preliminaries
Theorem 2.1. Let m,n   2. Then, Pm ⇥ Pn is C4-face-magic.
Proof. Label the vertex set of Pm ⇥ Pn as
V (Pm ⇥ Pn) = {(i, j) : 1  i  m, 1  j  n}
and its edge set as
E (Pm ⇥ Pn) = {{(i, j) , (i+ 1, j)} : 1  i < m, 1  j  n}
[ {{(i, j) , (i, j + 1)} : 1  i  m, 1  j < n} .
We will determine a label xi,j for each vertex (i, j) 2 V (Pm ⇥ Pn) and check that this
provides a C4-face-magic labeling.
S. J. Curran, R. M. Low and S. C. Locke: C4-face-magic toroidal labelings on Cm ⇥ Cn 3
Case 1. Assume m  n is even. Color the vertex (i, j) white if i+ j is even and black
if i+ j is odd. Note that the vertices (1, 1) and (m,n) are white. Let xi,j = i+m (j   1)
for each white vertex (i, j), and xi,j = (m  i+1)+m(n  j) for each black vertex (i, j).
An equivalent definition for {xi,j : (i, j) 2 V (Pm ⇥ Pn)} would be to write the number
i + m (j   1) in each cell (i, j), and then rotate the black vertices 180 degrees about the
center of the board.
Let Ci,j = {(i, j) , (i+ 1, j) , (i+ 1, j + 1) , (i, j + 1)}. If (i, j) is white, then the two
4-cycles Ci,j and Ci+1,j have the same face sum, since xi,j = xi+2,j   2 and xi,j+1 =
xi+2,j+1 + 2. If (i, j) is black, then the two 4-cycles Ci,j and Ci+1,j have the same face
sum, since xi,j = xi+2,j + 2 and xi,j+1 = xi+2,j+1   2. A similar proof for Ci,j and
Ci,j+1 shows that {xi,j} is C4- face-magic. The sum on each face must be the sum on
C1,1, which is 1+ (mn  1) + (m+ 2)+m (n  1) = 2mn+2. This completes Case 1.
Case 2. Without loss of generality, we may assume that m is even and n is odd. Let
m = 2m1 and n = 2n1   1 for some positive integers m1 and n1. Again, color vertex
(i, j) white if i+ j is even and black if i+ j is odd.
We first label the white vertices. Let xi,j = m(j   1) + i if both i and j are odd,
and xi,j = m(j   1) + i   1 if both i and j are even. We observe that x2k 1,2` 1 =
m(2`  2)+2k  1 for all 1  k  m1 and 1  `  n1, and x2k,2` = m(2`  1)+2k  1
for all 1  k  m1 and 1  ` < n1. Thus each odd label 1, 3, 5, . . . ,mn   1 is used
exactly once on a white vertex.
Next, we label the black vertices. Let xi,j = m(n j+1) i+1 if i is odd and j is even,
and xi,j = m(n  j+1)  i+2 if i is even and j is odd. We observe that whenever vertex
(i, j) is white, then vertex (m  i+ 1, n  j + 1) is black and xm i+1,n j+1 = xi,j + 1.
Thus each even label 2, 4, 6, . . . ,mn is used exactly once on a black vertex.
Let Ci,j = {(i, j) , (i+ 1, j) , (i+ 1, j + 1) , (i, j + 1)}. If (i, j) is white, then xi+2,j =
xi,j + 2 and xi,j+2 = xi,j + 2m. If (i, j) is black, then xi+2,j = xi,j   2 and xi,j+2 =
xi,j   2m. An argument similar to that in Case (i) shows that each 4-cycle Ci,j has the
same face sum as C1,1, which is 1+mn+(m+1)+m(n 1) = 2mn+2. This completes
Case 2.
Lemma 2.2. Let m and n be positive integers. A C4-face-magic labeling on P2m ⇥ P2n
always yields a C4-face-magic labeling on C2m ⇥ C2n with its natural embedding in the
torus. Furthermore, the C4-face-magic value is S = 2(4mn+ 1).
Proof. Let xi,j be the C4-face-magic labeling on vertex (i, j), for i = 1, 2, . . . , 2m and
j = 1, 2, . . . , 2n. Since S is the C4-face-magic value on P2m ⇥ P2n, we have xi,j +
xi+1,j +xi,j+1+xi+1,j+1 = S, for all i = 1, 2, . . . , 2m  1 and j = 1, 2, . . . , 2n  1. We
observe that
mnS =
mX
i=1
nX
j=1
(x2i 1,2j 1+x2i,2j 1+x2i 1,2j+x2i,2j) =
4mnX
i=1
i = 12 (4mn)(4mn+1).
Thus, S = 2(4mn + 1). Since xi,j + xi,j+1 + xi+1,j + xi+1,j+1 = S = xi+1,j +
xi+2,j + xi+1,j+1 + xi+2,j+1, we have xi,j + xi,j+1 = xi+2,j + xi+2,j+1. An induction
argument shows that xi,j + xi,j+1 = xi+2k,j + xi+2k,j+1. Since xi+2k,j + xi+2k,j+1 +
xi+2k+1,j + xi+2k+1,j+1 = S, we have xi,j + xi,j+1 + xi+2k+1,j + xi+2k+1,j+1 = S.
4 Art Discrete Appl. Math. 4 (2021) #P1.04
A similar argument shows that xi,j + xi+1,j + xi,j+2`+1 + xi+1,j+2`+1 = S. This yields
xi,j + xi+2k+1,j + xi,j+2`+1 + xi+2k+1,j+2`+1 = S. Hence, we have x1,j + x1,j+1 +
x2m,j+x2m,j+1 = S, for all j = 1, 2, . . . , 2n 1. Similarly, we have xi,1+xi+1,1+xi,2n+
xi+1,2n = S, for all i = 1, 2, . . . , 2m 1. Lastly, we have x1,1+x2m,1+x1,2n+x2m,2n =
S. Therefore, the C4-face-magic labeling on P2m ⇥ P2n yields a C4-face-magic labeling
on C2m ⇥ C2n with its natural embedding in the torus.
Lemma 2.3. Let m and n be integers such that m   3 and n   2. Suppose Pm ⇥ Cn is a
C4-face-magic cylindrical graph with the natural embedding of Pm ⇥ Cn on the cylinder.
Then, n is even.
Proof. For the purpose of contradiction, suppose n is odd, and let n = 2n1 + 1 for some
positive integer n1. Label the vertex set of Pm ⇥ Cn as
V (Pm ⇥ Cn) = {(i, j) : 1  i  m, 1  j  n}
and its edge set as
E (Pm ⇥ Cn) = {{(i, j) , (i+ 1, j)} : 1  i < m, 1  j  n}
[ {{(i, j) , (i, j + 1)} , {(i, n) , (i, 1)} : 1  i  m, 1  j < n} .
Let {xi,j : (i, j) 2 V (Pm ⇥ Cn)} be a C4-face-magic labeling on Pm⇥Cn with C4-face-
magic value S. Let Si = xi,1 + xi+1,1, for i = 1, 2. Equating the following C4-face sums
to each other:
xi,j + xi+1,j + xi,j+1 + xi+1,j+1 = S = xi,j+1 + xi+1,j+1 + xi,j+2 + xi+1,j+2,
we obtain xi,j + xi+1,j = xi,j+2 + xi+1,j+2, where the index j is taken modulo n. Thus,
xi,1 + xi+1,1 = xi,2j+1 + xi+1,2j+1 and
xi,2 + xi+1,2 = xi,2j + xi+1,2j , for j = 1, 2, . . . , n1.
Also, we have xi,n 1 + xi+1,n 1 = xi,n+1 + xi+1,n+1 = xi,1 + xi+1,1. Hence, Si =
xi,j + xi+1,j , for all i = 1, 2 and j = 1, 2, . . . , n. From the C4-face sum
S = (xi,j + xi+1,j) + (xi,j+1 + xi+1,j+1) = Si + Si = 2Si,
we have Si = 12S. Hence, x1,1 + x2,1 =
1
2S = x2,1 + x3,1, which in turn, implies that
x1,1 = x3,1. This is a contradiction. Therefore, n is even.
Proposition 2.4. Let m be an integer where m   2. Then, there is a C4-face-magic
toroidal labeling on Cm ⇥ C2.
Proof. Let xi,1 = i and xi,2 = 2m+1 i, for i = 1, 2, . . . ,m. Then, xi,1+xi,2 = 2m+1,
for i = 1, 2, . . . ,m. Thus, xi,1 +xi,2 +xi+1,1 +xi+1,2 = 2(2m+1), for i = 1, 2, . . . ,m.
Hence, {xi,j : (i, j) 2 V (Cm⇥C2)} is a C4-face-magic toroidal labeling on Cm⇥C2.
Proposition 2.5. Let m and n be integers where m,n   2. Then, Cm ⇥ Cn has a C4-
face-magic toroidal labeling if and only if either m = 2, or n = 2, or both m and n are
even.
S. J. Curran, R. M. Low and S. C. Locke: C4-face-magic toroidal labelings on Cm ⇥ Cn 5
Proof. ()) Suppose Cm ⇥ Cn has a C4-face-magic toroidal labeling. If either m = 2 or
n = 2, we are done. So assume that m,n   3. The C4-face-magic toroidal labeling on
Cm ⇥ Cn is simultaneously a C4-face-magic cylindrical labeling on both Cm ⇥ Pn and
Pm ⇥ Cn. By Lemma 2.3, both m and n are even.
( ) Suppose either m = 2, or n = 2, or both m and n are even. On the one hand, if
m = 2 or n = 2, by Proposition 2.4, Cm ⇥ Cn has a C4-face-magic toroidal labeling. On
the other hand, if both m and n are even, by Theorem 2.1, Pm ⇥ Pn has a C4-face-magic
labeling. By Lemma 2.2, the C4-face-magic labeling on Pm ⇥ Pn yields a C4-face-magic
toroidal labeling on Cm ⇥ Cn.
Throughout this paper, if {xi,j : (i, j) 2 V (C2m ⇥C2n)} is a labeling on C2m ⇥C2n,
then for convenience we consider the index i modulo 2m and the index j modulo 2n.
Definition 2.6. We say that the C4-face-magic torus labeling {xi,j : i = 1, 2, . . . , 2m and
j = 1, 2, . . . , 2n} on C2m ⇥ C2n is antipodal balanced if xi,j + xi+m,j+n = 4mn + 1,
for all integers i and j such that 1  i  2m and 1  j  2n.
Remark 2.7. We give a brief explanation for the term antipodal balanced. On the n-
sphere Sn j Rn+1, the antipodal map p : Sn ! Sn is given by p(x) =  x. Similarly,
on the torus T 2 = S1 ⇥ S1 j C2 ⇠= R4, we define the antipodal map p : T 2 ! T 2 by
p(ei✓1 , ei✓2) =  (ei✓1 , ei✓2) = (ei(✓1+⇡), ei(✓2+⇡)). Thus an antipodal balanced C4-face-
magic toroidal labeling on C2m ⇥C2n is one in which the sum of the labels at a vertex and
its antipodal vertex is constant for all vertices in C2m ⇥ C2n.
Lemma 2.8. Let m and n be positive integers. Let {xi,j : i = 1, 2, . . . , 2m and j =
1, 2, . . . , 2n} be an antipodal balanced C4-face-magic toroidal labeling on C2m ⇥ C2n.
For all integers i where 1  i  m, we define
di = xi(m 1)+1,in+1   x(i 1)(m 1)+1,(i 1)n+1.
Then for all integers i and j where 1  i  m and 1  j  2n, we have
xi(m 1)+1,in+j = x(i 1)(m 1)+1,(i 1)n+j + ( 1)j+1di.
Proof. By the definition of di, we have
xi(m 1)+1,in+1 = x(i 1)(m 1)+1,(i 1)n+1 + di.
We apply an induction argument on j. Thus we assume that
xi(m 1)+1,in+j 1 = x(i 1)(m 1)+1,(i 1)n+j 1 + ( 1)jdi. (2.1)
Since the labeling is antipodal balanced, we have
xi(m 1)+2,in+j 1 =
1
2S   x(i 1)(m 1)+1,(i 1)n+j 1, and (2.2)
xi(m 1)+2,in+j =
1
2S   x(i 1)(m 1)+1,(i 1)n+j . (2.3)
Since {xi,j} is a C4-face-magic toroidal labeling on C2m ⇥ C2n, we have
xi(m 1)+1,in+j 1 + xi(m 1)+1,in+j + xi(m 1)+2,in+j 1 + xi(m 1)+2,in+j = S. (2.4)
6 Art Discrete Appl. Math. 4 (2021) #P1.04
When we substitute the expressions from equations (2.1), (2.2) and (2.3) into equation
(2.4), we obtain
xi(m 1)+1,in+j = x(i 1)(m 1)+1,(i 1)n+j + ( 1)j+1di.
This completes the proof.
We next show that if C2m ⇥ C2n has an antipodal balanced C4-face-magic toroidal
labeling, then the parity of m and n are the same.
Lemma 2.9. Let m and n be positive integers. Let {xi,j : i = 1, 2, . . . , 2m and j =
1, 2, . . . , 2n} be an antipodal balanced C4-face-magic toroidal labeling on C2m ⇥ C2n.
Then, the parity of m and n are the same.
Proof. We may assume that m is even and n is odd. Let {xi,j} be an antipodal balanced
C4-face-magic toroidal labeling on C2m ⇥ C2n. We will show that this leads to a contra-
diction. For all integers i such that 1  i  m, we define
di = xi(m 1)+1,in+1   x(i 1)(m 1)+1,(i 1)n+1.
By Lemma 2.8, for all integers i and j such that 1  i  m and 1  j  2n, we have
xi(m 1)+1,in+j = x(i 1)(m 1)+1,(i 1)n+j + ( 1)j+1di.
Fix j such that 1  j  2n. The equations
xi(m 1)+1,in+j = x(i 1)(m 1)+1,(i 1)n+j + ( 1)j+1di for i = 1, 2, . . . ,m,
yield
xm+1,j = xm(m 1)+1,mn+j = x1,j + ( 1)j+1
 
d1 + d2 + · · ·+ dm
 
. (2.5)
Setting j = 1, equation (2.5) becomes
xm+1,1 = x1,1 +
 
d1 + d2 + · · ·+ dm
 
. (2.6)
Setting j = n+ 1, equation (2.5) becomes
xm+1,n+1 = x1,n+1  
 
d1 + d2 + · · ·+ dm
 
. (2.7)
Because {xi,j} is antipodal balanced, we have
x1,1 + xm+1,n+1 =
1
2S = xm+1,1 + x1,n+1. (2.8)
Substituting equations (2.6) and (2.7) into equation (2.8) and simplifying yields
d1 + d2 + · · ·+ dm = 0. (2.9)
This implies that xm+1,j = x1,j , for all j = 1, 2, . . . , 2n. This is a contradiction.
We next investigate how the conditions of Lemma 2.8 apply to the case C2m ⇥ C2n
where both m and n are even.
S. J. Curran, R. M. Low and S. C. Locke: C4-face-magic toroidal labelings on Cm ⇥ Cn 7
Lemma 2.10. Let m and n be positive even integers. Let {xi,j : i = 1, 2, . . . , 2m and j =
1, 2, . . . , 2n} be an antipodal balanced C4-face-magic toroidal labeling on C2m ⇥ C2n.
For all integers i such that 1  i  m, we define
di = xi(m 1)+1,in+1   x(i 1)(m 1)+1,(i 1)n+1.
Then by Lemma 2.8, for all integers i and j such that 1  i  m and 1  j  2n, we have
xi(m 1)+1,in+j = x(i 1)(m 1)+1,(i 1)n+j + ( 1)j+1di.
Also, for all integers i and j such that m+ 1  i  2m and 1  j  2n, we have
xi(m 1)+1,in+j = x(i 1)(m 1)+1,(i 1)n+j + ( 1)jdi m.
For all integers j such that 1  j  n, we define
d0j = xjm+1,j(n 1)+1   x(j 1)m+1,(j 1)(n 1)+1.
Then for all integers i and j such that 1  i  2m and 1  j  n, we have
xjm+i,j(n 1)+1 = x(j 1)m+i,(j 1)(n 1)+1 + ( 1)i+1d0j .
Also, for all integers i and j such that 1  i  2m and n+ 1  j  2n, we have
xjm+i,j(n 1)+1 = x(j 1)m+i,(j 1)(n 1)+1 + ( 1)id0j n.
Proof. By Lemma 2.8, for all integers i and j such that 1  i  m and 1  j  2n, we
have
xi(m 1)+1,in+j = x(i 1)(m 1)+1,(i 1)n+j + ( 1)j+1di. (2.10)
Since the indices in+ j and (i  1)n+ j are reduced modulo 2n, equation (2.10) holds for
all integers j. Let i be an integer such that m+ 1  i  2m. We replace i with i m and
j with n+ j in equation (2.10) to obtain
x(i m)(m 1)+1,(i m)n+n+j = x(i m 1)(m 1)+1,(i m 1)n+n+j + ( 1)n+j+1di m.
This reduces to
xi(m 1)+m+1,(i+1)n+j = x(i 1)(m 1)+m+1,in+j + ( 1)j+1di m. (2.11)
Since {xi,j} is antipodal balanced, we have
xi(m 1)+m+1,(i+1)n+j =
1
2S   xi(m 1)+1,in+j and (2.12)
x(i 1)(m 1)+m+1,in+j =
1
2S   x(i 1)(m 1)+1,(i 1)n+j . (2.13)
When we substitute the expressions in equations (2.12) and (2.13) into equation (2.11), we
have, for all integers i and j such that m+ 1  i  2m and 1  j  2n,
xi(m 1)+1,in+j = x(i 1)(m 1)+1,(i 1)n+j + ( 1)jdi m. (2.14)
When we interchange the roles of i and j in the previous argument, when 1  i  2m
and 1  j  n, we have
xjm+i,j(n 1)+1 = x(j 1)m+i,(j 1)(n 1)+1 + ( 1)i+1d0j ;
and when 1  i  2m and n+ 1  j  2n, we have
xjm+i,j(n 1)+1 = x(j 1)m+i,(j 1)(n 1)+1 + ( 1)id0j n.
8 Art Discrete Appl. Math. 4 (2021) #P1.04
We next investigate how the conditions of Lemma 2.8 apply to the case C2m ⇥ C2n
where both m and n are odd.
Lemma 2.11. Let m and n be positive odd integers. Let {xi,j : i = 1, 2, . . . , 2m and j =
1, 2, . . . , 2n} be an antipodal balanced C4-face-magic toroidal labeling on C2m ⇥ C2n.
For all integers i such that 1  i  m, we define
di = xi(m 1)+1,in+1   x(i 1)(m 1)+1,(i 1)n+1.
Then, for all integers i and j such that 1  i  m and 1  j  2n, we have
xi(m 1)+1,in+j = x(i 1)(m 1)+1,(i 1)n+j + ( 1)j+1di and
xi(m 1)+m+1,in+j = x(i 1)(m 1)+m+1,(i 1)n+j + ( 1)j+1di.
For all integers i such that 1  j  n, we define
d0j = xjm+1,j(n 1)+1   x(j 1)m+1,(j 1)(n 1)+1.
Then, for all integers i and j such that 1  i  2m and 1  j  n, we have
xjm+i,j(n 1)+1 = x(j 1)m+i,(j 1)(n 1)+1 + ( 1)i+1d0j and
xjm+i,j(n 1)+n+1 = x(j 1)m+i,(j 1)(n 1)+n+1 + ( 1)id0j .
Proof. By Lemma 2.8, for all integers i and j such that 1  i  m and 1  j  2n, we
have
xi(m 1)+1,in+j = x(i 1)(m 1)+1,(i 1)n+j + ( 1)j+1di. (2.15)
Because in + j and (i   1)n + j are reduced modulo 2n, equation (2.15) holds for all
integers j. Because {xi,j} is antipodal balanced, we have
xi(m 1)+1,in+j =
1
2S   xi(m 1)+m+1,(i+1)n+j and (2.16)
x(i 1)(m 1)+1,(i 1)n+j =
1
2S   x(i 1)(m 1)+m+1,in+j . (2.17)
When we substitute the expressions from equations (2.16) and (2.17) into equation (2.15),
we obtain
xi(m 1)+m+1,(i+1)n+j = x(i 1)(m 1)+m+1,in+j + ( 1)jdi. (2.18)
When we replace j with j+n in equation (2.18), we have, for all integers i and j such that
1  i  m and 1  j  2n,
xi(m 1)+m+1,in+j = x(i 1)(m 1)+m+1,(i 1)n+j + ( 1)j+1di.
When we interchange the roles of i and j in the above argument, we have for all integers
i and j such that 1  i  2m and 1  j  n,
xjm+i,j(n 1)+1 = x(j 1)m+i,(j 1)(n 1)+1 + ( 1)i+1d0j and
xjm+i,j(n 1)+n+1 = x(j 1)m+i,(j 1)(n 1)+n+1 + ( 1)id0j .
S. J. Curran, R. M. Low and S. C. Locke: C4-face-magic toroidal labelings on Cm ⇥ Cn 9
3 Results on C4 ⇥ C4
Curran and Low [7] determine all antipodal balanced C4-face-magic toroidal labelings on
C4 ⇥ C4 (up to symmetries on a torus). In order to state this result precisely, we must
introduce some definitions. This result, as stated in Theorem 3.5, is the basis for the in-
vestigation of antipodal balanced C4-face-magic toroidal labelings on C2m ⇥ C2n in this
paper.
Definition 3.1. Let n be a positive integer Let {xi,j : i, j = 1, 2, . . . , 2n} be a C4-face-
magic torus labeling on C2n ⇥ C2n. We say that the C4-face-magic labeling {xi,j} on
C2n ⇥ C2n is torus symmetric if all row sums, column sums, and diagonal sums have a
constant value S0. In other words, the sums
Ri =
2nX
j=1
xi,j = S
0 for all i = 1, 2, . . . , 2n,
Cj =
2nX
i=1
xi,j = S
0 for all j = 1, 2, . . . , 2n,
Dj =
2nX
i=1
xi,i+j = S
0 for all j = 1, 2, . . . , 2n and
D0j =
2nX
i=1
xi,j i = S
0 for all j = 1, 2, . . . , 2n,
are constant.
According to Lemma 3.2, a torus symmetric C4-face-magic toroidal labeling on C4 ⇥
C4 is equivalent to a C4-face-magic toroidal labeling on C4 ⇥ C4 in which the four 2 ⇥ 2
block sums given by Bi,j = xi,j + xi,j+2 + xi+2,j + xi+2,j+2, for all i, j = 1, 2, also add
up to the C4-face-magic value 34.
Lemma 3.2 ([7], Lemma 8). Consider the system of linear equations xi,j + xi+1,j +
xi,j+1 + xi+1,j+1 = 34 (S1), for all i = 1, 2, 3 and j = 1, 2, 3 for a C4-face-magic
labeling on P4 ⇥ P4. Let (S2) be the system S1 together with the equations Bi,j = xi,j +
xi,j+2 + xi+2,j + xi+2,j+2 = 34, for all i, j = 1, 2. If the labeling {xi,j} satisfies system
(S2), then {xi,j} is torus symmetric. Also, let (S3) be the system (S1) together with the
equations R1 = x1,1 + x1,2 + x1,3 + x1,4 = 34, C1 = x1,1 + x2,1 + x3,1 + x4,1 = 34,
and D4 = x1,1 + x2,2 + x3,3 + x4,4 = 34. Then, (S2) is equivalent to (S3).
Definition 3.3. Consider the natural embedding of C2n ⇥ C2n in the torus. We say that
two torus symmetric C4-face-magic toroidal labelings on C2n ⇥ C2n are torus equivalent
if there is a homeomorphism of the torus that maps C2n ⇥ C2n onto itself such that the
first C4-face-magic toroidal labeling on C2n⇥C2n is mapped to the second C4-face-magic
toroidal labeling on C2n ⇥ C2n.
By Lemma 3.4, a torus symmetric C4-face-magic toroidal labeling on C4 ⇥ C4 is an-
tipodal balanced.
Lemma 3.4 ([7], Lemma 13). Let {xi,j} be a torus symmetric C4-face-magic toroidal
labeling on C4⇥C4. Then, for all i and j, we have xi,j +xi+2,j+2 = 17 where the indices
are taken modulo four. In other words, the labeling {xi,j} is antipodal balanced.
10 Art Discrete Appl. Math. 4 (2021) #P1.04
All torus symmetric C4-face-magic labelings on C4 ⇥C4 (up to torus equivalence) are
given in the following theorem.
Theorem 3.5 ([7], Theorem 10). There are three distinct torus nonequivalent torus sym-
metric C4-face-magic toroidal labelings on C4 ⇥ C4. These three distinct torus nonequiv-
alent torus symmetric C4-face-magic toroidal labelings on C4 ⇥ C4 are given below:
1 8 13 12
14 11 2 7
4 5 16 9
15 10 3 6
Table 1: Torus symmetric C4-face-magic toroidal labeling A on C4 ⇥ C4.
1 8 11 14
12 13 2 7
6 3 16 9
15 10 5 4
Table 2: Torus symmetric C4-face-magic toroidal labeling B on C4 ⇥ C4.
1 12 7 14
8 13 2 11
10 3 16 5
15 6 9 4
Table 3: Torus symmetric C4-face-magic toroidal labeling C on C4 ⇥ C4.
An interesting observation about these three labelings is that the row sums and col-
umn sums of any two labelings are the 16 C4-face sums in the third labeling. In the re-
mark below, we indicate how the three torus nonequivalent torus symmetric C4-face-magic
toroidal labelings in Theorem 3.5 can be regarded as coming from one particular labeling
on C4 ⇥ C4.
Remark 3.6 ([7], Remark 24). Label the vertices of C4 ⇥ C4 with the elements from the
set {0, 1}4 so that the labelings on each C4 face adds up to (2, 2, 2, 2). This labeling is
given in Table 4. Then the corresponding C4-face-magic torus labelings on C4 ⇥ C4 are
given by xi,j = x1,1 + a1d1 + a2d2 + a3d3 + a4d4 where x1,1 = 1, (a1, a2, a3, a4) is
the labeling on vertex (i, j) in C4 ⇥ C4 given in Table 4, and (d1, d2, d3, d4) is one of
the three choices of either (1, 2, 4, 8), (1, 4, 2, 8) or (1, 8, 2, 4). The choices of (1, 2, 4, 8),
(1, 4, 2, 8) or (1, 8, 2, 4) for (d1, d2, d3, d4) result in the labelings A, B and C, respectively,
in Theorem 3.5.
S. J. Curran, R. M. Low and S. C. Locke: C4-face-magic toroidal labelings on Cm ⇥ Cn 11
(0, 0, 0, 0) (1, 1, 1, 0) (0, 0, 1, 1) (1, 1, 0, 1)
(1, 0, 1, 1) (0, 1, 0, 1) (1, 0, 0, 0) (0, 1, 1, 0)
(1, 1, 0, 0) (0, 0, 1, 0) (1, 1, 1, 1) (0, 0, 0, 1)
(0, 1, 1, 1) (1, 0, 0, 1) (0, 1, 0, 0) (1, 0, 1, 0)
Table 4: C4-face-magic toroidal labeling with elements from {0, 1}4 on C4 ⇥ C4.
4 Results on C2m ⇥ C2n
We first consider antipodal balanced C4-face-magic toroidal labelings on C6 ⇥ C6.
Lemma 4.1. Let {xi,j} be an antipodal balanced toroidal C4-face-magic labeling on C6⇥
C6. Let di = x2i+1,3i+1   x2i 1,3i 2, for i = 1, 2, 3, and let dj+3 = d0j = x3j+1,2j+1  
x3j 2,2j 2, for j = 1, 2, 3. Then the value of xi,j is determined by the values of x1,1 and
di, for i = 1, 2, 3, 4, 5, 6 as displayed in Table 5.
Proof. By Lemma 2.11, for integers i and j, we have
x2i+1,3i+j = x2i 1,3i 3+j + ( 1)j+1di,
when 1  i  3 and 1  j  6 and
x2i+4,3i+j = x2i+2,3i 3+j + ( 1)j+1di,
when 1  i  3 and 1  j  6. Thus x1,4 = x1,1 + d1 + d2 + d3. Furthermore, it
suffices to establish the result for the expressions x1,j and x4,j , for j = 1, 2, 3. Also, by
Lemma 4.1, for integers i and j, we have
x3j+i,2j+1 = x3j 3)+i,2j 1 + ( 1)i+1dj+3,
when 1  i  6 and 1  j  3 and
x3j+i,2j+4 = x3j 3+i,2j+2 + ( 1)idj+3,
when 1  i  6 and 1  j  3. Thus x4,1 = x1,1+d4+d5+d6 and x4,3 = x1,1+d4. From
x1,4 = x1,1+d1+d+2+d3 and x1,4 = x4,2 d6, we have x4,2 = x1,1+d1+d2+d3+d6.
From x4,1 = x1,1 + d4 + d5 + d6 and x1,3 = x4,1   d4, we have x1,1 = x1,1 + d5 + d6.
Lastly, from x1,4 = x1,1 + d1 + d + 2 + d3, x6,3 = x3,1 + d4, x3,5 = x6,3 + d5 and
x1,2 = x3,5   d1, we have x1,2 = x1, 1 + d1 + d2 + d3 + d4 + d5.
We next consider antipodal balanced C4-face-magic toroidal labelings on C8 ⇥ C8.
Lemma 4.2. Let {xi,j} be an antipodal balanced toroidal C4-face-magic labeling on C8⇥
C8. Let di = x3i+1,4i+1 x3i 2,4i 3, for i = 1, 2, 3, 4, and let dj+4 = d0j = x4j+1,3j+1 
x4j 3,3j 2, for j = 1, 2, 3, 4. Then the value of xi,j is determined by the values of x1,1
and di, for i = 1, 2, 3, 4, 5, 6, 7, 8 as displayed in Figure 1.
12 Art Discrete Appl. Math. 4 (2021) #P1.04
x1,1 x1,2 = x1,3 = x1,4 = x1,5 = x1,6 =
x1,1 + d1 x1,1 + d5 x1,1 + d1 x1,1 + d4 x1,1 + d1
+d2 + d3 +d6 +d2 + d3 +d5 +d2 + d3
+d4 + d5 +d5 + d6
x2,1 = x2,2 = x2,3 = x2,4 = x2,5 = x2,6 =
x1,1 + d1 x1,1 + d3 x1,1 + d1 x1,1 + d3 x1,1 + d1 x1,1 + d3
+d2 + d4 +d6 +d2 + d4 +d4 + d5 +d2 + d6 +d4
+d5 + d6 +d6
x3,1 = x3,2 = x3,3 = x3,4 = x3,5 = x3,6 =
x1,1 + d2 x1,1 + d1 x1,1 + d2 x1,1 + d1 x1,1 + d2 x1,1 + d1
+d3 +d4 + d5 +d3 + d5 +d3 + d4 +d5 + d6
+d6 +d5
x4,1 = x4,2 = x4,3 = x4,4 = x4,5 = x4,6 =
x1,1 + d4 x1,1 + d1 x1,1 + d4 x1,1 + d1 x1,1 + d6 x1,1 + d1
+d5 + d6 +d2 + d3 +d2 + d3 +d2 + d3
+d6 +d4 + d5 + d6 +d4
x5,1 = x5,2 = x5,3 = x5,4 = x5,5 = x5,6 =
x1,1 + d1 x1,1 + d3 x1,1 + d1 x1,1 + d3 x1,1 + d1 x1,1 + d3
+d2 +d4 + d5 +d2 + d5 +d2 + d4 +d5 + d6
+d6 +d5
x6,1 = x6,2 = x6,3 = x6,4 = x6,5 = x6,6 =
x1,1 + d2 x1,1 + d1 x1,1 + d2 x1,1 + d1 x1,1 + d2 x1,1 + d1
+d3 + d4 +d6 +d3 + d4 +d4 + d5 +d3 + d6 +d4
+d5 + d6 +d6
Table 5: C4-face-magic toroidal labeling involving the differences d1, d2, d3, d4, d5 and
d6 on C6 ⇥ C6.
Proof. By Lemma 2.10, for integers i and j, we have
x3i+1,j = x3i 2,j+4 + ( 1)j+1di,
when 1  i  4 and 1  j  8 and
x3i+1,j = x3i 2,j+4 + ( 1)jdi 4,
when 5  i  8 and 1  j  8. Thus x5,1 = x1,1 + d1 + d2 + d3 + d4. Furthermore, we
only need to verify the expressions for x1,j for 2  j  8. Also, by Lemma 2.10, for all
integers i and j such that we have
xi,3j+1 = xi+4,3j 2 + ( 1)i+1d0j ,
when 1  i  8 and 1  j  4 and
xi,3j+1 = xi+4,3j 2 + ( 1)id0j 4,
when 1  i  8 and 5  j  8. Hence, x1,7 = x1,1 + d5 + d6, x1,5 = x1,7 + d7 + d8,
x1,3 = x1,5 d5 d6. Thus x1,5 = x1,1+d5+d6+d7+d8 and x1,3 = x1,1+d7+d8. Also,
S. J. Curran, R. M. Low and S. C. Locke: C4-face-magic toroidal labelings on Cm ⇥ Cn 13
x1,4 = x5,1+ d5, x1,2 = x1,4+ d6+ d7, x1,8 = x1,2+ d8  d5 and x1,6 = x1,8  d6  d7.
Since x5,1 = x1,1 + d1 + d2 + d3 + d4, we have x1,4 = x1,1 + d1 + d2 + d3 + d4 + d5
x1,2 = x1,1+d1+d2+d3+d4+d5+d6+d7, x1,8 = x1,1+d1+d2+d3+d4+d6+d7+d8
and x1,6 = x1,1 + d1 + d2 + d3 + d4 + d8.
Lemma 4.3. Let {xi,j} be an antipodal balanced C4-face-magic toroidal labeling on
C4m ⇥ C4n. Then, we have
Ri =
4nX
j=1
xi,j = 2n(16mn+ 1), for all i = 1, 2, . . . , 4m
and
Cj =
4mX
i=1
xi,j = 2m(16mn+ 1), for all j = 1, 2, . . . , 4n.
Furthermore, if m = n, then we have
Di =
4nX
j=1
xj,i+j = 2n(16n
2 + 1), for all i = 1, 2, . . . , 4n
and
D0i =
4nX
j=1
xj,i j = 2n(16n
2 + 1), for all i = 1, 2, . . . , 4n.
Proof. Let S = 2(16mn + 1) be the C4-face-magic value of {xi,j}. We have, for all
i = 1, 2, . . . , 4m,
Ri +Ri+1 =
2nX
j=1
 
xi,2j 1 + xi,2j + xi+1,2j 1 + xi+1,2j
 
= 2nS = 4n(16mn+ 1).
Thus it suffices to show that R1 = 2n(16mn + 1). We first observe that for all j =
1, 2, . . . , n,
x2m+1,2j 1+2n = x1,2j 1+2n + (d1 + d2 + · · ·+ d2m).
Thus,
1
2S = x1,2j 1 + x2m+1,2j 1+2n = x1,2j 1 + x1,2j 1+2n + (d1 + d2 + · · ·+ d2m).
We next observe that for all j = 1, 2, . . . , n,
x2m+1,2j+2n = x1,2j+2n   (d1 + d2 + · · ·+ d2m).
Thus,
1
2S = x1,2j + x2m+1,2j+2n = x1,2j + x1,2j+2n   (d1 + d2 + · · ·+ d2m).
14 Art Discrete Appl. Math. 4 (2021) #P1.04
x1,1 x1,2 = x1,3 = x1,4 = x1,5 = x1,6 = x1,7 = x1,8 =
x1,1 + d1 x1,1 + d7 x1,1 + d1 x1,1 + d5 x1,1 + d1 x1,1 + d5 x1,1 + d1
+d2 + d3 +d8 +d2 + d3 +d6 + d7 +d2 + d3 +d6 +d2 + d3
+d4 + d5 +d4 + d5 +d8 +d4 + d8 +d4 + d6
+d6 + d7 +d7 + d8
x2,1 = x2,2 = x2,3 = x2,4 = x2,5 = x2,6 = x2,7 = x2,8 =
x1,1 + d1 x1,1 + d4 x1,1 + d1 x1,1 + d4 x1,1 + d1 x1,1 + d4 x1,1 + d1 x1,1 + d4
+d2 + d3 +d8 +d2 + d3 +d6 + d7 +d2 + d3 +d5 + d6 +d2 + d3 +d5
+d5 + d6 +d5 + d6 +d8 +d7 +d7 + d8
+d7 + d8
x3,1 = x3,2 = x3,3 = x3,4 = x3,5 = x3,6 = x3,7 = x3,8 =
x1,1 + d3 x1,1 + d1 x1,1 + d3 x1,1 + d1 x1,1 + d3 x1,1 + d1 x1,1 + d3 x1,1 + d1
+d4 +d2 + d5 +d4 + d7 +d2 + d5 +d4 + d5 +d2 + d8 +d4 + d5 +d2 + d6
+d6 + d7 +d8 +d6 + d7 +d6 +d7 + d8
+d8
x4,1 = x4,2 = x4,3 = x4,4 = x4,5 = x4,6 = x4,7 = x4,8 =
x1,1 + d1 x1,1 + d2 x1,1 + d1 x1,1 + d2 x1,1 + d1 x1,1 + d2 x1,1 + d1 x1,1 + d2
+d5 + d6 +d3 + d4 +d5 + d6 +d3 + d4 +d3 + d4 +d7 + d8 +d3 + d4
+d7 + d8 +d8 +d6 + d7 +d5 + d6 +d5
+d8 +d7
x5,1 = x5,2 = x5,3 = x5,4 = x5,5 = x5,6 = x5,7 = x5,8 =
x1,1 + d1 x1,1 + d5 x1,1 + d1 x1,1 + d5 x1,1 + d1 x1,1 + d8 x1,1 + d1 x1,1 + d6
+d2 + d3 +d6 + d7 +d2 + d3 +d2 + d3 +d2 + d3 +d7 + d8
+d4 +d4 + d7 +d4 + d5 +d4 + d5
+d8 +d6 + d7 + d8 +d6
x6,1 = x6,2 = x6,3 = x6,4 = x6,5 = x6,6 = x6,7 = x6,8 =
x1,1 + d4 x1,1 + d1 x1,1 + d4 x1,1 + d1 x1,1 + d4 x1,1 + d1 x1,1 + d4 x1,1 + d1
+d5 + d6 +d2 + d3 +d5 + d6 +d2 + d3 +d2 + d3 +d7 + d8 +d2 + d3
+d7 + d8 +d8 +d6 + d7 +d5 + d6 +d5
+d8 +d7
x7,1 = x7,2 = x7,3 = x7,4 = x7,5 = x7,6 = x7,7 = x7,8 =
x1,1 + d1 x1,1 + d3 x1,1 + d1 x1,1 + d3 x1,1 + d1 x1,1 + d3 x1,1 + d1 x1,1 + d3
+d2 +d4 + d5 +d2 + d7 +d4 + d5 +d2 + d5 +d4 + d8 +d2 + d5 +d4 + d6
+d6 + d7 +d8 +d6 + d7 +d6 +d7 + d8
+d8
x8,1 = x8,2 = x8,3 = x8,4 = x8,5 = x8,6 = x8,7 = x8,8 =
x1,1 + d2 x1,1 + d1 x1,1 + d2 x1,1 + d1 x1,1 + d2 x1,1 + d1 x1,1 + d2 x1,1 + d1
+d3 + d4 +d8 +d3 + d4 +d6 + d7 +d3 + d4 +d5 + d6 +d3 + d4 +d5
+d5 + d6 +d5 + d6 +d8 +d7 +d7 + d8
+d7 + d8
Figure 1: C4-face-magic toroidal labeling involving the differences d1, d2, d3, d4, d5, d6,
d7 and d8 on C8 ⇥ C8.
S. J. Curran, R. M. Low and S. C. Locke: C4-face-magic toroidal labelings on Cm ⇥ Cn 15
Hence, we have
R1 =
4nX
j=1
x1,j =
2nX
j=1
 
x1,2j 1 + x1,2j
 
=
nX
j=1
 
x1,2j 1 + x1,2j 1+2n
 
+
nX
j=1
 
x1,2j + x1,2j+2n
 
= nS = 2n(16mn+ 1).
By interchanging the roles of i and j, we have
Cj =
4mX
i=1
xi,j = 2m(16mn+ 1), for all j = 1, 2, . . . , 4n.
Finally, we assume that m = n. Then,
Di =
4nX
j=1
xj,i+j =
2nX
j=1
 
xj,i+j + xj+2n,i+j+2n
 
= (2n)
 
1
2S
 
= 2n(16n2 + 1).
A similar argument shows that for all i = 1, 2, . . . , 4n,
D0i =
4nX
j=1
xj,i j = 2n(16n
2 + 1).
Proposition 4.4. Let m and n be integers where m,n   3. Let {xi,j} be a C4-face-magic
labeling on Pm ⇥ Pn with face-magic value S. Suppose that for all integers i and j such
that 1  i  m  2 and 1  j  n  2, we have
xi,j + xi,j+2 + xi+2,j + xi+2,j+2 = S.
Then, m  4 and n  4.
Proof. For the purpose of contradiction, we assume that m   5. We first observe that for
all integers i and j such that 1  i  m  2 and 1  j  n  2, we have
xi,j + xi,j+1 + xi+1,j + xi+1,j+1 = S = xi,j+1 + xi,j+2 + xi+1,j+1 + xi+1,j+2.
Thus,
xi,j + xi+1,j = xi,j+2 + xi+1,j+2. (4.1)
Replacing i with i+ 1 in equation (4.1) yields
xi+1,j + xi+2,j = xi+1,j+2 + xi+2,j+2. (4.2)
When we subtract equation (4.2) from equation (4.1) and rearrange terms, we obtain
xi,j + xi+2,j+2 = xi+2,j + xi,j+2. (4.3)
16 Art Discrete Appl. Math. 4 (2021) #P1.04
Since
xi,j + xi,j+2 + xi+2,j + xi+2,j+2 = S,
we have
xi,j + xi+2,j+2 =
1
2S = xi+2,j + xi,j+2. (4.4)
Let i = 1 or 3, and j = 1. Then equation (4.4) yields
x1,1 + x3,3 =
1
2S = x5,1 + x3,3.
Hence, x1,1 = x5,1. This is a contradiction. Therefore, m  4. A similar argument shows
that n  4.
Proposition 4.5. Let m and n be even positive integers. Let yj , for j = 1, 2, . . . , 2n,
be a positive integer, and let d1, d2, . . . , dm be integers. We define a labeling {xi,j} on
C2m ⇥ C2n by letting, for all integers i and j such that 0  i  m and 1  j  2n,
xi(m 1)+1,in+j = yj + ( 1)j+1
iX
k=1
dk and
xi(m 1)+m+1,in+j = yj + ( 1)j+1
mX
k=i+1
dk.
Let A = {
Pi
k=1 dk,
Pm
k=i dk : 1  i  m}[{0}. For all integers j such that 1  j  2n,
let
Aj = {yj + ( 1)j+1a : a 2 A}.
Suppose that
1. for all integers j such that 1  j  n, yj + yj+n +( 1)j+1
 Pm
k=1 dk
 
= 4mn+1
and
2. the set {Aj : 1  j  2n} forms a partition of the set {k 2 Z : 1  k  4mn}.
Then, {xi,j} is an antipodal balanced C4-face-magic toroidal labeling on C2m ⇥ C2n.
Proof. We first show that the C4-face sum is preserved for all of the relevant C4 faces on
C2m ⇥ C2n. We note that m   1 is relatively prime to 2m. Thus, m   1 is a generator of
Z2m. Hence,
{xi(m 1)+1,in+j : i = 1, 2, . . . , 2m and j = 1, 2, . . . , 2n}
is the set {xi,j : i = 1, 2, . . . , 2m and j = 1, 2, . . . , 2n}. We observe that (i+m)(m 1)+
m+ 1 = i(m  1) + 1 (mod 2m). We have, for all integers i and j such that 0  i  m
and 1  j  2n,
xi(m 1)+1,in+j = yj + ( 1)j+1
iX
k=1
dk and (4.5)
xi(m 1)+m+1,in+j = yj + ( 1)j+1
mX
k=i+1
dk. (4.6)
S. J. Curran, R. M. Low and S. C. Locke: C4-face-magic toroidal labelings on Cm ⇥ Cn 17
We replace i with i  1 and j with j + n in equation (4.6) to obtain, for all integers i and j
such 1  i  m+ 1 and 1  j  2n,
x(i 1)(m 1)+m+1,(i 1)n+j+n = xi(m 1)+2,in+j = yj+n + ( 1)j+1
mX
k=i
dk.
Hence, for all integers i and j such 1  i  m and 1  j  2n, we have
xi(m 1)+1,in+j + xi(m 1)+1,in+j+1 + xi(m 1)+2,in+j + xi(m 1)+2,in+j+1
= yj + ( 1)j+1
✓ iX
k=1
dk
◆
+ yj+1 + ( 1)j+2
✓ iX
k=1
dk
◆
+ yj+n + ( 1)j+1
✓ mX
k=i
dk
◆
+ yj+n+1 + ( 1)j+2
✓ mX
k=i
dk
◆
=
✓
yj + yj+n + ( 1)j+1
  mX
k=1
dk
 ◆
+
✓
yj+1 + yj+n+1 + ( 1)j+2
  mX
k=1
dk
 ◆
= 12S +
1
2S = S.
Next, we replace i with i  1 and j with j + n in equation (4.5) to obtain, for all integers i
and j such 1  i  m+ 1 and 1  j  2n,
x(i 1)(m 1)+1,(i 1)n+j+n = xi(m 1)+m+2,in+j = yj+n + ( 1)j+1
i 1X
k=1
dk.
Hence, for all integers i and j such 1  i  m and 1  j  2n, we have
xi(m 1)+m+1,in+j + xi(m 1)+m+1,in+j+1 + xi(m 1)+m+2,in+j + xi(m 1)+m+2,in+j+1
= yj + ( 1)j+1
✓ mX
k=i+1
dk
◆
+ yj+1 + ( 1)j+2
✓ mX
k=i+1
dk
◆
+ yj+n + ( 1)j+1
✓i 1X
k=1
dk
◆
+ yj+n+1 + ( 1)j+2
✓i 1X
k=1
dk
◆
=
✓
yj + yj+n + ( 1)j+1
  mX
k=1
dk
 ◆
+
✓
yj+1 + yj+n+1 + ( 1)j+2
  mX
k=1
dk
 ◆
= 12S +
1
2S = S.
Hence, {xi,j} is a C4-face-magic toroidal labeling on C2m ⇥ C2n.
We show that each integer k, where 1  k  4mn, is used exactly once in the labeling
{xi,j}. For each integer j such that 1  j  2n, we show that {xi(m 1)+1,in+j : 1  i 
18 Art Discrete Appl. Math. 4 (2021) #P1.04
2m} = Aj . We have
{xi(m 1)+1,in+j : 1  i  2m} = {xi(m 1)+1,in+j , xi(m 1)+m+1,in+j : 1  i  m}
= {yj , yj + ( 1)j+1
  iX
k=1
dk
 
, yj + ( 1)j+1
  mX
k=i
dk
 
: 1  i  m}
= {yy + ( 1)j+1a : a 2 A} = Aj .
Since {Aj : 1  j  2n} is a partition of the set {k 2 Z : 1  k  4mn}, each integer k,
where 1  k  4mn, is used exactly once in the labeling {xi,j}.
Finally, we show that {xi,j} is an antipodal balanced labeling on C2m ⇥ C2n. When
we replace j with j + n in equation (4.6), we have
x(i+m)(m 1)+1,(i+m)n+j+n = xi(m 1)+1+m,in+j+n = yj+n + ( 1)j+1
  mX
k=i+1
dk
 
.
(4.7)
When we add equations (4.5) and (4.7), we have
xi(m 1)+1,in+j + xi(m 1)+1+m,in+j+n
= yj + ( 1)j+1
  iX
k=1
dk
 
+ yj+n + ( 1)j+1
  mX
k=i+1
dk
 
= yj + yj+n + ( 1)j+1
  mX
k=1
dk
 
= 12S = 4mn+ 1.
When we replace i with i+m in this equation, we have
x(i+m)(m 1)+1+m,(i+m)n+j + x(i+m)(m 1)+1,(i+m)n+j+n
= xi(m 1)+1,in+j + xi(m 1)+1+m,in+j+n = 4mn+ 1.
This completes the proof.
Proposition 4.6. We define a labeling {xi,j} on C4m ⇥ C4n in the following manner. For
integers i and j such that 1  i  2m and 1  j  2n, when j is odd, we let
xi(2m 1)+1,j+i(2n) = 4m(j   1) + 2i,
xi(2m 1)+1,j+(i+1)(2n) = 4m(4n  j) + 2i,
xi(2m 1)+2m+1,j+i(2n) = 4mj   2i+ 1 and
xi(2m 1)+2m+1,j+(i+1)(2n) = 4m(4n  j + 1)  2i+ 1;
and when j is even, we let
xi(2m 1)+1,j+i(2n) = 4mj   2i+ 1,
xi(2m 1)+1,j+(i+1)(2n) = 4m(4n  j + 1)  2i+ 1,
xi(2m 1)+2m+1,j+i(2n) = 4m(j   1) + 2i and
xi(2m 1)+2m+1,j+(i+1)(2n) = 4m(4n  j) + 2i.
Then, {xi,j} is an antipodal balanced C4-face-magic toroidal labeling on C4m ⇥ C4n.
Furthermore, by Lemma 4.3, {xi,j} is row-sum balanced and column-sum balanced, and
{xi,j} is torus symmetric whenever m = n.
S. J. Curran, R. M. Low and S. C. Locke: C4-face-magic toroidal labelings on Cm ⇥ Cn 19
Proof. In the notation of Proposition 4.5, we have d1 = 1 and dk = 2, for 2  k  2m.
Also, for integers j such that 1  j  2n, we have yj = 4m(j   1) + 1 and yj+2n =
4m(4n   j) + 1 when j is odd, and yj = 4mj and yj+2n = 4m(4n   j + 1) when j is
even. Then the labeling {xi,j} of C4m ⇥ C4n satisfies, for all integers i and j such that
1  i  2m and 1  j  2n, when j is odd, we have
xi(2m 1)+1,i(2n)+j = yj + ( 1)j+1
  iX
k=1
dk
 
=
 
4m(j   1) + 1
 
+ (2i  1) = 4m(j   1) + 2i and
xi(2m 1)+1,i(2n)+j+2n = yj+2n + ( 1)j+1
  iX
k=1
dk
 
=
 
4m(4n  j) + 1
 
+ (2i  1) = 4m(4n  j) + 2i,
and when j is even, we have
xi(2m 1)+1,i(2n)+j = yj + ( 1)j+1
  iX
k=1
dk
 
=
 
4mj
 
  (2i  1) = 4mj   2i+ 1 and
xi(2m 1)+1,i(2n)+j+2n = yj+2n + ( 1)j+1
  iX
k=1
dk
 
=
 
4m(4n  j + 1)
 
  (2i  1) = 4m(4n  j + 1)  2i+ 1.
We let i = 2m in each of the previous four equations. Thus, for all integers i and j such
that 1  i  2m and 1  j  2n, when j is odd, we have
x2m+1,j = 4mj and x2m+1,j+2n = 4m(4n  j + 1),
and when j is even, we have
x2m+1,j = 4m(j   1) + 1 and x2m+1,j+2n = 4m(4n  j) + 1.
Next, we observe that the labeling {xi,j} on C4m ⇥ C4n satisfies, for all integers i and j
such that 1  i  2m and 1  j  2n, when j is odd, we have
xi(2m 1)+2m+1,i(2n)+j = yj + ( 1)j+1
  2mX
k=i+1
dk
 
=
 
4m(j   1) + 1
 
+ 2(2m  i) = 4mj   2i and
xi(2m 1)+2m+1,i(2n)+j+2n = yj+2n + ( 1)j+1
  2mX
k=i+1
dk
 
=
 
4m(4n  j) + 1
 
+ 2(2m  i) = 4m(4n  j + 1)  2i,
20 Art Discrete Appl. Math. 4 (2021) #P1.04
and when j is even, we have
xi(2m 1)+2m+1,i(2n)+j = yj + ( 1)j+1
  2mX
k=i+1
dk
 
=
 
4mj
 
  2(2m  i) = 4m(j   1) + 2i and
xi(2m 1)+2m+1,i(2n)+j+2n = yj+2n + ( 1)j+1
  2mX
k=i+1
dk
 
=
 
4m(4n  j + 1)
 
  2(2m  i) = 4m(4n  j) + 2i.
We show that condition (1) of Proposition 4.5 is satisfied. Let j be an integer such that
1  j  2n. When j is odd, we have
yj+yj+2n+( 1)j+1
  2mX
k=1
dk
 
=
 
4m(j 1)+1
 
+
 
4m(4n j)+1
 
+(4m 1) = 16mn+1.
When j is even, we have
yj + yj+2n + ( 1)j+1
  2mX
k=1
dk
 
=
 
4mj
 
+
 
4m(4n  j + 1)
 
  (4m  1) = 16mn+ 1.
We show that condition (2) of Proposition 4.5 is satisfied. We first observe that
A = {
iX
k=1
dk,
2mX
k=i
dk : 1  k  2m} [ {0}
= {2i  1 : 1  i  2m} [ {4m  2i+ 2 : 2  i  2m} [ {0}
= {k 2 Z : 0  k  4m  1}.
Let j be an integer such that 1  j  2n. When j is odd, we have
Aj = {yj + ( 1)j+1a : a 2 A} = {4m(j   1) + 1 + i : i 2 Z, 0  i  4m  1}
= {k 2 Z : 4m(j   1) + 1  j  4mj}
and
Aj+2n = {yj+2n + ( 1)j+1a : a 2 A}
= {4m(4n  j) + 1 + i : i 2 Z, 0  i  4m  1}
= {k 2 Z : 4m(4n  j) + 1  j  4m(4n  j + 1)}.
When j is even, we have
Aj = {yj + ( 1)j+1a : a 2 A} = {4mj   i : i 2 Z, 0  i  4m  1}
= {k 2 Z : 4m(j   1) + 1  j  4mj}
S. J. Curran, R. M. Low and S. C. Locke: C4-face-magic toroidal labelings on Cm ⇥ Cn 21
and
Aj+2n = {yj+2n + ( 1)j+1a : a 2 A}
= {4m(4n  j + 1)  i : i 2 Z, 0  i  4m  1}
= {k 2 Z : 4m(4n  j) + 1  j  4m(4n  j + 1)}.
Hence, for all integers j, when 1  j  2n, we have
Aj = {k 2 Z : 4m(j   1) + 1  j  4mj}
and when 2n+ 1  j  4n, we have
Aj = {k 2 Z : 4m(4n  j) + 1  j  4m(4n  j + 1)}.
Therefore, {Aj : 1  j  4n} is a partition of the set {k 2 Z : 1  k  16mn}. This
completes the proof.
Example 4.7. We consider the example of Proposition 4.6 where m = n = 2. This
example is given in Table 6.
1 16 17 32 57 56 41 40
62 51 46 35 6 11 22 27
5 12 21 28 61 52 45 36
58 55 42 39 2 15 18 31
8 9 24 25 64 49 48 33
59 54 43 38 3 14 19 30
4 13 20 29 60 53 44 37
63 50 47 34 7 10 23 26
Table 6: An antipodal balanced C4-face-magic toroidal labeling on C8 ⇥ C8.
We next prove the converse to Proposition 4.5.
Proposition 4.8. Let m and n be even positive integers. Let {xi,j : (i, j) 2 V (C2m ⇥
C2n)} be an antipodal balanced C4-face-magic toroidal labeling on C2m ⇥ C2n. For all
integers j such that 1  j  2n, let yj = x1,j . For all integers i such that 1  i  m, let
di = xi(m 1)+1,in+1   x(i 1)(m 1)+1,(i 1)n+1.
Then, for all integers i and j such that 0  i  m and 1  j  2n,
xi(m 1)+1,in+j = yj + ( 1)j+1
iX
k=1
dk and (4.8)
xi(m 1)+m+1,in+j = yj + ( 1)j+1
mX
k=i+1
dk. (4.9)
22 Art Discrete Appl. Math. 4 (2021) #P1.04
Let A = {
Pi
k=1 dk,
Pm
k=i dk : 1  i  m}[{0}. For all integers j such that 1  j  2n,
let Aj = {yj + ( 1)j+1a : a 2 A}. Then
1. for all integers j such that 1  j  n, yj + yj+n +( 1)j+1
 Pm
k=1 dk
 
= 4mn+1
and
2. the set {Aj : 1  j  2n} forms a partition of the set {k 2 Z : 1  k  4mn}.
Proof. By Lemma 2.10, for all integers i and j such that 1  i  m and 1  j  2n, we
have
xi(m 1)+1,in+j = x(i 1)(m 1)+1,(i 1)n+j + ( 1)j+1di. (4.10)
For all integers i and j such that 1  i  m and 1  j  2n, repeated use of equation
(4.10) yields
xi(m 1)+1,in+j = x1,j + ( 1)j+1
iX
k=1
dk. (4.11)
Thus equation (4.8) holds. When we let i = m in equation (4.11), we have
xm+1,j = x1,j + ( 1)j+1
mX
k=1
dk. (4.12)
By Lemma 2.10, for all integers i and j such that m + 1  i  2m and 1  j  2n, we
have
xi(m 1)+1,in+j = x(i 1)(m 1)+1,(i 1)n+j + ( 1)jdi m. (4.13)
Let 1  i  m. Replacing i with i+m in equation (4.13) yields
xi(m 1)+m+1,in+j = x(i 1)(m 1)+m+1,(i 1)n+j + ( 1)jdi. (4.14)
For all integers i and j such that 1  i  m and 1  j  2n, repeated use of equation
(4.14) yields
xi(m 1)+m+1,in+j = xm+1,j + ( 1)j
iX
k=1
dk. (4.15)
When we combine equations (4.12) and (4.15), we have
xi(m 1)+m+1,in+j = x1,j + ( 1)j+1
mX
k=i+1
dk. (4.16)
Thus equation (4.9) holds.
Since {xi,j} is an antipodal balanced C4-face-magic toroidal labeling on C2m ⇥ C2n,
we have x1,n+j + xm+1,j = 4mn + 1 for all integers j such that 1  j  n. By
equation (4.12), we have xm+1,j = yj + ( 1)j+1
Pm
k=1 dk. Since x1,n+j = yj+n, we
have yj + yj+n + ( 1)j+1
 Pm
k=1 dk
 
= x1,n+j + xm+1,j = 4mn+ 1.
By equations (4.8) and (4.9), for integers j such that 1  j  2n, we have
Aj = {xi(m 1)+1,in+j , xi(m 1)+m+1,in+j : 0  i  m  1}.
Thus the set {Aj : 1  j  2n} forms a partition of the set {xi,j : 1  i  2m and 1 
j  2n} = {k 2 Z : 1  k  4mn}.
S. J. Curran, R. M. Low and S. C. Locke: C4-face-magic toroidal labelings on Cm ⇥ Cn 23
Proposition 4.9. Let m and n be odd positive integers. Let yj and zj , for j = 1, 2, . . . , n,
be a positive integers, and let d1, d2, . . . , dm be integers. We define a labeling {xi,j} on
C2m ⇥ C2n by letting, for all integers i and j such 0  i  m and 1  j  n,
xi(m 1)+1,in+j = yj + ( 1)j+1
iX
k=1
dk, (4.17)
xi(m 1)+1,(i+1)n+j = yj + ( 1)j+1
mX
k=i+1
dk, (4.18)
xi(m 1)+m+1,in+j = zj + ( 1)j+1
iX
k=1
dk and (4.19)
xi(m 1)+m+1,(i+1)n+j = zj + ( 1)j+1
mX
k=i+1
dk. (4.20)
Let A = {
Pi
k=1 dk,
Pm
k=i dk : 1  i  m}[ {0}. For all integers j such that 1  j  n,
let
Aj = {yj + ( 1)j+1a : a 2 A} and Aj+n = {zj + ( 1)j+1a : a 2 A}.
Suppose
1. for all integers j such that 1  j  n, yj + zj + ( 1)j+1
 Pm
k=1 dk
 
= 4mn + 1,
and
2. the set {Aj : 1  j  2n} forms a partition of the set {k 2 Z : 1  k  4mn}.
Then, {xi,j} is an antipodal balanced C4-face-magic toroidal labeling on C2m ⇥ C2n.
Proof. We first show that the C4-face sum is preserved for all relevant C4 faces on C2m ⇥
C2n. Since gcd(2m,m  1) = 2, m  1 generates the subgroup h2i of Z2m. Hence,
{xi(m 1)+1,in+j , x(i+1)(m 1)+1,in+j , xi(m 1)+m+1,in+j , x(i+1)(m 1)+m+1,in+j :
i = 1, 2, . . . ,m and j = 1, 2, . . . , n}
is the set
{xi,j : i = 1, 2, . . . 2,m and j = 1, 2, . . . , 2n}.
We replace i with i  1 in equations (4.17), (4.18), (4.19) and (4.20) to obtain
xi(m 1)+m+2,(i+1)n+j = yj + ( 1)j+1
i 1X
k=1
dk, (4.21)
xi(m 1)+m+2,in+j = yj + ( 1)j+1
mX
k=i
dk, (4.22)
xi(m 1)+2,(i+1)n+j = zj + ( 1)j+1
i 1X
k=1
dk and (4.23)
xi(m 1)+2,in+j = zj + ( 1)j+1
mX
k=i
dk. (4.24)
24 Art Discrete Appl. Math. 4 (2021) #P1.04
We replace j with j+1 in equations (4.17), (4.18), (4.19), (4.20), (4.21), (4.22), (4.23) and
(4.24) to obtain
xi(m 1)+1,in+j+1 = yj+1 + ( 1)j+2
iX
k=1
dk, (4.25)
xi(m 1)+1,(i+1)n+j+1 = yj+1 + ( 1)j+2
mX
k=i+1
dk, (4.26)
xi(m 1)+m+1,in+j+1 = zj+1 + ( 1)j+2
iX
k=1
dk, (4.27)
xi(m 1)+m+1,(i+1)n+j+1 = zj+1 + ( 1)j+2
mX
k=i+1
dk, (4.28)
xi(m 1)+m+2,(i+1)n+j+1 = yj+1 + ( 1)j+2
i 1X
k=1
dk, (4.29)
xi(m 1)+m+2,in+j+1 = yj+1 + ( 1)j+2
mX
k=i
dk, (4.30)
xi(m 1)+2,(i+1)n+j+1 = zj+1 + ( 1)j+2
i 1X
k=1
dk and (4.31)
xi(m 1)+2,in+j+1 = zj+1 + ( 1)j+2
mX
k=i
dk. (4.32)
When we add equations (4.17), (4.25), (4.24) and (4.32) together, for 1  i  m and
1  j  n  1, we obtain
xi(m 1)+1,in+j + xi(m 1)+1,in+j+1 + xi(m 1)+2,in+j + xi(m 1)+2,in+j+1
=
✓
yj + ( 1)j+1
  iX
k=1
dk
 ◆
+
✓
yj+1 + ( 1)j+2
  iX
k=1
dk
 ◆
+
✓
zj + ( 1)j+1
  mX
k=i
dk
 ◆
+
✓
zj+1 + ( 1)j+2
  mX
k=i
dk
 ◆
=
✓
yj + zj + ( 1)j+1
  mX
k=1
dk
 ◆
+
✓
yj+1 + zj+1 + ( 1)j+2
  mX
k=1
dk
 ◆
= S.
When we add equations (4.18), (4.26), (4.23) and (4.31) together, for 1  i  m and
1  j  n  1, we obtain
xi(m 1)+1,(i+1)n+j + xi(m 1)+1,(i+1)n+j+1
+ xi(m 1)+2,(i+1)n+j + xi(m 1)+2,(i+1)n+j+1 = S.
S. J. Curran, R. M. Low and S. C. Locke: C4-face-magic toroidal labelings on Cm ⇥ Cn 25
Similar C4-face-magic sums occur when we add equations (4.19), (4.49), (4.22), and (4.52)
together; or when we add equations (4.20), (4.50), (4.21) and (4.51) together.
Suppose i is odd and let j = n in equations (4.18), (4.20), (4.21), and (4.23), we have
xi(m 1)+1,n = yn + ( 1)n+1
mX
k=i+1
dk, (4.33)
xi(m 1)+m+1,n = zn + ( 1)n+1
mX
k=i+1
dk, (4.34)
xi(m 1)+m+2,n = yn + ( 1)n+1
i 1X
k=1
dk and (4.35)
xi(m 1)+2,n = zn + ( 1)n+1
i 1X
k=1
dk. (4.36)
Suppose i is odd and let j = 1 in equations (4.17), (4.19), (4.22), and (4.24), we have
xi(m 1)+1,n+1 = y1 + ( 1)2
iX
k=1
dk, (4.37)
xi(m 1)+m+1,n+1 = z1 + ( 1)2
iX
k=1
dk, (4.38)
xi(m 1)+m+2,n+1 = y1 + ( 1)2
mX
k=i
dk and (4.39)
xi(m 1)+2,n+1 = z1 + ( 1)2
mX
k=i
dk. (4.40)
When we add equations (4.33), (4.37), (4.36) and (4.40) together, we have
xi(m 1)+1,n + xi(m 1)+1,n+1 + xi(m 1)+2,n + xi(m 1)+2,n+1
=
✓
yn +
  mX
k=i+1
dk
 ◆
+
✓
y1 +
  iX
k=1
dk
 ◆
+
✓
zn +
 i 1X
k=1
dk
 ◆
+
✓
z1 +
  mX
k=i
dk
 ◆
=
✓
y1 + yn +
  mX
k=1
dk
 ◆
+
✓
z1 + zn +
  mX
k=1
dk
 ◆
= S.
When we add equations (4.34), (4.38), (4.35) and (4.39) together, we have
xi(m 1)+m+1,n + xi(m 1)+m+1,n+1 + xi(m 1)+m+2,n + xi(m 1)+m+2,n+1 = S.
26 Art Discrete Appl. Math. 4 (2021) #P1.04
Suppose i is even and let j = n in equations (4.17), (4.19), (4.22) and (4.24), we have
xi(m 1)+1,n = yn + ( 1)n+1
iX
k=1
dk, (4.41)
xi(m 1)+m+1,n = zn + ( 1)n+1
iX
k=1
dk, (4.42)
xi(m 1)+m+2,n = yn + ( 1)n+1
mX
k=i
dk and (4.43)
xi(m 1)+2,n = zn + ( 1)n+1
mX
k=i
dk. (4.44)
Suppose i is even and let j = 1 in equations (4.18), (4.20), (4.21) and (4.23), we have
xi(m 1)+1,n+1 = y1 + ( 1)2
mX
k=i+1
dk, (4.45)
xi(m 1)+m+1,n+1 = z1 + ( 1)2
mX
k=i+1
dk, (4.46)
xi(m 1)+m+2,n+1 = y1 + ( 1)2
i 1X
k=1
dk and (4.47)
xi(m 1)+2,n+1 = z1 + ( 1)2
i 1X
k=1
dk. (4.48)
When we add equations (4.41), (4.45), (4.44) and (4.48) together, we have
xi(m 1)+1,n + xi(m 1)+1,n+1 + xi(m 1)+2,n + xi(m 1)+2,n+1
=
✓
yn +
  iX
k=1
dk
 ◆
+
✓
y1 +
  mX
k=i+1
dk
 ◆
+
✓
zn +
  mX
k=i
dk
 ◆
+
✓
z1 +
 i 1X
k=1
dk
 ◆
=
✓
y1 + yn +
  mX
k=1
dk
 ◆
+
✓
z1 + zn +
  mX
k=1
dk
 ◆
= S.
When we add equations (4.42), (4.46), (4.43) and (4.47) together, we have
xi(m 1)+m+1,n + xi(m 1)+m+1,n+1 + xi(m 1)+m+2,n + xi(m 1)+m+2,n+1 = S.
S. J. Curran, R. M. Low and S. C. Locke: C4-face-magic toroidal labelings on Cm ⇥ Cn 27
For 1  j  n, by equations (4.17) and (4.18), we have
{xi(m 1)+1,in+j : i = 0, 1, . . . , 2m  1}
= {xi(m 1)+1,in+j , x(i+m)(m 1)+1,(i+m)n+j : i = 0, 1, . . . ,m  1}
= {xi(m 1)+1,in+j , xi(m 1)+1,(i+1)n+j : i = 1, 2, . . . ,m}
= {yj + ( 1)j+1
iX
k=1
dk, yj + ( 1)j+1
mX
k=i+1
dk : i = 0, 1, . . . ,m  1}
= {yj + ( 1)j+1a : a 2 A} = Aj .
For 1  j  n, by equations (4.19) and (4.20), we have
{xi(m 1)+m+1,in+j : i = 0, 1, . . . , 2m  1}
= {xi(m 1)+m+1,in+j , x(i+m)(m 1)+m+1,(i+m)n+j : i = 0, 1, . . . ,m  1}
= {xi(m 1)+m+1,in+j , xi(m 1)+m+1,(i+1)n+j : i = 1, 2, . . . ,m}
= {zj + ( 1)j+1
iX
k=1
dk, zj + ( 1)j+1
mX
k=i+1
dk : i = 0, 1, . . . ,m  1}
= {zj + ( 1)j+1a : a 2 A} = Aj+n.
Since {Aj , Aj+n : j = 1, 2, . . . , n} is a partition of {k 2 Z : 1  k  4mn}, {xi,j : i =
1, 2, . . . , 2m and j = 1, 2, . . . , 2n} = {k 2 Z : 1  k  4mn}. By Lemma 2.2, {xi,j} is
a C4-face-magic labeling on C2m ⇥ C2n.
We need to show that {xi,j} is antipodal balanced. Let i and j be integers such that
0  i  m and 1  j  n. We add equations (4.17) and (4.20) together to obtain
xi(m 1)+1,in+j + xi(m 1)+m+1,(i+1)n+j =
= yj + ( 1)j+1
iX
k=1
dk + zj + ( 1)j+1
mX
k=i+1
dk
= yj ++zj + ( 1)j+1
mX
k=1
dk =
1
2S = 4mn+ 1.
We add equations (4.18) and (4.19) together to obtain
xi(m 1)+1,(i+1)n+j + xi(m 1)+m+1,in+j =
= yj + ( 1)j+1
iX
k=i+1
dk + zj + ( 1)j+1
iX
k=1
dk
= yj ++zj + ( 1)j+1
mX
k=1
dk =
1
2S = 4mn+ 1.
Hence, {xi,j} is an antipodal balanced C4-face-magic toroidal labeling on C2m⇥C2n.
28 Art Discrete Appl. Math. 4 (2021) #P1.04
Proposition 4.10. Suppose m and n are odd positive integers. Suppose i and j are integers
such that 1  i  m, 1  j  n, and j is odd. Let
xi(m 1)+1,in+j = 2m(j   1) + 2i,
xi(m 1)+1,(i+1)n+j = 2mj   2i+ 1,
xi(m 1)+m+1,in+j = 2m(2n  j) + 2i and
xi(m 1)+m+1,(i+1)n+j = 2m(2n  j + 1)  2i+ 1.
Suppose i and j are integers such that 1  i  m, 1  j  n, and j is even. Let
xi(m 1)+1,in+j = 2mj   2i+ 1,
xi(m 1)+1,(i+1)n+j = 2m(j   1) + 2i,
xi(m 1)+m+1,in+j = 2m(2n  j + 1)  2i+ 1 and
xi(m 1)+m+1,(i+1)n+j = 2m(2n  j) + 2i.
Then, {xi,j} is an antipodal balanced C4-face-magic toroidal labeling on C2m ⇥ C2n.
Proof. This labeling {xi,j} corresponds to the labeling in Proposition 4.9 where for inte-
gers j such that 1  j  n and j is odd, yj = 2m(j   1) + 1 and zj = 2m(2n  j) + 1;
for integers j such that 1  j  n and j is even, yj = 2mj and zj = 2m(2n  j +1); and
d1 = 1 and dk = 2 for integers k such that 2  k  m.
Let j be an integer such that 1  j  n and j is odd. Then,
yj + zj + ( 1)j+1
  mX
k=1
dk
 
=
 
2m(j   1) + 1
 
+
 
2m(2n  j) + 1
 
+ (2m  1) = 4mn+ 1.
Let j be an integer such that 1  j  n and j is even. Then,
yj + zj + ( 1)j+1
  mX
k=1
dk
 
= 2mj + 2m(2n  j + 1)  (2m  1) = 4mn+ 1.
Thus condition (1) of Proposition 4.9 is satisfied.
We have
A = {
iX
k=1
dk,
mX
k=i+1
dk : i = 1, 2, . . . ,m} [ {0}
= {2i  1, 2(m  i) : i = 1, 2, . . . ,m} [ {0} = {k 2 Z : 0  k  2m  1}.
Let j be an integer such that 1  j  n and j is odd. Then,
Aj = {yj + ( 1)j+1a : a 2 A} = {2m(j   1) + 1 + a : a 2 A}
and
Aj+n = {zj + ( 1)j+1a : a 2 A} = {2m(2n  j) + 1 + a : a 2 A}.
S. J. Curran, R. M. Low and S. C. Locke: C4-face-magic toroidal labelings on Cm ⇥ Cn 29
Let j be an integer such that 1  j  n and j is even. Then,
Aj = {yj + ( 1)j+1a : a 2 A} = {2mj   a : a 2 A}
= {2m(j   1) + 1 + a0 : a0 2 A}
and
Aj+n = {zj + ( 1)j+1a : a 2 A} = {2m(2n  j + 1)  a : a 2 A}
= {2m(2n  j) + 1 + a0 : a0 2 A}.
Hence, {Aj : 1  j  2n} forms a partition of {k 2 Z : 1  k  4mn}. Thus condition
(2) of Proposition 4.9 is satisfied. Therefore by Proposition 4.9, {xi,j} is an antipodal
balanced C4-face-magic toroidal labeling on C2m ⇥ C2n.
Example 4.11. We consider the example of Proposition 4.10 where m = n = 3. This
example is given in Table 7.
1 12 13 6 7 18
34 27 22 33 28 21
5 8 17 2 11 14
31 30 19 36 25 24
4 9 16 3 10 15
35 26 23 32 29 20
Table 7: An antipodal balanced C4-face-magic toroidal labeling on C6 ⇥ C6.
We need the following converse to Proposition 4.9 in order to prove our last result in
this paper.
Proposition 4.12. Let m and n be odd positive integers. Let {xi,j : (i, j) 2 V (C2m ⇥
C2n)} be an antipodal balanced C4-face-magic toroidal labeling on C2m ⇥ C2n. For all
integers j such that 1  j  n, let yj = x1,j and zj = xm+1,j . For all integers i such that
1  i  m, let
di = xi(m 1)+1,in+1   x(i 1)(m 1)+1,(i 1)n+1.
Then, for all integers i and j such that 0  i  m and 1  j  n, we have
xi(m 1)+1,in+j = yj + ( 1)j+1
iX
k=1
dk, (4.49)
xi(m 1)+1,(i+1)n+j = yj + ( 1)j+1
mX
k=i+1
dk, (4.50)
xi(m 1)+m+1,in+j = zj + ( 1)j+1
iX
k=1
dk and (4.51)
xi(m 1)+m+1,(i+1)n+j = zj + ( 1)j+1
mX
k=i+1
dk. (4.52)
30 Art Discrete Appl. Math. 4 (2021) #P1.04
Let A = {
Pi
k=1 dk,
Pm
k=i dk : 1  i  m}[ {0}. For all integers j such that 1  j  n,
let
Aj = {yj + ( 1)j+1a : a 2 A} and Aj+n = {zj + ( 1)j+1a : a 2 A}.
Then
1. for all integers j such that 1  j  n, yj + zj + ( 1)j+1
 Pm
k=1 dk
 
= 4mn + 1,
and
2. the set {Aj : 1  j  2n} forms a partition of the set {k 2 Z : 1  k  4mn}.
Proof. By Lemma 2.11, for all integers i and j such that 1  i  m and 1  j  2n, we
have
xi(m 1)+1,in+j = x(i 1)(m 1)+1,(i 1)n+j + ( 1)j+1di and (4.53)
xi(m 1)+m+1,in+j = x(i 1)(m 1)+m+1,(i 1)n+j + ( 1)j+1di. (4.54)
For integers i and j such that 1  i  m and 1  j  n, repeated use of equation (4.53)
yields
xi(m 1)+1,in+j = x1,j + ( 1)j+1
iX
k=1
dk. (4.55)
Thus equation (4.49) holds. When we let i = m in equation (4.55), we have
x1,n+j = x1,j + ( 1)j+1
mX
k=1
dk. (4.56)
When we replace j with j + n in equation (4.53), for integers i and j such that 1  i  m
and 1  j  n, we have
xi(m 1)+1,(i+1)n+j = x(i 1)(m 1)+1,in+j + ( 1)jdi. (4.57)
For integers i and j such that 1  i  m and 1  j  n, repeated use of equation (4.57)
yields
xi(m 1)+1,(i+1)n+j = x1,n+j + ( 1)j
iX
k=1
dk. (4.58)
Combining equation (4.58) with equation (4.56) yields
xi(m 1)+1,(i+1)n+j = x1,j + ( 1)j+1
mX
k=i+1
dk.
Thus equation (4.50) holds. A similar proof using equation (4.54) shows that equations
(4.51) and (4.52) hold.
A proof similar to that in Proposition 4.8 shows that, for all integers j such that 1 
j  n, we have yj + zj +( 1)j+1
 Pm
k=1 dk
 
= 4mn+1, and the set {Aj : 1  j  2n}
forms a partition of the set {k 2 Z : 1  k  4mn}.
Proposition 4.13. Let m and n be positive odd integers. Then, C2m⇥C2n has no antipodal
balanced C4-face-magic toroidal labeling that is both row-sum balanced and column-sum
balanced.
S. J. Curran, R. M. Low and S. C. Locke: C4-face-magic toroidal labelings on Cm ⇥ Cn 31
Proof. Let {xi,j} be an antipodal balanced C4-face-magic toroidal labeling on C2m⇥C2n
that is both row-sum balanced and column-sum balanced. For all integers j such that
1  j  n, let yj = x1,j and zj = xm+1,j . For all integers i such that 1  i  m, let
di = xi(m 1)+1,in+1   x(i 1)(m 1)+1,(i 1)n+1.
By Proposition 4.12, for all integers i such that 1  i  m, equations (4.49), (4.50), (4.51),
and (4.52) hold.
Let
T =
nX
j=1
✓
yj + zj + ( 1)j+1
  mX
k=1
dk
 ◆
.
By Proposition 4.12, we have yj + zj + ( 1)j+1
 Pm
k=1 dk
 
= 4mn+ 1. Then
T =
nX
j=1
✓
yj + zj + ( 1)j+1
  mX
k=1
dk
 ◆
=
nX
j=1
 
4mn+ 1
 
= n(4mn+ 1) ⌘ 1 (mod 2).
For i = 1, 2, . . . , 2m, let Ri =
P2n
j=1 xi,j be the row sum of the labels on the vertices
in row i. Also, for j = 1, 2, . . . , 2n, let Cj =
P2m
i=1 xi,j be the column sum of the labels
on the vertices in column j. Let m1 be the integer such that m = 2m1 + 1. Then for any
integer j such that 1  j  n, we have
Cj =
mX
i=1
 
xi(m 1)+1,j + xi(m 1)+m+1,j
 
=
m1+1X
i=1
 
x(2i 1)(m 1)+1,j + x(2i 1)(m 1)+m+1,j
 
+
m1X
i=1
 
x(2i)(m 1)+1,j + x(2i)(m 1)+m+1,j
 
=
m1+1X
i=1
✓
yj + ( 1)j+1
  mX
k=2i
dk
 
+ zj + ( 1)j+1
  mX
k=2i
dk
 ◆
+
m1X
i=1
✓
yj + ( 1)j+1
  2iX
k=1
dk
 
+ zj + ( 1)j+1
  2iX
k=1
dk
 ◆
= m(yj + zj) + ( 1)j+1
  mX
k=1
(m+ ( 1)k)dk
 
.
32 Art Discrete Appl. Math. 4 (2021) #P1.04
We also have
Cn+j =
mX
i=1
 
xi(m 1)+1,n+j + xi(m 1)+m+1,n+j
 
=
m1+1X
i=1
 
x(2i 1)(m 1)+1,n+j + x(2i 1)(m 1)+m+1,n+j
 
+
m1X
i=1
 
x(2i)(m 1)+1,n+j + x(2i)(m 1)+m+1,n+j
 
=
m1+1X
i=1
✓
yj + ( 1)j+1
 2i 1X
k=1
dk
 
+ zj + ( 1)j+1
 2i 1X
k=1
dk
 ◆
+
m1X
i=1
✓
yj + ( 1)j+1
  mX
k=2i+1
dk
 
+ zj + ( 1)j+1
  mX
k=2i+1
dk
 ◆
= m(yj + zj) + ( 1)j+1
  mX
k=1
(m+ ( 1)k+1)dk
 
.
Since Cj = Cn+j , we have
Pm
k=1( 1)kdk = 0.
We have
R1 =
nX
j=1
 
x1,j + x1,n+j
 
=
nX
j=1
✓
yj + yj + ( 1)j+1
mX
k=1
dk
◆
= 2
nX
j=1
yj +
mX
k=1
dk.
We also have
Rm+1 =
nX
j=1
 
xm+1,j + xm+1,n+j
 
=
nX
j=1
✓
zj + zj + ( 1)j+1
mX
k=1
dk
◆
= 2
nX
j=1
zj +
mX
k=1
dk.
Since R1 = Rm+1, we have
Pn
j=1 yj =
Pn
j=1 zj . Thus
T =
nX
j=1
✓
yj + zj + ( 1)j+1
  mX
k=1
dk
 ◆
=
nX
j=1
yj +
nX
j=1
zj +
mX
k=1
dk +
mX
k=1
( 1)kdk
⌘ 2
  nX
j=1
yj
 
+
m1X
i=1
2d2i ⌘ 0 (mod 2).
This contradicts T ⌘ 1 (mod 2).
ORCID iDs
Stephen J. Curran https://orcid.org/0000-0002-3523-7601
S. J. Curran, R. M. Low and S. C. Locke: C4-face-magic toroidal labelings on Cm ⇥ Cn 33
Richard M. Low https://orcid.org/0000-0003-2320-8296
Stephen C. Locke https://orcid.org/0000-0002-2957-8071
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10.1016/0016-0032(89)90010-0.
[3] M. Bača, On magic labelings of grid graphs, Ars Combin. 33 (1992), 295–299.
[4] M. Bača, On magic labelings of honeycomb, Discrete Math. 105 (1992), 305–311, doi:10.
1016/0012-365X(92)90153-7.
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Co., Inc., New York, 1976.
[6] S. I. Butt, M. Numan and A. Semaničová-Feňovčı́ková, Face antimagic labelings of toroidal
and klein bottle grid graphs, AKCE Int. J. Graphs Comb. (2020), 1–9, doi:10.1016/j.akcej.2018.
09.005.
[7] S. J. Curran and R. M. Low, C4-face-magic torus labelings on C4 ⇥ C4, Cong. Numer. 233
(2019), 79–94.
[8] J. A. Gallian, A dynamic survey of graph labeling, Electron. J. Combin. 5 (1998), Dynamic Sur-
vey 6, 43, doi:10.37236/27, https://www.combinatorics.org/ojs/index.php/
eljc/issue/view/Surveys.
[9] J. Hsieh, S.-M. Lee, P.-J. Liang and R. M. Low, On C4 face-(1, 0, 0) magic polyomino graphs
and their related graphs, manuscript.
[10] K. Kathiresan and S. Gokulakrishnan, On magic labelings of type (1, 1, 1) for the special
classes of plane graphs, Util. Math. 63 (2003), 25–32.
[11] A. Kotzig and A. Rosa, Magic valuations of finite graphs, Canad. Math. Bull. 13 (1970), 451–
461, doi:10.4153/CMB-1970-084-1.
[12] K.-W. Lih, On magic and consecutive labelings of plane graphs, Utilitas Math. 24 (1983),
165–197.
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ISSN 2590-9770
The Art of Discrete and Applied Mathematics 4 (2021) #P1.05
https://doi.org/10.26493/2590-9770.1280.4da
(Also available at http://adam-journal.eu)
An alternate proof of the monotonicity of the
number of positive entries in nonnegative matrix
powers*
Slobodan Filipovski†
FAMNIT, University of Primorska, Glagoljaška 8, Koper, Slovenia
Comenius University, Mlynská dolina, 842 48, Bratislava, Slovakia
Received 6 December 2018, accepted 19 July 2020, published online 23. January 2021
Abstract
Let A be a nonnegative real matrix of order n and f(A) denote the number of positive
entries in A. In 2018, Xie proved that if f(A)  3 or f(A)   n2   2n + 2, then the
sequence (f(Ak))1k=1 is monotone for positive integers k. In this note we give an alternate
proof of this result by counting walks in a digraph of order n.
Keywords: Digraphs, walks, monotonicity, adjacency matrix.
Math. Subj. Class. (2020): 05C20, 05C81, 15B48
1 Introduction
A matrix is nonnegative (respectively, positive) if all its entries are nonnegative (re-
spectively, positive) real numbers. Nonnegative matrices are widely applied in science,
engineering and technology, see [1] and [2]. A nonnegative square matrix A is said to be
primitive if there exists a positive integer k such that Ak is positive. By f(A) we denote
the number of positive entries in A. In [4] Šidák proved that there exists a primitive matrix
A of order 9 satisfying f(A) = 18 > f(A2) = 16. Motivated by this observation, in [5]
Xi proved that if f(A)  3 or f(A)   n2   2n + 2, then the sequence (f(Ak))1k=1 is
monotone for positive integers k. The proof of this result relies on linear algebra approach
considering A as a 0   1 square matrix, that is, a matrix from the vector space Mn(R)
whose entries are either 0 or 1. Recall, Mn(R) is the set of all square matrices of size n
under the ordinary addition and scalar multiplication of matrices. Clearly, the above re-
striction on the entries of A is valid since the value of each positive entry in A does not
*We would like to thank the anonymous referee for helpful comments which have improved the paper.
†
Supported by VEGA 1/0423/20.
E-mail address: slobodan.filipovski@famnit.upr.si (Slobodan Filipovski)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Art Discrete Appl. Math. 4 (2021) #P1.05
effect f(Ak) for all positive integers k. In this note we give an alternate proof of this result
using counting method from graph theory.
By a digraph we mean a structure G = (V,A), where V (G) is a finite set of vertices,
and A(G) is a set of ordered pairs (u, v) of vertices u, v 2 V (G) called arcs. The order of
the digraph G is the number of vertices in G. An in-neighbour of a vertex v in a digraph G
is a vertex u such that (u, v) 2 A(G). Similarly, an out-neighbour of a vertex v is a vertex
w such that (v, w) 2 A(G). The in-degree, respectively out-degree, of a vertex v 2 V (G)
is the number of its in-neighbours, respectively out-neighbours, in G. A walk w of length
k in G is an alternating sequence (v0a1v1a2 . . . akvk) of vertices and arcs in G such that
ai = (vi 1, vi) for each i. If the arcs a1, a2, . . . , ak of a walk w are distinct, w is called
a trail. A cycle Ck of length k is a closed trial of length k > 0 with all vertices distinct
(except the first and the last).
If a digraph G has n vertices v1, v2, . . . , vn, a useful way to represent it is with an n⇥n
matrix of zeros and ones called its adjacency matrix, AG. The ij-th entry of the adjacency
matrix, (AG)ij , is 1 if there is an arc from vertex vi to vertex vj and 0 otherwise. That is,
(AG)ij =
⇢
1, if (vi, vj) 2 A(G)
0, otherwise
The length-k walk counting matrix for an n-vertex digraph G is the n ⇥ n matrix C such
that
Cuv := the number of length-k walks from u to v.
The main result in this note is based on the following well-known result:
Theorem 1.1 ([3]). The length-k counting matrix of a digraph, G, is (AG)k, for all k 2 N.
2 Main results
In the following proposition we reprove Theorem 1 and Theorem 2 from [5].
Proposition 2.1. Let A be a 0   1 matrix of order n. If f(A)  3, then the sequence
(f(Ak))1k=1 is monotone.
Proof. Let G be a digraph on n vertices v1, v2, . . . , vn corresponding to the adjacency
matrix A, that is, there is an arc from vertex vi to vertex vj in G (vi ! vj) if (A)ij = 1.
We deal with four possible cases.
1. The case when f(A) = 0 is trivial. Since Ak = On, then f(Ak) = 0 for any positive
integer k.
2. If f(A) = 1, then G contains exactly one arc a = (vi, vj).
• If vi = vj , then for any positive integer k there exists a unique k-walk from
vi to vi. Therefore (Ak)ii = 1. Moreover, since there exists no other k-walk
between the vertices of G, the remaining n
2   1 entries of Ak are zeros. In this
case, for any positive integer k we have f(Ak) = 1.
• If vi 6= vj , then (A)ij = 1. It is easy to see that G does not contain a walk of
length k   2, that is, for any k   2 Ak is a zero matrix. Therefore, for any
k   2 we obtain 1 = f(A) > f(Ak) = 0.
S. Filipovski: An alternate proof of the monotonicity of the number of positive entries 3
3. Let f(A) = 2, i.e., let a1 = (vi, vj) and a2 = (vr, vs) be two distinct arcs of G.
If G contains two loops, then we consider one possible case:
• Let vi = vj 6= vr = vs. For any positive integer k   1 there exists exactly
one k-walk from vertex vi to vertex vj and exactly one k-walk from vertex vr
to vertex vs. It yields f(Ak) = 2.
If G contains one loop, we consider the following three cases:
• If vi = vj = vr 6= vs, then f(Ak) = 2 for any positive integer k   1.
• If vi = vj = vs 6= vr, then f(Ak) = 2 for any positive integer k   1.
• If vi = vj , vr 6= vs, vi 6= vr and vi 6= vs, then f(Ak) = 1 for any positive
integer k   2.
If G does not contain loops, then we focus on the cases when at least one of the
vertices vi, vj , vr and vs has positive in-degree and positive out-degree. Otherwise,
G does not contain a k-walk for k   2.
• If vi 6= vj = vr 6= vs and vi 6= vs, then G contains exactly one 2-walk from
vi to vs. Moreover, there is no k-walk when k   3. Thus 2 = f(A) > 1 =
f(A2) > f(Ak) = 0 for any positive integer k   3.
• If vi 6= vj = vr 6= vs and vi = vs, then f(Ak) = 2 for any positive integer k.
4. The proof when f(A) = 3 follows the same reasoning as the previous cases.
Let a1 = (vi, vj), a2 = (vr, vs) and a3 = (vp, vt) be three distinct arcs of G.
If G contains three loops, then we have:
• Let vi = vj , vr = vs and vp = vt. It is easy to see that f(Ak) = 3 for any
positive integer k   1.
Similarly, if G contains two loops, we treat the following cases.
• If vi = vj , vr = vs, vp 6= vt and if there is no common vertex between the arcs
a1, a2 and a3, then f(Ak) = 2 for any positive integer k   2.
• If vi = vj = vp 6= vt = vr = vs, then f(Ak) = 3 for any positive integer
k   1.
• If vi = vj = vp 6= vt 6= vr = vs, then f(Ak) = 3 for any positive integer
k   1.
• If vi = vj = vt 6= vp 6= vr = vs, then f(Ak) = 3 for any positive integer
k   1.
If G contains one loop, we obtain the following cases.
• If vi = vj , vr = vt 6= vs = vp and vi 6= vr, vi 6= vs, then f(Ak) = 3 for any
positive integer k   1.
• If vi = vj , vr 6= vs, vp 6= vt and if there is no a common vertex between the
arcs a1, a2 and a3, then f(Ak) = 1 for any positive integer k   2.
• If vi = vj , vr 6= vs = vp 6= vt, vr 6= vt and if there is no common vertex
between a1 and a2 and a1 and a3, then f(A2) = 2 and f(Ak) = 1 for any
positive integer k   3.
4 Art Discrete Appl. Math. 4 (2021) #P1.05
• If vi = vj 6= vr = vp 6= vt, vr 6= vs, vs 6= vt, vi 6= vs and vi 6= vt, then
f(Ak) = 1 for any positive integer k   2.
• If vi = vj , vr 6= vs = vt 6= vp, vr 6= vp and if there is no common vertex
between a1 and a2 and a1 and a3, then f(Ak) = 1 for any positive integer
k   2.
• If vi = vj = vr 6= vs, vp 6= vt and if there is no common vertex between a1
and a3 and between a2 and a3 , then f(Ak) = 2 for any positive integer k   2.
• If vi = vj = vs 6= vr, vp 6= vt and if there is no common vertex between a1
and a3 and between a2 and a3 , then f(Ak) = 2 for any positive integer k   2.
• If vi = vj = vr 6= vs = vp 6= vt and vi 6= vt, then f(Ak) = 3 for any positive
integer k   1.
• If vi = vj = vs 6= vr = vp 6= vt and vi 6= vt, then f(Ak) = 2 for any positive
integer k   2.
• If vi = vj = vs 6= vr = vt 6= vp and vi 6= vp, then f(Ak) = 3 for any positive
integer k   1.
• If vi = vj = vr 6= vs = vt 6= vp and vi 6= vp, then f(Ak) = 2 for any positive
integer k   2.
• If vi 6= vj = vr = vs = vp 6= vt and vi 6= vt, then f(Ak) = 3 for any positive
integer k   1.
• If vi 6= vj = vr = vs = vt 6= vp, then f(Ak) = 3 for any positive integer
k   1.
• If vj 6= vi = vr = vs = vp 6= vt, then f(Ak) = 3 for any positive integer
k   1.
• If vi 6= vj = vr = vs = vp 6= vt and vi = vt, then f(Ak) = 4 for any positive
integer k   2.
If G does not contain loops, then each k-walk of G, k   3, contains at least two
vertices of positive in-degree and positive out-degree. Based on this observation we
consider the following cases.
• If vi = vs 6= vj = vr, vp 6= vt and if there is no common vertex between the
arcs a1 and a3, then f(Ak) = 2 for any positive integers k   2.
• If vi 6= vj , vr 6= vs, vp 6= vt, vj = vr, vs = vp and vt = vi, then f(Ak) = 3
for any positive integer k   1.
• If vi 6= vj , vr 6= vs, vp 6= vt, vj = vr, vs = vp and vi 6= vt 6= vj , then
f(A2) = 2, f(A3) = 1 and f(Ak) = 0 for any positive integer k   4.
• If vt 6= vp = vs = vi 6= vj = vr and vj 6= vt, then f(Ak) = 3 for any positive
integer k   1.
• If vp 6= vt = vs = vi 6= vj = vr and vj 6= vp, then f(Ak) = 3 for any positive
integer k   1.
The following result is a reproof of Theorem 5 from [5].
Theorem 2.2. Let A be a 0 1 matrix of order n. If f(A)   n2 2n+2, then the sequence
(f(Ak))1k=1 is non-decreasing.
S. Filipovski: An alternate proof of the monotonicity of the number of positive entries 5
Proof. Let G be a digraph on n vertices v1, v2, . . . , vn which corresponds to the matrix A
(A is the adjacency matrix of G consisting of at most 2n 2 zeros). According to Theorem
1.1, proving f(Ak+1)   f(Ak) for every positive integer k, is equivalent to proving that
the number of pairs of vertices of G for which there exists at least one (k + 1)-walk is
greater or equal than the number of pairs of vertices of G for which there exists at least one
k-walk.
Let us suppose that G contains a walk of length k, i.e. let w = (vi, vi+1, . . . , vj) be a
k-walk from vi to vj = vi+k. Thus (Ak)ij   1. We prove the following five claims.
Claim 1: If w contains at least four distinct vertices, then there exists at least one (k+ 1)-
walk from vi to vj . Therefore (Ak+1)ij   1.
Let w = (vi, vi+1, . . . , vj) contain at least four distinct vertices vi, vt, vs and vj . If
w contains a loop, then G contains at least one (k + 1)-walk from vi to vj . There-
fore we assume that (A)ii = (A)tt = (A)ss = (A)jj = 0. Thus vi 6= vi+1 and
vi+1 6= vi+2. If there exists no (k + 1)-walk from vi to vj , then for each vertex
v 2 V (G)\{vi, vi+1}, G does not contain 2-walks of type (vi, v, vi+1). Other-
wise we obtain (k + 1)-walk (vi, v, vi+1, vi+2, . . . , vj). This implies an existence
of at least n  2 non-connected pairs of vertices among (vi, v) and (v, vi+1), where
v 2 V (G)\{vi, vi+1}. Similarly, for each vertex v 2 V (G)\{vi+1, vi+2}, G does
not contain 2-walks of type (vi+1, v, vi+2). Otherwise we obtain (k + 1)-walk
(vi, vi+1, v, vi+2, . . . , vj).This implies an existence of at least n   3 non-connected
pairs of vertices among (vi+1, v) and (v, vi+2), where v 2 V (G)\{vi, vi+1, vi+2}.
Since G does not contain at least four loops, we obtain at least (n 2)+(n 3)+4 =
2n  1 non-connected pairs of vertices in G, which is not possible.
Claim 2: If k   3 and w contains three distinct vertices, then there exists at least one
(k + 1)-walk from vi to vj . Therefore (Ak+1)ij   1.
We proceed similarly as in the previous case. Let w = (vi, vi+1, . . . , vj) contain
three distinct vertices vi, vt and vj . If w contains a loop, then there exists at least
one (k + 1)-walk from vi to vj . Therefore we suppose (A)ii = (A)tt = (A)jj = 0.
Clearly vi+1 6= vi and vt 6= vt+1. Without loss of generality let vi+1 = vt. If G does
not contain a (k + 1)-walk from vi to vj , then for each v 2 V (G)\{vi, vt, vj} there
exist no walks of type (vi, v, vi+1) and (vt, v, vt+1). Otherwise we obtain the walks
(vi, v, vi+1, . . . , vj) and (vi, vi+1, . . . , vt, v, vt+1, . . . , vj), both of length k+1. The
non-existence of the walks (vi, v, vi+1) and (vt, v, vt+1) implies an existence of at
least 2(n   3) non-connected pairs of vertices among the pairs (vi, v), (v, vi+1 =
vt), (vt, v) and (v, vt+1).
Let vi+2 = vi. We suppose that the walks (vi, vj , vt) and (vt, vj , vi) do not exist.
Otherwise we obtain (k + 1)-walks from vi to vj (vi, vj , vi+1, vi+2, . . . , vj) and
(vi, vi+1, vj , vi+2, . . . , vj), respectively. This yields an existence of at least two non-
connected pairs among the pairs (vi, vj), (vj , vt), (vt, vj) and (vj , vi). In this case G
contains at least 2n  1 = 3 + 2(n  3) + 2 non-connected pairs of vertices, which
is not possible.
Let vi+2 = vj . Similarly as in the previous case, we conclude that there exists no
a walk (vi, vj , vt). Otherwise we obtain the walk (vi, vj , vi+1, vi+2, . . . , vj). This
yields an existence of at least one non-connected pair among the pairs (vi, vj) and
(vj , vt). In this case G contains at least 2n   2 non-connected pairs of vertices.
6 Art Discrete Appl. Math. 4 (2021) #P1.05
Since A contains at most 2n   2 zeros, we obtain that vt and vj are connected to
vi. For any even k   4 we obtain a k-walk (vi, vt, vi, vt, . . . , vi, vt, vj). Similarly,
if k = 5 we obtain the walk (vi, vt, vj , vi, vt, vj). If k   7 is an odd number, then
k = 2s + 1 = (2s   4) + 5 where s   3. In this case we obtain a k-walk from vi
to vj by connecting the walk (vi, vt, vi, vt, . . . , vt, vi) of length 2s  4 and the walk
(vi, vt, vj , vi, vt, vj) of length 5.
Claim 3: If k = 2 and w = (vi, vt, vj), then (A3)ij   1 or the number of positive entries
of A
3
at (i, i), (i, t), (i, j), (t, i), (t, t), (t, j), (j, i), (j, t) and (j, j) position is greater
or equal than the number of positive entries of the matrix A
2
at the same positions.
Let G does not contain 3-walk from vi to vj and let v 2 V (G)\{vi, vt, vj}. If G con-
tains walks of type (vi, v, vt) and (vt, v, vj), then there exist 3-walks (vi, v, vt, vj)
and (vi, vt, v, vj). In this case (A3)ij   1.
On the other hand, the non-existence of the walks (vi, v, vt) and (vt, v, vj) implies
an existence of at least 2(n  3) non-connected pairs among the pairs (vi, v), (v, vt),
(vt, v) and (v, vj). Now, if vi is connected to vj , then vj is not connected to vi and
vt. Otherwise we obtain the walks (vi, vj , vi, vj) and (vi, vj , vt, vj). Since (A)ji =
(A)jt = 0 the matrix A contains at least 3+2(n 3)+2 = 2n 1 zeros. This is not
possible. If vi is not connected to vj , then A contains at least 2n 2 zeros. Therefore
vj is connected to vi and vt, and vt is connected to vi. By counting 2-walks between
the vertices vi, vt and vj , we find that the matrix A
2
consists of seven positive entries
and two zeros at (i, i), (i, t), (i, j), (t, i), (t, t), (t, j), (j, i), (j, t) and (j, j) position.
On the other hand, by counting the 3-walks between the vertices vi, vt and vj we
conclude that A
3
consists eight positive entries and one zero at the same positions.
Claim 4: Let w = (vi, vi+1, . . . , vj) contain two distinct vertices vi and vj . The number
of positive entries of A
k+1
at (i, i), (i, j), (j, i) and (j, j) position is greater or equal
than the number of positive entries of the matrix A
k
at the same positions.
Let k   2. If the walk w contains a loop, then it is easy to conclude that G contains
a (k + 1)-walk from vi to vj . In this case (Ak)ij   1 implies (Ak+1)ij   1.
If w does not contain loops, then k is an odd number. We observe that G contains a k-
walk from vertex vj to vertex vi, which implies (Ak)ji   1. If there exists no k-walk
from vi to vi and if there exists no k-walk from vj to vj , then (Ak)ii = (Ak)jj = 0.
Since k + 1 is an even number, G contains (k + 1)-walks from vi to vi and from vj
to vj , that is, (Ak+1)ii   1 and (Ak+1)jj   1. Moreover, the digraph G does not
contain (k+1)-walk from vertex vi to vertex vj and from vertex vj to vertex vi, that
is, (Ak+1)ij = (Ak+1)ji = 0. Thus, the matrices Ak and Ak+1 contain two zeros
and two positive entries at (i, i), (i, j), (j, i) and (j, j) position.
Similarly, (Ak)ii   1 implies (Ak+1)ij   1 and (Ak)jj   1 implies (Ak+1)ji   1.
Let k = 1. If vj is connected to vi, we have the same case as k   2. If vj is not
connected to vi, then there exists at least one 2-walk from vj to vi or from vi to vj .
Otherwise we have at least 2n   1 non-connected pairs of vertices in G, that is, at
least 2n  1 zeros in A, a contradiction.
Claim 5: If w contains exactly one vertex vi, then there exists a (k + 1)-walk from vi to
vi. Therefore (Ak+1)ii   1.
In this case the walk w is obtained repeating the loop vi ! vi k-times. Thus, there
exists a (k + 1)-walk from vi to vi.
S. Filipovski: An alternate proof of the monotonicity of the number of positive entries 7
As a conclusion, in the four cases (whether the k-walk from vertex vi to vertex vj
contains one, two, three or more distinct vertices), we obtain that the number of
positive entries in A
k+1
is greater or equal than the number of positive entries in A
k
,
that is, f(Ak+1)   f(Ak).
ORCID iDs
Slobodan Filipovski https://orcid.org/0000-0002-7286-4954
References
[1] R. B. Bapat and T. E. S. Raghavan, Nonnegative matrices and applications, volume 64 of En-
cyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1997,
doi:10.1017/CBO9780511529979.
[2] A. Berman and R. J. Plemmons, Nonnegative matrices in the mathematical sciences, volume 9
of Classics in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM),
Philadelphia, PA, 1994, doi:10.1137/1.9781611971262, revised reprint of the 1979 original.
[3] E. Lehman, F. T. Leighton and A. R. Meyer, Mathematics for Computer Science, 12th Media
Services, 2017.
[4] Z. Šidák, On the number of positive elements in powers of a non-negative matrix, C̆asopis Pěst.
Mat. 89 (1964), 28–30.
[5] Q. Xie, Monotonicity of the number of positive entries in nonnegative matrix powers, J. Inequal.
Appl. (2018), Paper No. 255, 5, doi:10.1186/s13660-018-1833-5.
ISSN 2590-9770
The Art of Discrete and Applied Mathematics 4 (2021) #P1.06
https://doi.org/10.26493/2590-9770.1376.a29
(Also available at http://adam-journal.eu)
Semi-perimeter and inner site-perimeter of
k-ary words and bargraphs
Toufik Mansour
Department of Mathematics, University of Haifa, 3498838 Haifa, Israel
Received 31 January 2020, accepted 26 July 2020, published online 24 January 2021
Abstract
Given a bargraph B, a border cell of B is a cell of B that shares at least one common
edge with an outside cell of B. Clearly, the inner site-perimeter of B is the number of
border cells of B. A tangent cell of B is a cell of B which is not a border cell of B and
shares at least one vertex with an outside cell of B. In this paper, we study the generating
function for the number of k-ary words, represented as bargraphs, according to the number
of horizontal steps, up steps, border cells and tangent cells. This allows us to express
some cases via Chebyshev polynomials of the second kind. Moreover, we find an explicit
formula for the number of bargraphs according to the number of horizontal steps, up steps,
and tangent cells/inner site-perimeter. We also derive asymptotic estimates for the mean
number of tangent cells/inner site-perimeter.
Keywords: Bargraphs, Chebyshev polynomials, k-ary words, semi-perimeter, inner site-perimeter.
Math. Subj. Class.: 05A15, 05A16, 60C05
1 Introduction
The solid–on–solid (SOS) model has received a lot of attention. The SOS model arose from
the consideration of the boundary between oppositely magnetized phases in the Ising model
[9, 23]. The linear SOS model with a magnetic field and wall interaction was solved in [19].
Later, in [20], the restricted SOS (RSOS) has been considered, where the interface takes on
a restricted subset of configurations and the interactions of field and single wall interaction
in the half-plane. Then, in [21] is presented the solution of the linear RSOS model confined
to a slit. Each configuration of the RSOS presented by a k-bounded bargraph (see below)
with allowing horizontal steps in the x-axis, where the exact solution in [21] is presented
by studying the generating function for the number of k-bounded bargraphs according to
the semi-perimeter.
E-mail address: tmansour@univ.haifa.ac.il (Toufik Mansour)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Art Discrete Appl. Math. 4 (2021) #P1.06
A bargraph is a column-convex polyomino where all the columns are bottom justified.
Throughout this paper, we represent a bargraph as a lattice path in Z2, starting at the origin
and after ending upon first return to the x-axis, the progression ends at the origin (0, 0).
The allowed steps are the up step u = (0, 1), the down step d = (0, 1), the horizontal
step h = (1, 0), and the left horizontal step ( 1, 0). The first step has to be an up step,
the horizontal steps must all lie above the x-axis, the left horizontal steps must all lie
on x-axis, and an up step cannot follow a down step and vice versa. Alternatively, we
identify a bargraph by the 1 ⇥ 1 squares (called cells) that lie inside its lattice path. For
instance, Figure 1 represents the bargraph uhuhhuhuhdhhddd (as lattice path) and the
word  1 2 · · · 7 = 1223433 where the ith column contains exactly  i cells. For a given
bargraph B, we define the semi-perimeter of B to be the number of up steps and horizontal
steps in the lattice path, the site-perimeter of B to be the number of nearest-neighbour
cells outside the boundary of B, and the inner site-perimeter to be the number of cells
inside B that have at least one edge in common with an outside cell. Figure 1 represents a
s
s
s s
s
s
s s
s
s
s
sssssss
bbbbbbb
b b t
b t
b
b b
b
Figure 1: The bargraph of the word 1223433
corresponding bargraph B of the word 1223433 with semi-perimeter 11, site-perimeter 18
(the sum of the cells marked by s), and inner site-perimeter 10 (the sum of the cells marked
by b).
In the last decades, the enumeration of bargraphs according to statistics has received
a lot of attention (following the interest in Statistical physics as we said at the beginning
of the introduction). Earlier authors, such as Prellberg and Brak [22] and Feretić [10] (see
also [6, 12]), have found that the generating function that counts all bargraphs (including
the empty bargraph) is given by
B(x, y) =
1 + x  y   xy  
p
(1  x  y   xy)2   4x2y
2x
, (1.1)
where x counts number of horizontal steps and y counts the number of up steps. Note
that the generating function for bargraphs according to the semi-perimeter, often called the
isotropic generating function, is given by B(x, x). To find the asymptotics of the coefficient
of xn in B(x, x), one computes the dominant singularity ⇢ which is the positive root of
1  4x+ 2x2 + x4 = 0. Thus, by singularity analysis (for example, see [11]) we have
[xn]B(x, x) ⇠ 1
2
s
1  ⇢  ⇢3
⇡⇢n3
⇢ n (1.2)
with
⇢ =
1
3
✓
 1  2
8/3
(13 + 3
p
3)1/3
+ 21/3(13 + 3
p
33)1/3
◆
⇡ 0.295598 · · · . (1.3)
T. Mansour: Semi-perimeter and inner site-perimeter of k-ary words and bargraphs 3
Later, Blecher et al. refined the generating function B(x, y) by considering a third
statistic such as: levels [1], peaks [3], area [2], descents [2] and height [5]. In [6] is studied
the number of bargraphs according to the site-perimeter. In [14] is considered the number
of bargraphs according to the inner site-perimeter. In [4] it is derived a functional equation
for the generating function that counts the number of k-ary words according to the size
of the rightmost letter, the number of letters and the site-perimeter. For more details and
motivations related to statistical physics and enumerative combinatorics, we refer the reader
to [15] and references therein. The aim of this paper, is to refine some of these results.
For a given bargraph B, we define a border cell of B as a cell of B that has at least
one edge in common with an outside cell of B. Clearly, the inner site-perimeter of B is the
number of border cells of B. Further, we define a tangent cell of B to be a cell of B which
is not a border cell of B and that has at least one vertex in common with an outside cell of
B (see Figure 1). In what follows throughout this paper, whenever we refer to k-ary words,
we mean their corresponding bargraph representation.
The paper aims to study the generating function for the number of k-ary words (bounded
bargraphs that lie below the line y = k), according to the number of horizontal steps, up
steps, tangent cells and border cells. More precisely: (1) We find an explicit formula for
the generating function for the number of bargraphs according to the number of horizon-
tal steps, the number of up steps, inner site-perimeter, and the number of tangent cells.
In particular, we present the average number of tangent cells/inner site-perimeter as the
semi-perimeter n of the bargraph tends to infinity. (2) We find an explicit formula for the
generating function for the number of k-ary words according to the number of horizontal
steps and up steps in terms of Chebyshev polynomials of the second kind, then we rederive
(1.1). (3) We find an explicit formula for the generating function for the number of k-ary
words according to the number of horizontal steps, up steps and tangent cells in terms of
Chebyshev polynomials of the second kind. (4) We also study the generating function for
the number of k-ary words according to the number of horizontal steps, up steps and inner
site-perimeter.
2 Results
Define [x]d = 1 x
d
1 x = 1 + x + · · · + x
d 1 for all d   0. Let Ck = Ck(x, y, p, q) be
the generating function for the number of k-ary words of length n according to the number
of horizontal steps (marked by x), up steps (marked by y), border cells (marked by p) and
tangent cells (marked by q). We decompose each k-ary word ⇡ as
⇡ = ⇡(0)1⇡(1) · · · 1⇡(s), s   0
where ⇡(j) is a word over alphabet {2, 3, . . . , k}, for all j = 0, 1, . . . , s. Then
Ck = Dk +
X
s 1
(xp)s(y +Dk   1)(1 + (Dk   1)/y)s
= Dk +
xp(y +Dk   1)(1 + (Dk   1)/y)
1  px(1 + (Dk   1)/y)
, (2.1)
where Dk = Dk(x, y, p, q) is the generating function for the number of words of length n
over alphabet {2, 3, . . . , k}, according to the number of horizontal steps, up steps, border
cells and tangent cells.
4 Art Discrete Appl. Math. 4 (2021) #P1.06
Next we write an equation for the generating function Dk with k   2. Clearly, D2 =
1+ xy
2p2
1 xp2 . In order to write a recurrence relation for Dk, we decompose each word ⇡ over
alphabet {2, 3, . . . , k} as
⇡ = ⇡(0)2⇡(1) · · · 2⇡(s), s   0
where ⇡(j) is a word over alphabet {3, 4, . . . , k}, for all j = 0, 1, . . . , s.
The case s = 0 contributes
F0 = 1 + x(yp)
3[yp]k 2 + yp
2(Dk 1   1  x(yp)2[yp]k 2),
where first, second and third term counts the empty word, words with one letter, words with
at least two letters, respectively.
Similarly, the case s = 1 contributes
F1 = xp
2(y2 + x(yp)3[yp]k 2 + ypq(Dk 1   1  x(yp)2[yp]k 2))·
(1 + xyp3[yp]k 2 + pq/y(Dk 1   1  x(yp)2[yp]k 2)),
where xp2(y2 + x(yp)3[yp]k 2 + ypq(Dk 1   1   x(yp)2[yp]k 2)) counts the words of
the form ⇡(0)2, and 1 + xyp3[yp]k 2 + pq/y(Dk 1   1   x(yp)2[yp]k 2) counts either
the empty word or the nonempty words of the form ⇡(1) without first two up steps.
For s   2, we have the contribution
F1(xp
2)s 1(1 + xyp2q[yp]k 2 + pq
2/y(Dk 1   1  x(yp)2[yp]k 2))s 1.
Therefore, by summing all the contributions, we obtain the following result.
Lemma 2.1. The generating function Dk = Dk(x, y, p, q) satisfies
Dk = 1 + x(yp)
3[yp]k 2 + yp
2(Dk 1   1  x(yp)2[yp]k 2)
+ xp2(y2 + x(yp)3[yp]k 2 + ypq(Dk 1   1  x(yp)2[yp]k 2))·
· (1 + xyp3[yp]k 2 +
pq
y
(Dk 1   1  x(yp)2[yp]k 2))·
·
✓
1  xp2(1 + xyp2q[yp]k 2 +
pq2
y
(Dk 1   1  x(yp)2[yp]k 2))
◆ 1
with D2 = 1 +
xy2p2
1 xp2 .
Hence, by (2.1), we have the following formula for the generating function Ck(x, p, q).
Theorem 2.2. Let k   1. Then
Ck(x, y, p, q) = Dk(x, y, p, q)+
xp(y +Dk(x, y, p, q)  1)(1 + (Dk(x, y, p, q)  1)/y)
1  px(1 + (Dk(x, y, p, q)  1)/y)
,
where Dk(x, y, p, q) is given in Lemma 2.1.
2.1 Bargraphs
Clearly, C(x, y, p, q) = limk!1 Ck(x, y, p, q) is the generating function for the number
of bargraphs according to the number of horizontal steps, number of up steps, inner site-
perimeter, and the number of tangent cells. Similarly, D(x, y, p, q) = limk!1 Dk(x, y, p, q)
T. Mansour: Semi-perimeter and inner site-perimeter of k-ary words and bargraphs 5
is the generating function for the number of bargraphs such that each nonempty column
contains at least two cells according to the number of horizontal steps, number of up steps,
inner site-perimeter, and number of tangent cells. By taking k ! 1, Lemma 2.1 gives
D(x, y, p, q)  1 =  ↵1 +
p
↵21   4↵0↵2
2↵2
(2.2)
= p2xy2 + p3xy3 + p4x2y2 + p4xy4 + 2p5x2y3 + p5xy5 + · · · ,
where
↵0 =  p2y3(p2y   1)(py   1)x+ p5y4(py   1)(p  2q + 1)x2
+ p7y5(p2q2   2pq2 + pq   p+ q)x3,
↵1 =  y(py   1)2(p2y   1) + p2y(py   1)2(p2y   2pqy   1)x
+ p4qy2(py   1)(2p3qy   3p2qy + p2y   pq + 1)x2,
↵2 =  p3q2x(1 + py   p2y)(py   1)2.
On the other hand, by taking k ! 1, Theorem 2.2 and (2.2) imply the following result.
Theorem 2.3. The generating function for the number of bargraphs according to the num-
ber of horizontal steps, number of up steps, inner site-perimeter, and the number of tangent
cells is given by
C(x, y, p, q)
= 1 +
 ↵1 +
p
↵21   4↵0↵2
2↵2
+
xp(2y↵2   ↵1 +
p
↵21   4↵0↵2)2
2↵2(2y↵2   px(2y↵2   ↵1 +
p
↵21   4↵0↵2))
= 1 + pxy + p2xy2 + p2x2y + p3xy3 + 2p3x2y2 + p3x3y + · · · .
We illustrate the above theorem through following 2 examples.
Example 2.4. By Theorem 2.3 we have that the generating function C(1, 1, 1, p) for the
number of bargraphs according to inner site-perimeter is given by
2p6   2p5 + 2p4   2p2 + 1 
p
4p11   4p9 + 4p5 + 4p4   4p3   4p2 + 1
p( 2p6 + 2p5   4p4 + 2p3 + 4p2   1 +
p
4p11   4p9 + 4p5 + 4p4   4p3   4p2 + 1)
.
In the case of counting bargraphs according to site-perimeter, we refer the reader to [6].
Next we define a strong inner site-perimeter to be the number of border cells and the
number of tangent cells.
Example 2.5. By Theorem 2.3 we have that the generating function C(1, 1, p, p) for the
number of bargraphs according to strong inner site-perimeter is given by
2p8   3p7 + p6 + p5   p4   2p2 + 1 
p
 )
p( 2p8 + 3p7   3p6 + p5 + 3p4 + 2p2   1 +
p
 )
,
where   = p14   2p13 + 3p12   5p10 + 6p9 + p8   2p7 + 2p6   2p5 + 2p4   4p2 + 1.
6 Art Discrete Appl. Math. 4 (2021) #P1.06
In particular, the generating function for the bargraphs according to semi-perimeter and
the number of tangent cells is given by C(x, x, 1, q). Differentiating C(x, x, 1, q) with
respect to q and evaluating it at q = 1 gives
@
@q
C(x, x, 1, q) |q=1 =
x8   x7 + 7x6   10x5 + 20x4   25x3 + 24x2   12x+ 2
2x2
p
x4 + 2x2   4x+ 1
  x
7   2x6 + 7x5   13x4 + 19x3   18x2 + 10x  2
2x2(x  1) .
In what follows ⇢ is as defined by equation (1.3). By direct calculations and (1.2), we
have
lim
x 7!⇢
@
@q
C(x, x, 1, q) |q=1 (1  x/⇢)1/2 = 1 
5
2
⇢  3
2
⇢2.
Hence, we have the following result.
Corollary 2.6. The average number of tangent cells is asymptotic to
(2  5⇢  3⇢3)
p
1  ⇢  ⇢3
2
p
⇢
n
as the semi-perimeter n of the bargraph tends to infinity.
Moreover, by differentiating C(x, x, p, 1) with respect to p, evaluating at p = 1 and
using (1.2) gives
lim
x 7!⇢
@
@
C(x, x, p, 1) |p=1 (1  x/⇢)1/2 =
25  15⇢  25⇢2   21⇢3
16
.
Hence, we have the following result.
Corollary 2.7. The average inner site-perimeter is asymptotic to
(25  15⇢  25⇢2   21⇢3)
p
1  ⇢  ⇢3
16
p
⇢
n
as the semi-perimeter n of the bargraph tends to infinity.
2.2 Semi-perimeter and k-ary words
Define Bk(x, y) = Ck(x, y, 1, 1) and Ek(x, y) = Dk(x, y, 1, 1)   1, for all k   1. Then
Theorem 2.2 with p = q = 1 gives
Bk(x, y) =
y(1  x+ xy) + (xy   x+ y)Ek(x, y)
y(1  x)  xEk(x, y)
, (2.3)
where
Ek(x, y) =
xy3 + (1 + x)y2Ek 1(x, y)
y(1  x)  xEk 1(x, y)
.
T. Mansour: Semi-perimeter and inner site-perimeter of k-ary words and bargraphs 7
Recall that the Chebyshev polynomials of the second kind Um(t) satisfy the recurrence
relation Um(t) = 2tUm 1(t) Um 2(t) with the initial conditions U0(t) = 1 and U1(t) =
2t. By induction ok k, we have
Ek(x, y) =
xy
p
yUk 2
⇣
1 x+y+xy
2
p
y
⌘
Uk 1
⇣
1 x+y+xy
2
p
y
⌘
  (1 + x)pyUk 2
⇣
1 x+y+xy
2
p
y
⌘ , (2.4)
where Um(t) is the m-th Chebyshev polynomials of the second kind. Substituting into
(2.3) gives the following result.
Theorem 2.8. The generating function for the number of k-ary words, k   2, according
to the number of horizontal steps and up steps is given by
Bk(x, y) =
(1  x+ xy)Uk 1
⇣
1 x+y+xy
2
p
y
⌘
 pyUk 2
⇣
1 x+y+xy
2
p
y
⌘
(1  x)Uk 1
⇣
1 x+y+xy
2
p
y
⌘
 pyUk 2
⇣
1 x+y+xy
2
p
y
⌘ .
Note that limk 7!1
Uk 1(
1
2
p
t
)
p
tUk(
1
2
p
t
)
= C(t), where C(t) = 1 
p
1 4t
2t (for example, see [17]
and references therein). Thus, Theorem 2.8 shows that
lim
k 7!1
Bk(x, y) =
1  x+ xy  py limk 7!1
Uk 2
⇣
1 x+y+xy
2
p
y
⌘
Uk 1
⇣
1 x+y+xy
2
p
y
⌘
1  x py limk 7!1
Uk 2
⇣
1 x+y+xy
2
p
y
⌘
Uk 1
⇣
1 x+y+xy
2
p
y
⌘
=
1  x+ xy   y1 x+y+xyC
⇣
y
(1 x+y+xy)2
⌘
1  x  y1 x+y+xyC
⇣
y
(1 x+y+xy)2
⌘
=
(1  x+ xy)(1  x+ y + xy)  yC
⇣
y
(1 x+y+xy)2
⌘
(1  x)(1  x+ y + xy)  yC
⇣
y
(1 x+y+xy)2
⌘
=
1 + x  y   xy  
p
(1  x  y   xy)2   4x2y
2x
,
which agrees with (1.1).
2.3 Tangent cells
Theorem 2.2 with p = 1 gives
Ck(x, y, 1, q) =
y(1  x+ xy) + (xy   x+ y)Ek(x, y, 1, q)
y(1  x)  xEk(x, y, 1, q)
, (2.5)
with
xq2Ek(x, y, 1, q) =  y2(1  x+ 2xq + q(1  q)x2y[y]k 2) + ek,
where ek =
y3(1 x+qx)2
y(1 x)+y2(1+qx)(1 x+qx) ek 1 and e2 =
y2(1 x+qx)2
1 x .
8 Art Discrete Appl. Math. 4 (2021) #P1.06
By induction on k and the definition of Chebyshev polynomials of the second kind, we
obtain
ek =
y
p
y(1  x+ qx)( (1 x)
2 qx2y(1 x+qx)p
y(1 x+qx) Uk 4(t)  (1  x)Uk 5(t))
(1 x)2 qx2y(1 x+qx)p
y(1 x+qx) Uk 3(t)  (1  x)Uk 4(t)
,
where t = (1 x)(1+y+qxy)+qxy(1+qx)2py(1 x+qx) . By substituting into (2.5), we obtain the following
result.
Theorem 2.9. The generating function for the number of k-ary words, k   2, according
to the number of horizontal steps, up steps and tangent cells is given by
Ck(x, y, 1, q) = 1 +
xy
1  x +
y
1  x ·
Ek(x, y, 1, q)
y(1  x)  xEk(x, y, 1, q)
,
where
xq2Ek(x, y, 1, q)
=  y2(1  x+ 2xq + q(1  q)x2y[y]k 2)
+
y
p
y(1  x+ qx)
⇣
(1 x)2 qx2y(1 x+qx)p
y(1 x+qx) Uk 4(t)  (1  x)Uk 5(t)
⌘
(1 x)2 qx2y(1 x+qx)p
y(1 x+qx) Uk 3(t)  (1  x)Uk 4(t)
and t = (1 x)(1+y+qxy)+qxy(1+qx)2py(1 x+qx) .
Note that by taking q = 1 into Theorem 2.9 gives Theorem 2.8, as expected.
Next we turn our attention in finding the generating function for the total number of
tangent cells over all k-ary words according to number horizontal steps and up steps. Define
Cqk(x, y) =
@
@qCk(x, y, 1, q) |q=1 and Eqk(x, y) =
@
@qEk(x, y, 1, q) |q=1. Differentiating
(2.5) with respect to q and evaluating at q = 1 gives
Cqk(x, y) =
y2
(y(1  x)  xEk(x, y))2
Eqk(x, y), (2.6)
where
Eqk(x, y)
=
y3
(y(1  x)  xEk 1(x, y))2
Eqk 1(x, y)
+
xy2(y + Ek 1(x, y))
(y(1  x)  xEk 1(x, y))2
 
xy2(x  2)[y]k 2 + (2 + x2y[y]k 2)Ek 1(x, y)
 
.
with Eq2(x, y) = 0.
By induction on k, we have
Eqk(x, y)
=
k 1X
j=2
xy3(k j) 1(y + Ej(x, y))
 
xy2(x  2)[y]j 1 + (2 + x2y[y]j 1)Ej(x, y)
 
Qk 1
i=j (y(1  x)  xEi(x, y))2
.
Thus, by (2.6), we obtain the following result.
T. Mansour: Semi-perimeter and inner site-perimeter of k-ary words and bargraphs 9
Theorem 2.10. Let k   2. The generating function for the total number of tangent cells
over all k-ary words according to the number horizontal steps and up steps is given by
Cqk(x, y)
=
k 1X
j=2
xy3(k j)+1(y + Ej(x, y))
 
xy2(x  2)[y]j 1 + (2 + x2y[y]j 1)Ej(x, y)
 
Qk
i=j(y(1  x)  xEi(x, y))2
.
where Ek(x, y) is given in (2.4).
For instance, Theorem 2.10 gives Cq2(x, y) = 0 (as expected, since there are no tan-
gent cells in 2-ary words) and
Cq3(x, y) =
x3y3(xy   x+ 3)
(x2y   x2 + xy + 2x  1)2(xy   x+ 1)
= 3x3y3 + 14x4y3 + 4x4y4 + 40x5y3 + 90x6y3 + 34x5y4 + · · · .
We emphasize in bold, the three 3-ary words with three horizontal steps and three up steps
as bargraphs of 232, 233 and 332 with 1 + 1 + 1 = 3 tangent cells.
2.4 Border cells
Theorem 2.2 with q = 1 gives
Ck(x, y, p, 1) = Ek(x, y, p, 1) + 1 +
xp(y + Ek(x, y, p, 1))(1 + Ek(x, y, p, 1)/y)
1  px(1 + Ek(x, y, p, 1)/y)
,
where
Ek(x, y, p, 1) = xy
3(p3   p4)[yp]k 2 + yp2Ek 1(x, y, p, 1)
+
xp2(y + pE2k 1(x, y, p, 1))
1  xp2(1 + xy(p2   p3)[yp]k 2 + pyEk 1(x, y, p, 1))
with E2(x, y, p, 1) = xy
2p2
1 xp2 .
Now we find the generating function for the total inner site-perimeter (the number of
border cells) over all k-ary words according to the number horizontal steps and up steps.
Define Cpk(x, y) = @@pCk(x, y, p, 1) |p=1 and Epk(x, y) =
@
@pEk(x, y, p, 1) |p=1. Differ-
entiating with respect to p and evaluating at p = 1 gives
Cpk(x, y) =
y2
(y(1  x)  xEk(x, y))2
Epk(x, y) +
xy(y + Ek(x, y))2
(y(1  x)  xEk(x, y))2
, (2.7)
where
Epk(x, y) =
y3
(y(1  x)  xEk(x, y))2
Epk 1(x, y) + Fk(x, y)
with
Fk(x, y) =
y4(3  x  x2y[y]k 2)
x(y(1  x)  xEk 1(x, y))2
  y
3(5  2x2y[y]k 2)
x(y(1  x)  xEk 1(x, y))
+
y2(3  x  2x2y[y]k 2)
x
  y
x
(y(1  x)  xEk 1(x, y)). (2.8)
10 Art Discrete Appl. Math. 4 (2021) #P1.06
By induction on k with Ep2(x, y) = 2xy
2
(1 x)2 , we obtain
Epk(x, y) =
kX
j=2
y3(k j)Fj(x, y)Qk 1
i=j (y(1  x)  xEi(x, y))2
.
Hence, by (2.7), we have the following result.
Theorem 2.11. Let k   2. The generating function for the total inner site-perimeter (the
number of border cells) over all k-ary words according to the number of horizontal steps
and up steps is given by
Cpk(x, y) =
kX
j=2
y3(k j)+2Fj(x, y)Qk
i=j(y(1  x)  xEi(x, y))2
+
xy(y + Ek(x, y))2
(y(1  x)  xEk(x, y))2
,
where Ek(x, y) and Fk(x, y) are given in (2.4) and (2.8), respectively.
For instance, Theorem 2.11 gives
Cp2(x, y) =
xy(x2(y   1)2 + 2x(y   1) + 2y + 1)
(x2(y   1) + 2x  1)2
= xy + 2x2y + 2xy2 + 10x2y2 + 3x3y + 28x3y2 + 4x4y + · · · .
We emphasize in bold, the three 2-ary words with two horizontal steps and two up steps as
bargraphs of 12, 21 and 22 with inner site perimeter 3 + 3 + 4 = 10.
We end this paper by the following comment on the relation between bargraphs and
Chebyshev polynomials. We recall that a Dyck path of semi-length n is a lattice path
that starts at (0, 0), ends at (2n, 0), remains weakly above the x-axis, and consists of up
steps (1, 1) and down steps (1, 1). Apparently, for the first time the relation between
restricted permutations and Chebyshev polynomials was discovered by Chow and West in
[7], then explored in [18], and characterized as Dyck paths in [13]. Chebyshev polynomi-
als of the second kind also occur in the enumeration of height-restricted Dyck paths, and
they are much more natural there (for instance, see [13, 18]). On the other hand, Deutsch
and Elizalde [8] established a bijection ⇢ between Dyck paths and bargraphs, where the
semi-length of a Dyck path becomes the semi-perimeter minus the number of peaks of
the corresponding bargraph (a peak in a bargraph B is an occurrence of uhjd for some
j   1). Besides that, as discussed in [15], due to the geometric nature of bargraphs, we
tried to study the statistics tangent cells, semi-perimeter and inner-site perimeter directly
on bargraphs, and not to transfer our statistics via the bijection ⇢. We followed this ap-
proach since sometimes the bargraph statistics can not be transferred to nice statistics in
Dyck paths, and sometimes the enumeration of the statistics in Dyck paths requires the
same amount of work as working directly in bargraphs. It is the main reason that directs
us to choose by our techniques rather then bijection ⇢. In our present study, transferring
the statistics tangent cells, inner-perimeter in bargraphs to statistics in Dyck paths remains
a nice point of exploration for the interested readers.
ORCID iDs
Toufik Mansour https://orcid.org/0000-0001-8028-2391
T. Mansour: Semi-perimeter and inner site-perimeter of k-ary words and bargraphs 11
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ISSN 2590-9770
The Art of Discrete and Applied Mathematics 4 (2021) #P1.07
https://doi.org/10.26493/2590-9770.1387.a84
(Also available at http://adam-journal.eu)
A simple construction of exponentially many
nonisomorphic orientable triangular
embeddings of K12s
Vladimir P. Korzhik
Pidstryhach Institute for Applied Problems in Mechanics and Mathematics of National
Academy of Sciences of Ukraine, Lviv, Ukrain
Received 27 August 2019, accepted 24 July 2020, published online 26 January 2021
Abstract
Using an index one current graph with the cyclic current group we give a simple con-
struction of 22s 7 nonisomorphic orientable triangular embeddings of the complete graph
K12s, s   4. These embeddings have no nontrivial automorphisms.
Keywords: Topological embedding, complete graph, nonisomorphic embeddings, triangular embed-
ding.
Math. Subj. Class.: 05C10, 05C15
1 Introduction
In the present paper, by an embedding of a graph we mean a cellular embedding of the
graph in an orientable surface. An embedding of a graph is triangular if all faces are 3-
gonal. Euler’s formula allows the possibility for a complete graph Kn to have a triangular
embedding if n ⌘ 0, 3, 4 or 7 (mod 12). Constructing triangular embeddings of complete
graphs was a major step in proving the Map Color Theorem [11].
Let K be a graph without loops and multiple edges. An m-gonal face of an embed-
ding of K will be designated as a cyclic sequence (v1, v2, . . . , vm) of vertices obtained by
listing the incident vertices when traversing the boundary walk of the face in some chosen
direction. The sequences (v1, v2, . . . , vm) and (vm, . . . , v2, v1) designate the same face.
One can differentiate embeddings of graphs as labeled objects (in this case we speak
about different labeled embeddings and they have different face sets) and as unlabeled ob-
jects (in this case we speak about nonisomorphic embeddings). Two triangular embeddings
f1 and f2 of Kn are isomorphic if there is a bijection  between the vertices of Kn such
E-mail address: korzhikvp@gmail.com (Vladimir P. Korzhik)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Art Discrete Appl. Math. 4 (2021) #P1.07
that (w1, w2, w3) is a face of f1 if and only if ( (w1), (w2), (w3)) is a face of f2. The
bijection  is called an isomorphism from the embedding f1 onto the embedding f2.
During the proof of the Map Color Theorem, one triangular embedding was constructed
for every complete graph Kn, n ⌘ 0, 3, 4 or 7 (mod 12). In this paper we consider the
natural question on the rate of growth of the number of nonisomorphic triangular embed-
dings of complete graphs. At present there are two approaches to construct many such
embeddings.
The first approach uses recursive constructions that generate a face 2-colorable triangu-
lar embedding of a complete graph from a face 2-colorable triangular embedding of a com-
plete graph of lesser order. First it was shown [1, 3] that there are at least 2an
2 o(n2) (where
a is a positive constant) nonisomorphic face 2-colorable triangular embeddings of Kn for
some families of values of n such that n ⌘ 3 or 7 (mod 12), namely, for n ⌘ 7 or 19
(mod 36), n ⌘ 15 (mod 60), n ⌘ 15 or 43 (mod 84), etc. Later it was shown [2, 4, 5]
that there are at least nbn
2 o(n2) nonisomorphic face 2-colorable triangular embeddings of
Kn for an infinite, but rather sparse set of values of n (where n ⌘ 3 or 7 (mod 12)). This
approach having to do with face 2-colorable triangular embeddings does not work in the
case of complete graphs of even order.
The second approach [7, 9, 10] uses the current graph technique. Within the limits of
the approach, it was shown that there are constants M, c > 0, b   1/12 such that for
every n   M , n ⌘ 0, 3, 4 or 7 (mod 12), there are at least c2bn nonisomorphic triangular
embeddings of Kn. In the case n ⌘ 0 (mod 12), this approach (see [9]) gives 2s 6
nonisomorphic triangular embeddings of K12s, s   6, and, up to the present time, this
result was the only known result on the number of nonisomorphic triangular embeddings
of K12s. This result was obtained by using index four current graphs with the cyclic current
group Z12s, and the constructions involved are rather complicated.
In the present paper we give a simple construction of 22s 7 nonisomorphic triangular
embeddings of K12s, s   4. We use an index one current graph with current group Z12s 4
that was constructed by T. Sun [12] and which generates an embedding of K12s 4, s   4,
that can be modified into a triangular embedding of K12s (thereby providing a simple
construction of a a triangular embedding of K12s, s   3). In the present paper, following
the approach used in [7, 9, 10], changing rotations of some vertices of the current graph, we
obtain 22s 7 different current graphs generating 22s 7 different embeddings of K12s 4 that
can be modified into 22s 7 different triangular embeddings of K12s. Analyzing faces of the
embeddings, we show (Theorem 3.1) that all these 22s 7 different triangular embeddings
of K12s, s   4, are nonisomorphic, thereby providing a much simpler construction of
exponentially many nonisomorphic orientable triangular embeddings of K12s.
2 Index one current graphs
In this section we describe index one current graphs which generate embeddings of K12s 4
that can be modified into triangular embeddings of K12s.
First we briefly review some material about index one current graphs in the form used in
the paper. The reader is referred to [6, 11] for a more detailed development of the material
sketched herein. We assume the reader is familiar with current graphs and embeddings
generated by current graphs.
Let G be a connected graph (multiple edges and loops are allowed) with the vertex set
V (G) whose edges have been given plus and minus direction. Hence each edge e gives rise
V. P. Korzhik: A simple construction of triangular embeddings of K12s 3
to two reverse arcs e+ and e  of G. The involutary permutation ✓ of the arc set A(G) of the
graph G that permutes reverse arcs is called the involution of G. By a current assignment
on G we mean a function   from A(G) into the set of nonzero elements of a group Zn
such that  (e ) =   (e+) for every edge e. The values of   are called currents and the
group Zn is called the current group. If an edge e is incident with a onevalent vertex w and
 (e ) =  (e+) (that is,  (e+) is of order 2 in Zn), then the arcs e+ and e  are identified
and this arc is called an end arc (and in this case we do not consider w to be a vertex of G).
A rotation D of G is a permutation of A(G) whose orbits cyclically permute the arcs
directed outwards from each vertex. The rotation D can be represented as D = {Dw :
w 2 V (G)}, where Dw, called a rotation of the vertex v, is a cyclic permutation of the
arcs directed outwards from v. Consider the permutation D✓ of A(G). It is easy to see
that the terminal vertex of an arc a is the initial vertex of the arc D✓a, hence a cycle
(a1, a2, . . . , am) of D✓ can be considered as an oriented path in G called a circuit induced
by the rotation D of G. By a one-rotation of G we mean a rotation of G inducing exactly
one circuit.
A triple hG, , Di is called a current graph. The index of the current graph is the
number of circuits induced by D. By the log of a circuit (a1, a2, . . . , am) of the current
graph we mean the cyclic sequence ( (a1), (a2), . . . , (am)).
A current graph hG, , Di can be represented as a figure of G where the rotations of
vertices are indicated. The black vertices denote a clockwise rotation and the white vertices
a counterclockwise rotation. Each pair of reverse arcs is represented by one of the arcs with
the current indicated. The end arc, as is customary, is depicted as a straight line without an
arrow, with a vertex at one end and without a vertex at the other end.
If (a1, a2, . . . , at) is the rotation of a vertex of a current graph hG, , Di, where  (ai) =
"i for i = 1, 2, . . . , t, then the cyclic sequence ("1, "2, . . . , "t) is called the current rotation
of the vertex and the element "1 + "2 + · · ·+ "t is the excess of the vertex. If the excess of
a vertex equals zero, we say that the vertex satisfies Kirchhoff’s Current Law (KCL).
Figure 1(a) shows (for now ignore the labels x, y, z, and w, and the boxes connected by
lines with edges of the graph) an index one current graph hG, , Di with the current group
Z12s 4, s   4, having the following properties (A1)-(A6):
(A1) G has two onevalent vertices, one twovalent vertex, and all other vertices are triva-
lent.
(A2) The log of the circuit contains every nonzero element of Z12s 4 exactly once.
(A3) G has exactly one end arc which has current 6s  2.
(A4) Every trivalent vertex satisfies KCL.
(A5) The two onevalent vertices have excess  1 and 6s+1, respectively (each of the two
excesses has order 12s  4 in Z12s 4).
(A6) The twovalent vertex has current rotation (1, 3).
The fragment of the current graph lying inside the dashed box is shown in Figure 1(b). The
current graph is slightly different from the current graph given in [12]: we changed the
rotations of some vertices for present purposes.
The current graph generates an embedding f(D) of the graph K12s 4 whose vertex set
is the set V (s) = {0, 1, . . . , 12s   5} of all elements of Z12s 4. There is a mapping from
the face set onto the vertex set of the current graph. Given a vertex of the current graph,
4 Art Discrete Appl. Math. 4 (2021) #P1.07
Figure 1: An index one current graph.
the faces mapping onto the vertex are called the faces induced by the vertex, and they are
determined by Theorem 4.4.1 of [6]. In the case of the current graph hG, , Di satisfying
(A1)-(A6) we have the following. A trivalent vertex with current rotation ("1, "2, "3) in-
duces 12s   4 triangular faces (u, u + "1, u + "1 + "2), u 2 V (s). The onevalent vertex
with excess  1 (resp. 6s + 1) induces one (12s   4)-gonal face shown in Figure 2(a)
(resp. (b)) (now ignore the dashed edges in Figure 2). The twovalent vertex induces two
(12s  4)-gonal faces shown in Figure 2(c).
The log of the circuit of the current graph hG, , Di (where we ignore the letters x, y,
z, and w) determines the cyclic order in which the vertices adjacent to the vertex 0 of G are
arranged on the surface around the vertex 0 in f(D).
The fragment of the current graph shown in Figure 1(b) has exactly 2s   7 vertical
edges.
Lemma 2.1 ([8, Lemma 2]). Let a rotation D of a graph G induce exactly one circuit. Let
an edge e of G be incident with distinct trivalent vertices v and w. Then there are two ways
to choose rotations of v and w without changing the rotations of other vertices, such that
the obtained rotation of G induces exactly one circuit.
V. P. Korzhik: A simple construction of triangular embeddings of K12s 5
Figure 2: The (12s  4)-gonal faces of the embedding f(Q).
6 Art Discrete Appl. Math. 4 (2021) #P1.07
Denote by L(s) the set of the vertices of the current graph lying inside the dashed box
in Figure 1(a). Now we fix the indicated rotations of the vertices of the current graph in
Figure 1(a) that do not lie inside the dashed box, and then, applying Lemma 2.1 to the
2s  7 vertical edges in Figure 1(b), we can choose the rotations of the vertices of L(s) in
22s 7 different ways such that for the corresponding 22s 7 different one-rotations Q of G,
we obtain index one current graphs hG, , Qi satisfying (A1)-(A6). Denote by D the set of
all such 22s 7 different one-rotations Q of G.
The embedding f(Q) of K12s 4 generated by hG, , Qi, Q 2 D, has four (12s   4)-
gonal faces, and all other faces are triangular. Inserting four new vertices in the four (12s 
4)-gonal faces, respectively, we obtain a triangular embedding f 0(Q) of K12s  K4 which
(by attaching one additional handle to gain adjacencies between the new vertices) can be
modified into a triangular embedding f(Q) of K12s. All embeddings f 0(Q) and f(Q),
Q 2 D, have the same vertex set V (s)
S
R, where R = {x, y, z, w} is the set of the four
new vertices.
We will show (Theorem 3.1) that all 22s 7 triangular embeddings f(Q), Q 2 D, are
nonisomorphic. Two faces of an embedding are adjacent if they share a common edge.
To prove Theorem 3.1 we need to know pairs of adjacent faces of the embeddings f(Q),
Q 2 D.
A link joining two vertices u and u0 of an embedding is every pair (u, u1, u2), (u1, u2, u0)
of adjacent triangular faces of the embedding; we say that the vertices u and u0 are incident
with the link, and that u has the link with u0. By a link [u, u0] we mean a link between u
and u0.
Figure 3: A link of an embedding.
If an edge of hG, , Qi, Q 2 D, joins two trivalent vertices with current rotations
(↵, ,  ) and (",  ,  ), respectively (see Figure 3(a)), then the type of the edge is   =
  + ". We define the type of an edge up to inversion. Since KCL holds at the vertices, we
have   + " =  (↵+  ), hence the type is well defined. The two adjacent trivalent vertices
induce faces of f(Q) that form 12s   4 links shown in Figure 3(b) where u goes through
the values 0, 1, 2, . . . , 12s  5; we say that the 12s  4 links are induced by the edge with
type   =   + ". Now we have the following.
(B) For any vertex u of f(Q), among the links induced by an edge with type  , there
are exactly two links incident with u: one of them is a link [u, u+ ] shown in Figure 3(b),
and another link is a link [u, u  ] shown in Figure 3(c).
Since any two adjacent triangular faces of f(Q) are induced by adjacent trivalent ver-
tices of hG, , Qi, every link of f(Q) joining two vertices u and u+µ is induced by exactly
V. P. Korzhik: A simple construction of triangular embeddings of K12s 7
one edge of the current graph and the type of the edge is µ.
3 Links and nonisomorphic embeddings of K12s
To prove Theorem 3.1 we use the fact that in the embeddings f(Q), Q 2 D, some pairs of
vertices have a large number of links joining the vertices, and some pairs of vertices have a
small number of links joining the vertices.
Below we describe the modification of f(Q) into f(Q), and in so doing we study links
of the obtained embeddings.
First we describe links in f(Q). In Figure 1(a) there are 13 edges with their types
indicated (the type of an edge is given inside a box connected by a line with the edge). A
list of the types of the 13 edges is
1, 1, 1, 10, 3s  8, 3s  5, 3s+ 1, 3s+ 2, 3s+ 4, 3s+ 6, 3s+ 7, 6s  10, 6s  9.
It is easy to check that for s > 6, s = 6, s = 5, and s = 4, the list contains, respectively,
11, 10, 9, and 10 different types. Hence hG, , Qi contains at least 9 edges having different
types. The current graph hG, , Qi has exactly 6s   2 edges, and exactly 6s   6 of them
join two trivalent vertices, hence at most (6s   6)   8 edges of hG, , Qi have the same
type, and, by (B), we obtain the following.
(C) In f(Q), Q 2 D, every vertex of V (s) has at most 6s   14 links with any other
vertex of V (s).
In what follows an edge joining vertices u and u0 is denoted by (u, u0).
Now insert new vertices x, y, z and w in the four (12s   4)-gonal faces of f(Q) as
shown in Figure 2 as dashed lines. (As is customary, in Figure 1(a), if a onevalent or
twovalent vertex is labeled by letters, then the letters denote the new vertices that we insert
in the faces induced by the vertex.) We obtain a triangular embedding f 0(Q) of K12s K4.
Note that the boundary cycle of the two (12s   4)-gonal faces in Figure 2(c) contain all
edges (u, u + 1) and (u, u   3), u 2 V (s), and we insert a new vertex x (resp. z) in
the face whose boundary cycle contains all edges (2i, 2i + 1) and (2i + 1, 2i   2) (resp.
(2i+ 1, 2i+ 2) and (2i, 2i  3), i = 0, 1, . . . , 6s  3.
Every link of f(Q) is a link of f 0(Q). After we insert a new vertex in a (12s 4)-gonal
face, every vertex of V (s) lying on the boundary cycle of the face gains a new link with two
different vertices of V (s) lying on the cycle. Now, considering Figure 2 where we depict
all triangular faces incident with the edges of the boundary cycles of the (12s   4)-gonal
faces, we obtain the following:
(D) The links of f 0(Q) which are not links of f(Q) are as follows: the vertex y has
6s   2 links with each of x and z; the vertex w has exactly one link with every vertex of
V (s); the vertex x (resp. z) has exactly one link with each even (resp. odd) vertex of V (s);
every vertex u 2 V (s) has three new links [u, u + 2], three new links [u, u   2], one new
link [u, u+ 6] and one new link [u, u  6].
The triangular embedding f(Q), Q 2 D, of K12s is obtained from the embedding
f 0(Q) of K12s   K4 in the following way. The log of the circuit of hG, , Qi (follow-
ing [11], the letters x, y, z, w enter the log) determines the cyclic order in which the vertices
adjacent to the vertex 0 are arranged on the surface around the vertex 0 in f 0(Q). As easily
8 Art Discrete Appl. Math. 4 (2021) #P1.07
Figure 4: Attaching a handle.
V. P. Korzhik: A simple construction of triangular embeddings of K12s 9
seen, the log is of the form
(. . . , 9s, 3, x, 1, y, 12s  5, z, 12s  7, 9s  3, . . . , 6s  7, 6s  5, w, 6s+ 1, 12s  6, . . .)
so that the faces of f 0(Q) incident with the vertex 0 are arranged as shown in Figure 4(a).
Now, in Figure 4(a), we delete edges (0, 12s   5), (0, y), (0, 1), and then, as shown in
Figure 4(b), using a handle (depicted as two blank cycles with the letter H inside; the cycles
are to be identified, and the edge ends labeled by the same Greek letter ↵, ,  ,  , ", ⇣, ⌘ are
to be identified as well) we gain adjacencies
(x, y), (x,w), (x, z), (y, w), (y, 0), (y, 6s  5), (z, w), (0, 12s  5), (w, 12s  5).
In Figure 4 (and as in what follows in Figure 5) the shaded faces are faces of f 0(Q) that
remain unchanged when modifying f 0(Q) into f(Q). As a result, we obtain a triangular
embedding of the graph K12s without two edges (y, z) and (0, 1), but with two extra edges
(w, 12s  5) and (y, 6s  5). Note that the faces shown in Figure 4(b) are the same for all
Q 2 D.
The embedding f 0(Q) contains five pairs of adjacent faces shown as non-shaded faces
in Figures 5(a) – (e), respectively. For each of the five pairs, we show a fragment of
hG, , Qi whose vertices induce the faces of the pair and all the other (shaded) faces adja-
cent to the faces of the pair (the vertices of the fragment are not vertices of L(s), so that
the faces shown in Figures 5(a) – (e) are the same for all Q 2 D). The reader can consult
Figures 2 and 3 when checking pairs of adjacent faces in Figures 5(a) – (e).
The diagonal flips in the pairs of adjacent non-shaded faces shown in Figures 5(a), (b),
and (c), replace the edges (w, 12s  5), (6s, 6s  6), and (12s  8, 12s  9) by the edges
(6s, 6s   6), (12s   8, 12s   9), and (y, z), respectively, depicted in dashed line. As a
result, we lose an extra edge (w, 12s   5) and gain a missing edge (y, z). The diagonal
flips in the pairs of adjacent non-shaded faces shown in Figures 5(d) and (e) replace the
edges (y, 6s 5) and (6s 6, 6s 4) by the edges (6s 6, 6s 4) and (0, 1), respectively.
As a result, we lose an extra edge (y, 6s   5) and gain a missing edge (0, 1). We obtain
the triangular embedding f(Q) of K12s. Note that the diagonal flips do not affect the faces
shown in Figure 4(b), hence all faces shown in Figure 4(b) are faces of f(Q).
Now we need to know what new links we gain and what links incident with vertices of
R we lose when modifying f 0(Q) into f(Q).
In Figure 4(b), a new additional handle is attached to the 6-gonal face (0, z, 12s  
5, y, 1, x) and the triangular face (w, 0, 6s 5), and then some new edges are embedded. If
we actually identify the two cycles with the letter H inside, then the faces of f(Q) incident
with the new edges shown in Figure 4(b) can be redrawn as shown in Figure 4(c). Every lost
link contains a face that we lose during the modification, hence all lost links are incident
with vertices incident with lost faces. Every new link contains a new face, hence all new
links are incident with vertices incident with new faces. The reader can consult Figure 2
when checking faces in Figure 4.
When considering the five diagonal flips shown in Figures 5(a) – (e), it is easy
to see that if in Figure 5(f) we replace the edge (a, b) by (c, d), then we lose links
[c, d], [a, h], [a, f ], [b, g], [b, e] and gain new links [a, b], [c, e], [c, f ], [d, g], [d, h].
By inspection of Figures 4(c) and 5(a) – (e), the reader can check that during the modi-
fication of f 0(Q) into f(Q): the vertex y lost two links with each of x and z; each of x and
z gained at most one link with any vertex of V (s); the vertex w gained at most three new
10 Art Discrete Appl. Math. 4 (2021) #P1.07
Figure 5: Diagonal flips.
V. P. Korzhik: A simple construction of triangular embeddings of K12s 11
links with any vertex of V (s), one new link with y, and no links with each of x and z; any
two vertices of V (s) gained at most one new link; any vertex of V (s) gained at most four
new links with vertices of R.
Now, taking into account (C) and (D), we obtain the following.
(E) For any Q 2 D, in f(Q), we have the following: the vertex y has 6s  4 links with
each of x and z; each of x and z has 6s   4 links with y only; the vertex w has at most 4
links with any vertex of V (s)
S
R, and has no links with x and z; every vertex of V (s) has
a link either with x or z, and has less than 6s 4 links with every other vertex of V (s)
S
R.
By an automorphism of f(Q) we mean any isomorphism from f(Q) onto f(Q).
Theorem 3.1. All 22s 7 embeddings f(Q), Q 2 D, of K12s, s   4, are nonisomorphic
and each of them has no nontrivial automorphisms.
Proof. Suppose there is an isomorphism  of f(Q1) onto f(Q2), where Q1, Q2 2 D. If
two adjacent faces (u1, u2, u3) and (u2, u3, u4) are a link in f(Q1), then the two adjacent
faces ( (u1), (u2), (u3)) and ( (u2), (u3), (u4)) are a link in f(Q2). Since f(Q1)
and f(Q2) have the same number of links, namely, the number of edges of K12s, it follows
that the number of links between any two vertices u and u0 in f(Q1) equals the number of
links between any two vertices  (u) and  (u0) in f(Q2).
Figure 6: Common faces of all f(Q), Q 2 D.
By (E), the vertex y is the only vertex in each of f(Q1) and f(Q2) that has 6s   4
links with each of two vertices, hence  (y) = y. Since x and z are the only vertices in
each of f(Q1) and f(Q2) such that each of the vertices has 6s   4 links with exactly one
other vertex, we have { (x), (z)} = {x, z}. Since w is the only vertex of V (s)
S
{w} in
each of f(Q1) and f(Q2) that has no links with x and z, we have  (w) = w. Considering
Figure 4(c), we see that f(Q1) and f(Q2) have the same faces shown in Figure 6. Since
( (w), (x), (y)) = (w, (x), y) is a face of f(Q2), and  (x) 2 {x, z}, we obtain (see
Figure 6) that  (x) = x, and then  (z) = z.
If f(Q1) and f(Q2) have common adjacent faces ( (u1), (u2), (u3)) and
( (u2), (u3), (u4)), where  (uj) = uj for j = 1, 2, 3, then  (u4) = u4. The
faces incident with w (the faces are the same for all f(Q), Q 2 D), form a sequence
F1, F2, . . . , F12s 1 where:
(i) F1 = (w, x, y) and  (w) = w,  (x) = x,  (y) = y;
(ii) for j = 1, 2, . . . , 12s  1, the faces Fj and Fj+1 (here F12s = F1) share a common
edge (w, bj), where {b1, b2, . . . , b12s 1} = (V (s)
S
R) \ {w}.
It follows that  (u) = u for every u 2 V (s)
S
R, hence f(Q1) and f(Q2) have the same
faces. If Q1 = Q2, then we obtain that  is a trivial automorphism, hence f(Q1) does not
have nontrivial automorphisms.
12 Art Discrete Appl. Math. 4 (2021) #P1.07
Suppose, for a contradiction, that Q1 6= Q2. Since f(Q1) and f(Q2) have the same
faces, considering the modification of f(Q) into f(Q), we see that f(Q1) and f(Q2) have
the same faces as well, hence we have:
(a) The cyclic order in which the vertices adjacent to the vertex 0 are arranged on the
surface around the vertex 0 in f(Q1) is (up to reversal) the cyclic order in which the vertices
adjacent to the vertex 0 are arranged on the surface around the vertex 0 in f(Q2).
The embeddings f(Q1) and f(Q2) are generated by the current graphs hG, , Q1i and
hG, , Q2i, respectively. Since Q1 6= Q2, a trivalent vertex v (resp. w) of G has the
same rotation (resp. different rotations) in Q1 and Q2. Then the circuit of hG, , Q1i
(resp. hG, , Q2i) is of the form (a1, a2, . . . , b1, b2, . . .) (resp. (a1, a2, . . . , b1, b3, . . .))
where a1 and a2 are arcs incident with v, and b1, b2, b3 are arcs incident with
w, where b2 6= b3. Hence the two cascades have different logs of their circuits,
namely, ( (a1), (a2), . . . , (b1), (b2), . . .) and ( (a1), (a2), . . . , (b1), (b3), . . .)
where  (b2) 6=  (b3), contrary to (a) (note that in the cascades,  (a) 6=  (a0) for dif-
ferent arcs a and a0).
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ISSN 2590-9770
The Art of Discrete and Applied Mathematics 4 (2021) #P1.08
https://doi.org/10.26493/2590-9770.1370.445
(Also available at http://adam-journal.eu)
Two new families of non-CCA groups
⇤
Brandon Fuller , Joy Morris†
Department of Mathematics and Computer Science, University of Lethbridge
Lethbridge, AB T1K 3M4, Canada
Received 24 May 2020, accepted 11 August 2020, published online 28 January 2021
Abstract
We determine two new infinite families of Cayley graphs that admit colour-preserving
automorphisms that do not come from the group action. By definition, this means that
these Cayley graphs fail to have the CCA (Cayley Colour Automorphism) property, and the
corresponding infinite families of groups also fail to have the CCA property. The families
of groups consist of the direct product of any dihedral group of order 2n where n   3 is
odd, with either itself, or the cyclic group of order n. In particular, this family of examples
includes the smallest non-CCA group that does not fit into any previous family of known
non-CCA groups.
Keywords: Cayley graphs, automorphisms, colour preserving, CCA
Math. Subj. Class.: 05C25
1 Introduction
All groups and graphs in this paper are finite. All of our graphs are simple, undirected, and
have no loops.
A Cayley graph of G with respect to C (a subset of G\{e}) is the graph Cay(G,C)
whose vertices are the elements of G, with an edge from g to gc if and only if g 2 G, c 2
C. The set C is known as the connection set of Cay(G,C). This connection set gives
a natural colouring of the edges where we colour the edge from g to gc (which is the
same as the edge from gc to g) with a colour associated to {c, c 1}. A colour-preserving
automorphism of Cay(G,C) is a permutation of the vertices that preserves edges and non-
edges as well as edge colour. A Cayley graph Cay(G,C) is said to have the Cayley Colour
⇤The authors thank the anonymous referees for their careful reading of this paper, and for their helpful sug-
gestions.
†This work was supported by the Natural Science and Engineering Research Council of Canada (grant RGPIN-
2017-04905).
E-mail addresses: brandon.fuller621@gmail.com (Brandon Fuller), joy.morris@uleth.ca (Joy Morris)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Art Discrete Appl. Math. 4 (2021) #P1.08
Automorphism (CCA) property if every colour-preserving automorphism of the graph is
an affine function on G. The group G is said to be CCA if every connected Cayley graph
of G is CCA.
The study of this property has only come up recently in history. In 2012, M. Con-
der, T. Pizanski and A. Žitnik [1] proposed a question about the permutations of circulant
graphs that preserve a certain edge colouring that the second author [7] answered. The
second author showed that for any connected Cayley graph on the cyclic group Cn, all
colour-preserving automorphisms that fix the identity are automorphisms of Cn. In 2014,
A. Hujdurović, K. Kutnar, D. W. Morris, and J. Morris [3] extended the original ques-
tion by looking at Cayley graphs, using the natural edge colouring described. In early
2017, L. Morgan, J. Morris and G. Verret [5, 6] gave new results for finite simple groups
and Sylow cyclic groups that generalized results produced by E. Dobson, A. Hujdurović,
K. Kutnar, and J. Morris in [2]. The problem of determining colour-preserving and colour-
permuting automorphisms for directed Cayley graphs has already been studied and is well
understood: see for example [10], where the authors showed that for a connected Cayley
digraph, every colour-preserving automorphism is a left-translation by some element of the
group.
In his M.Sc. thesis, the first author produced code using GAP [9] and Sage [8] that
determines whether or not a group or graph has the CCA property, and ran this code on all
groups of order up to 200 (excluding orders 128 and 192). With this data in hand, a logical
step was to look for theoretical methods to explain some of the small non-CCA groups that
were not previously understood, and if possible to find new infinite families of non-CCA
groups using this method.
In this paper, we use results from [5] to show that whenever n   3, the groups Cn⇥D2n
and D2n ⇥D2n are non-CCA groups. Section 2 contains some basic background, defini-
tions, and notation, along with the statements of the results we need from [5]. Section 3
provides proofs of our main results.
2 Background
The following notation is used for the remainder of this paper. We use Cn to represent the
cyclic group of order n, and D2n (for n   3) to represent the dihedral group of order 2n.
We also have Q8 as the quaternion group of order 8.
The notation   = (V ( ), E( )) represents a graph of finite order, consisting of a set
V = V ( ) of vertices and a set E = E( ) ✓ {{u, v} | u, v 2 V } of edges. The set of
vertices that are adjacent to a vertex v, denoted  (v), is called the neighbourhood of v. We
use L( ) to indicate the line graph of the graph  , and S( ) is the subdivision graph of the
graph  .
If G acts on a graph   and S ✓ V ( ) is fixed setwise under the action of G, then
G
S is the restriction of the action of G to S. We use Gv to denote the stabiliser subgroup
(elements of G that fix v).
Definition 2.1 ([3, Definition. 2.6]). For an abelian group A of even order and an involution
y 2 A, the corresponding generalized dicyclic group is
Dic(A, y) = hx,A | x2 = y, x 1ax = a 1, 8a 2 Ai.
B. Fuller and J. Morris: Two new families of non-CCA groups 3
Definition 2.2 ([3, Definition. 5.1]). The generalized dihedral group over an abelian
group A is the group
Dih(A) = h , A |  2 = e, a  = a 1, 8a 2 Ai
Definition 2.3 ([5, Definition 4.5]). Let B be a permutation group and G a regular subgroup
of B. Let A0 be the colour-preserving automorphism group of the complete Cayley colour
graph KG = Cay(G,G \ {e}), and let bG be the subgroup of A0 consisting of all left
translations by elements of G. We say that (G,B) is a complete colour pair if B is a
subgroup of A0 and G is one of the following:
• G is abelian but not an elementary abelian 2-group, and A0 ⇠= Dih(G).
• G ⇠= Dic(A, y) but not of the form Q8 ⇥ Cn2 and A0 = bG o h i, where   is the
permutation that fixes A pointwise and maps every element of the coset Ax to its
inverse.
• G ⇠= Q8 ⇥ Cn2 and A0 = h bG, i, j , ki, where  ↵ is the permutation of Q8 ⇥ Cn2
that inverts every element of {±↵}⇥ Cn2 and fixes every other element.
The importance of Definition 2.3 comes from the fact that if (G,B) is a complete
colour pair, then in each case we have a colour-preserving automorphism of KG that is not
an element of bG.
An arc is an orientation for an edge in a graph. So the edge {u, v} admits two possible
orientations: (u, v), or (v, u).
Definition 2.4. Let   be a graph and G a permutation group acting on the vertices of  . We
say that   is a G-arc-regular graph if for each pair of arcs e1 = (u, v) and e2 = (w, x)
(each an oriented edge from E( )), there exists a unique element of G that maps u to w
and v to x, so that it maps the chosen orientation for e1 to the chosen orientation for e2.
Notation 2.5. For the remainder of this paper we use the following notation. Consider
the complete bipartite graph Kn,n. We define ⇢1 to be a cyclic permutation on one of
the bipartition sets, and ⇢2 be a cyclic permutation on the other bipartition set, with ⌧
an involution that commutes with ⇢1⇢2 and switches the bipartition sets. Let  1 be an
involution acting on the first bipartition set that inverts ⇢1, and  2 an involution acting on
the second bipartition set that inverts ⇢2.
We label the edges of S(Kn,n) as follows. Use v to denote the unique vertex in the
second bipartition set of Kn,n that is fixed under the action of  2. Now in S(Kn,n) label
the edge from ⌧(v) to the vertex subdividing {v, ⌧(v)} with the identity element e of G,
and label each other edge by the unique element of G that maps the edge e to that edge.
This produces a labeling that shows us that L(S(Kn,n)) is a Cayley graph on G. From this
it is straightforward to observe that the connection set C (which consists of all neighbours
of e) is {⌧} [ {⇢i2 : 1  i  n  1}.
Corollary 2.6 ([5, Corollary 4.10]). Let   be a connected G-arc-regular graph. If H is a
group of automorphisms of   such that:
• G  H , and
• (G (v)v , H (v)v ) is a complete colour pair for every vertex v of  ,
4 Art Discrete Appl. Math. 4 (2021) #P1.08
then H is a colour-preserving group of automorphisms of L(S( )) viewed as a Cayley
graph on G.
The real point of this corollary is that if we show that some element of H is not an
affine function, then this implies that L(S( )) is a non-CCA graph, and so G is a non-
CCA group. The fact that (G (v)v , H
 (v)
v ) is a complete colour pair is what allows us to
produce the desired non-affine element of H .
3 Main results
In our main result, we show that Kn,n is a (connected) Cn ⇥ D2n-arc-regular graph and
therefore if we take   = Kn,n, G = Cn⇥D2n, and H = D2n oC2 then all of the conditions
of Corollary 2.6 are satisfied. For clarity, we are using D2n o C2 to denote the semidirect
product (D2n⇥D2n)oC2, where the C2 is acting on the coordinates in the direct product.
Hence D2n o C2 is a colour-preserving group of automorphisms of L(S(Kn,n)). With this
we find an element in D2n o C2, a colour-preserving automorphism, that is a non-affine
function to show that L(S(Kn,n)) is non-CCA. The proof is not particularly difficult; the
difficulty of this result lies in finding an arc-regular graph and corresponding permutation
groups to which we can apply Corollary 2.6.
Theorem 3.1. The graph L(S(Kn,n)) viewed as a Cayley graph on Cn⇥D2n is non-CCA
whenever n   3 is odd.
Specifically, if G = h⇢1, ⇢2, ⌧i and C = {⌧} [ {⇢i2 : 1  i  n   1}, then  2 is a
non-affine colour-preserving automorphism of Cay(G,C).
Proof. We use Notation 2.5 and the labelling that is given in the paragraph following that
notation to view L(S( )) as a Cayley graph on G. Observe that G = h⇢1, ⇢2, ⌧i =
h⇢1⇢2, ⇢1⇢ 12 , ⌧i ⇠= Cn ⇥ D2n since n is odd so that h⇢22i = h⇢2i. Notice that G acts
regularly on the arcs of Kn,n, so that Kn,n is G-arc-regular.
Consider now the group H = h⇢1, ⇢2, ⌧, 1, 2i ⇠= D2n o C2 where each copy of D2n
acts independently on one of the bipartition sets of Kn,n, and the C2 (generated by ⌧ )
exchanges the coordinates. The first copy of D2n is generated by ⇢1 and  1. The second
copy is generated by ⇢2 and  2. It is clear that G  H since ⇢1, ⇢2, ⌧ 2 H .
Let v be an arbitrary vertex of the second bipartition set. The neighbours of v are all the
elements of the first bipartition set. We notice that G (v)v is the subgroup of G that fixes v
and its action is restricted to the bipartition set that v is not in. We see that ⇢1 is the cyclic
permutation of  (v). Since G = h⇢1, ⇢2, ⌧i, it is not hard to observe that G (v)v = h⇢1i ⇠=
Cn. Similarily since H = h⇢1, ⇢2, ⌧, 1, 2i we have that H (v)v = h⇢1, 1i ⇠= D2n. Thus
we only need to show that (Cn, D2n) is a complete colour pair.
We can see (Cn, D2n) is a complete colour pair using Definition 2.3. Let A0 be the
colour-preserving automorphism group for the Cayley graph K
G (v)v
. We know that A0 =
D2n = Dih(Cn) and thus since Cn is abelian and is not an elementary abelian 2-group
(n   3), all the properties of the first possibility for a complete colour pair are met. (In this
case, B = D2n = A0.) We thus conclude (using Corollary 2.6) that every element of H is
a colour-preserving automorphism of L(S(Kn,n)) viewed as a Cayley graph on G.
It remains to show that some element of H is not affine. We claim that  2 (acting on
G as an automorphism of the Cayley graph) is such an element. In order to prove this,
we show that   12 ⌧ 2 is not an element of G. Let v be the unique vertex in the second
B. Fuller and J. Morris: Two new families of non-CCA groups 5
bipartition set that is fixed by  2. Clearly,   12 ⌧ 2 =  2⌧ 2 maps the arc (v, ⌧(v)) to the
arc (⌧(v), v), since  2 fixes both v and ⌧(v). Since G is acting arc-regularly, it has a unique
element that maps (v, ⌧(v)) to the arc (⌧(v), v), and we know that this element is ⌧ . So if
 2 normalises G, we must have  2⌧ 2 = ⌧ . It is straightforward to verify that this is not the
case. For example, ⌧⇢2(v) = ⌧⇢1⇢2(v) = ⇢1⇢2⌧(v) = ⇢1⌧(v) (the first equality follows
from the fact that ⇢1 fixes the bipartition set that contains v; the second equality from the
fact that ⌧ and ⇢1⇢2 commute, and the third from the fact that ⇢2 fixes the bipartition set
that does not contain v). However,  2⌧ 2⇢2(v) = ⌧ 2⇢2(v) = ⌧⇢ 12  2(v) = ⌧⇢
 1
2 (v) (the
first equality follows because  2 fixes the bipartition set that does not contain v; the second
because h 2, ⇢2i ⇠= D2n, so  2 inverts ⇢2; and the third because  2 fixes v). However, since
n   3, ⌧⇢ 12 (v) is not the same as ⌧⇢2(v), because the order of ⇢2 is n. Thus,  2 2 H
does not normalise G, as claimed.
Corollary 3.2. The group Cn ⇥D2n is non-CCA whenever n   3 is odd.
We use the above result to show that D2n ⇥D2n is not CCA whenever n   3 is odd.
Proposition 3.3. The group D2n ⇥D2n is non-CCA whenever n   3 is odd.
Proof. Let G = h⇢1, ⇢2, ⌧i where these permutations are as defined in Notation 2.5. Define
H = hG,  i, where   is an involution that commutes with ⌧ and with ⇢ 11 ⇢2, and inverts
⇢1⇢2. Notice that this implies H ⇠= D2n ⇥D2n.
By Theorem 3.1, if G = h⇢1, ⇢2, ⌧i and C = {⌧} [ {⇢i2 : 1  i  n   1}, then  2
is a non-affine automorphism of Cay(G,C) (in its action on G as an automorphism of this
Cayley graph). We use this to produce a non-affine colour-preserving automorphism ' on
  = Cay(H,C [ { }).
Define ' by '(g) =  2(g), and '(g ) =  2(g)  for every g 2 G. We first show that
' is colour-preserving on  .
Consider any edge e of  . If both endpoints of e are in G then '(e) =  2(e) and since
 2 preserves colours, so does '.
If one endpoint of e is in G and the other is not, then it must be the case that e is
coloured  , and its endpoints are g and g  for some g 2 G. Furthermore, by definition of
' we have '(g ) = '(g) , so there is an edge between '(g) and '(g ), and its colour is
 . Thus ' also preserves the colour of any such edge.
The final case to consider is if both endpoints of e are in G . Suppose the endpoints of
e are ⇢i11 ⇢
i2
2 ⌧
f1  and ⇢j11 ⇢
j2
2 ⌧
f2 , where 0  i1, i2, j1, j2  n   1, and 0  f1, f2  1.
Since there is an edge between these vertices, we must have  ⌧f1⇢j1 ii1 ⇢
j2 i2
2 ⌧
f2  2 C
(recall that   and ⌧ are both involutions). Note that ⇢a1⇢b2 = (⇢1⇢2)(a+b)/2(⇢
 1
1 ⇢2)
(b a)/2;
we want to use this because we know that   commutes with ⌧ and with ⇢ 11 ⇢2 but inverts
⇢1⇢2. So we have
 ⌧
f1(⇢1⇢2)
(j1+j2 i1 i2)/2(⇢ 11 ⇢2)
(j2+i1 i2 j1)/2⌧f2 
= ⌧f1(⇢1⇢2)
(i1+i2 j1 j2)/2(⇢ 11 ⇢2)
(j2+i1 i2 j1)/2⌧f2
= ⌧f1⇢i2 j21 ⇢
i1 j1
2 ⌧
f2 2 C.
Since we know the elements of C, this implies one of three possibilities:
• the element is ⌧ , so that i2 = j2 and i1 = j1, and {f1, f2} = {0, 1};
6 Art Discrete Appl. Math. 4 (2021) #P1.08
• f1 = f2 = 0 and the element is ⇢j2 for some 1  j  n   1, so i2 = j2, and
j = i1   j1); or
• f1 = f2 = 1 and the element is ⇢j2 for some 1  j  n   1, so (using the above
equation and the fact that ⌧ commutes with ⇢1⇢2 and inverts ⇢ 11 ⇢2) i1 = j1, and
j = j2   i2.
We now need to understand the images of the endpoints of e under '. Recall from the
labelling established immediately following Notation 2.5 that we choose the vertex v to
be the unique vertex in the second bipartition set that is fixed by  2, and in S(Kn,n) we
label the edge from ⌧(v) to the vertex subdividing {v, ⌧(v)} with the identity element of
G. This means that the edge from ⇢i11 ⌧(v) to the vertex subdividing {⇢
i1
1 ⌧(v), ⇢
i2
2 (v)} will
be the image of the edge labelled with the identity under the action of ⇢i11 ⇢
i2
2 , so is labelled
⇢
i1
1 ⇢
i2
2 . Similarly, since the edge from v to the vertex subdividing {v, ⌧(v)} has the label ⌧ ,
the edge from ⇢i22 (v) to the vertex subdividing {⇢
i1
1 ⌧(v), ⇢
i2
2 (v)} will be labelled ⇢
i1
1 ⇢
i2
2 ⌧ .
This is the other “half” of the same subdivided edge from Kn,n. It should now be apparent
that  2(⇢i11 ⇢
i2
2 ) = ⇢
i1
1 ⇢
 i2
2 and therefore  2(⇢
i1
1 ⇢
i2
2 ⌧) = ⇢
i1
1 ⇢
 i2
2 ⌧ (the other half of the
same subdivided edge from Kn,n). Thus, the images of the endpoints of e under ' are
⇢
i1
1 ⇢
 i2
2 ⌧
f1  and ⇢j11 ⇢
 j2
2 ⌧
f2 .
Now using similar calculations to those above, the colour of the edge between these
images is
⌧
f1⇢
j2 i2
1 ⇢
i1 j1
2 ⌧
f2
(together with its inverse). Taking the three possibilities identified above in turn, if i1 = j1,
i2 = j2, and {f1, f2} = {0, 1} then this colour is ⌧ as before, so ' has preserved the
colour. If f1 = f2 = 0, i2 = j2, and the colour of e was {⇢j2, ⇢
 j
2 } where j = i1   j1, then
the colour of this edge is also {⇢j2, ⇢
 j
2 }. Finally, if f1 = f2 = 1, i1 = j1, and the colour
of e was {⇢j2, ⇢
 j
2 } where j = j2   i2, then the colour of this edge is {⇢
j
2, ⇢
 j
2 }. So in all
cases the colour of e is preserved under the action of '. This completes the proof that ' is
colour-preserving.
Since Cay(H,C) has two connected components (on G and G ), any colour-preserving
automorphism of   must preserve these components. Therefore, if ' is affine then its re-
striction to G (which is  2) would have to be affine on G. By Theorem 3.1 this is not the
case. Thus, ' is a colour-preserving automorphism of   that is not affine, and therefore  
and H are not CCA.
ORCID iDs
Brandon Fuller https://orcid.org/0000-0002-0757-8097
Joy Morris https://orcid.org/0000-0003-2416-669X
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2008, https://books.google.com/books?id=4ayMmAEACAAJ.
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[6] L. Morgan, J. Morris and G. Verret, A finite simple group is CCA if and only if it has no element
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[7] J. Morris, Automorphisms of circulants that respect partitions, Contrib. Discrete Math. 11
(2016), 1–6, doi:10.11575/cdm.v11i1.62390.
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[9] The GAP Group, Gap – groups, algorithms and programming, 2017, https://www.
gap-system.org.
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ISSN 2590-9770
The Art of Discrete and Applied Mathematics 4 (2021) #P1.09
https://doi.org/10.26493/2590-9770.1356.d19
(Also available at http://adam-journal.eu)
Transit sets of two-point crossover
⇤
Manoj Changat
Department of Futures Studies, University of Kerala, Trivandrum, IN 695 581, India
Prasanth G. Narasimha-Shenoi
Department of Mathematics, Government College Chittur, Palakkad, IN 678 104, India
Ferdoos Hossein Nezhad
Department of Futures Studies, University of Kerala, Trivandrum, IN 695 581, India
Matjaž Kovše
School of Basic Sciences, IIT Bhubaneswar, Bhubaneswar, India
Shilpa Mohandas
Department of Futures Studies, University of
Kerala, Trivandrum, IN 695 581, India
Abisha Ramachandran
Department of Mathematics, Sree Narayana
College, Sivagiri, Varkala, IN 695145, India
Peter F. Stadler
Bioinformatics Group, Department of Computer Science & Interdisciplinary Center for
Bioinformatics, Universität Leipzig, Härtelstraße 16-18, D-04107 Leipzig, Germany; German
Centre for Integrative Biodiversity Research (iDiv) Halle-Jena-Leipzig, Competence Center for
Scalable Data Services and Solutions Dresden-Leipzig, Leipzig Research Center for Civilization
Diseases, and Centre for Biotechnology and Biomedicine at Leipzig University at Universität
Leipzig; Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, D-04103 Leipzig,
Germany; Institute for Theoretical Chemistry, University of Vienna, Währingerstrasse 17, A-1090
Wien, Austria; Facultad de Ciencias at Universidad National de Colombia, Bogotá, Colombia;
Santa Fe Insitute, 1399 Hyde Park Rd., Santa Fe NM 87501, USA
Received 13 February 2020, accepted 21 September 2020, published online 06 February 2021
Abstract
Genetic Algorithms typically invoke crossover operators to produce offsprings that are
a “mixture” of two parents x and y. On strings, k-point crossover breaks parental geno-
types at at most k corresponding positions and concatenates alternating fragments for the
⇤This work was supported in part by the Department of Science and Technology of India (SERB project file no.
MTR/2017/000238 “Axiomatics of betweenness in discrete structures” to MC), and the German Academic Ex-
change Service (DAAD) through the bilateral Slovenian-German project “Mathematical Foundations of Selected
Topics in Science”.
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Art Discrete Appl. Math. 4 (2021) #P1.09
two parents. The transit set Rk(x, y) comprises all offsprings of this form. It forms the
tope set of an uniform oriented matroid with Vapnik-Chervonenkis dimension k + 1. The
Topological Representation Theorem for oriented matroids thus implies a representation in
terms of pseudosphere arrangements. This makes it possible to study 2-point crossover in
detail and to characterize the partial cubes defined by the transit sets of two-point cross-
over.
Keywords: Genetic algorithms, recombination, transit functions, oriented matroids, Vapnik-Chervo-
nenkis dimension.
Math. Subj. Class.: 05C62, 05C75
1 Introduction
Genetic Algorithms, Evolutionary Algorithms, and Genetic Programming are heuristics
commonly employed to solve complex optimization problems. A key component are cross-
over operators, which generate offsprings that are a mixture of two parents [16, 18, 22, 25].
Here we consider crossover operators on the set X = An strings with a fixed length n over
some alphabet A. A k-mask m is a binary string of length n with a most k break points
between consecutive runs of 0s and 1s. That is, there are 0  h  k < n “break points”
0 < i1 < i2 < · · · < ih < n, such that (with i0 := 0 and ih+1 = n) m satisfies mi = 0
for ij < i  ij+1 for even j and mi = 1 for ij < i  ij+1 for odd j. By definition, every
k-mask starts with 0. For example, for n = 15 and i1 = 3, i2 = 5, i3 = 8, i4 = 12, we
have the 4-mask
m = 000110001111000.
Note that m is also a k-mask for 4  k  15. A k-mask thus is a binary string with at most
k + 1 alternating runs of 0s and 1.
Definition 1.1. A string z 2 X is a k-point crossover offspring of x, y 2 X if there is
k-mask m such that either zi = xi if mi = 0 and zi = yi if mi = 1 for 1  i  n, or
zi = yi if mi = 0 and zi = xi if mi = 1 for 1  i  n.
For instance, given two parents x and y, as well as the 4-mask m, we obtain the two
offsprings z1 and z2 as follows:
x =++-++-++-++-+++
y =-+--++--++--+--
m =000110001111000
z1 =++--+-++++--+++
x =++-++-++-++-+++
y =-+--++--++--+--
m =000110001111000
z2 =-+-+++---++-+--
Intuitively, k-point crossover subdivides the parents x and y into at most k+ 1 consec-
utive fragments that alternate in the offspring z. There is a rich literature on various aspects
of k-point crossover operators. Algebraic properties are the focus of [7, 21, 24], disruption
analysis is studied in [5], the relation between search spaces of crossover and mutation is
discussed in [4, 23], coordinate transformation are explored in [8, 15]. The recombination
E-mail addresses: mchangat@gmail.com (Manoj Changat), prasanthgns@gmail.com (Prasanth G.
Narasimha-Shenoi), ferdows.h.n@gmail.com (Ferdoos Hossein Nezhad), matjaz.kovse@gmail.com (Matjaž
Kovše), rshilpam@gmail.com (Shilpa Mohandas), cyanabisha@gmail.com (Abisha Ramachandran),
studla@bioinf.uni-leipzig.de (Peter F. Stadler)
M. Changat et al.: Transit sets of two-point crossover 3
sets Rk(x, y) of possible crossover offsprings z of two parents x and y under k-point cross-
over. The function Rk : X ⇥ X ! 2X satisfies, for all x, y 2 X , (T1) x, y 2 Rk(x, y),
(T2) Rk(x, y) = Rk(y, x), and (T3) Rk(x, x) = {x} [14]. These three axioms define
transit functions [19], forming a common framework to describe intervals, convexities, and
betweenness. In [3], we studied properties of the transit functions Rk deriving from k-point
crossover. Convexity as a property of crossover operators is studied e.g. in [11, 12].
Here, we focus on the transit sets Rk(x, y) themselves. Since Rk(x, y) depends only
on the positions in which x and y differ, it suffices to consider a two-letter alphabet A =
{+, } and thus X = {+, }n. We therefore interpret X as the vertex set of the n-
dimensional Boolean Hypercube, and Rk(x, y) as an induced subgraph of X . It is shown in
[3, Cor. 4.2] that Rk(x, y) is a partial cube, that is, an isometric subgraph of n-dimensional
Boolean Hypercube [6].
The Hamming distance on X is the number d(x, y) of positions in which x and y differ.
Any two vertices x and y span a sub-hypercube Q(x, y) of X with dimension d(x, y),
which coincides with the set of all crossover offsprings Rk(x, y) whenever d(x, y)  k.
Otherwise, Rk(x, y) is an induced subgraph of Q(x, y). Its cardinality
|Rk(x, y)| =
(
2t if t  k
2 k(t  1) if t > k
(1.1)
depends only on the Hamming distance t := d(x, y) and the parameter k [3, 14], where
 h(n) :=
Ph
i=0
 n
i
 
. In fact, the graphs Rk(x, y) depend only on k and the Hamming
distance d(x, y):
Lemma 1.2. Let x, y 2 {+, }n and x0, y0 2 {+, }n
0
. Then Rk(x, y) and Rk(x0, y0)
are isomorphic if and only if d(x, y) = d(x0, y0).
Proof. Since every coordinate i for which xi = yi is constant in Rk(x, y) we know that
Rk(x, y) is an isometric subgraph of the subcube spanned by the d := d(x, y) coordinates i
with xi 6= yi. Relabeling the coordinates on {+, }d is an isomorphism, hence Rk(x, y) is
isomorphic to Rk( d,+d), where  d and +d are the strings of length d with all coordinates
being   and +, respectively. Thus Rk(x, y) and Rk(x0, y0) are isomorphic if d(x, y) =
d(x0, y0). On the other hand, Rk( d,+d) and Rk( d
0
,+d
0
) cannot be isomorphic if d 6= d0
since the diameter of the graphs differs.
In this contribution, we show that the transit set of k-point forms the tope set of an
uniform oriented matroid, which provides a means of gaining further insight into their
structure and allows a characterization of the transit sets of two-point crossover.
2 VC-Dimension of Recombination Sets Rk(x, y)
The Vapnik-Chervonenkis dimension (VC-dimension) quantifies the complexity of set sys-
tems [26, 27]. Given some base set Y of cardinality n := |Y |, a family H ✓ 2Y forms
an induced subgraph G of the Boolean hypercube {+, }n: for A 2 H, we identify
y 2 A ✓ Y with the y-coordinate of the corresponding point being +, while y /2 A
corresponds to  . A set C ✓ Y is said to be shattered by H if {Q \ C|Q 2 H} = 2C .
The V C-dimension of H is the largest integer dV C such that there is a set C ✓ Y of car-
dinality dV C shattered by H. By convention, dV C =  1 for H = ;. Clearly, Y is always
shattered by H = 2Y . Thus the VC-dimension of the Boolean hypercube {+, }n itself is
4 Art Discrete Appl. Math. 4 (2021) #P1.09
n. Analogously, every subset Y 0 ✓ Y is shattered by 2Y
0
and thus the VC-dimension of a
sub-hypercube of dimensions |Y | = n0 is dV C = n0.
As noted in [14], the 1-point crossover recombination set R1(x, y) is an isometrically
embedded cycle C2t for t   2. It is not hard to check that dV C = 2 in this case. For
a partial cube G with d cuts the VC-dimension equals the dimension of the largest cube-
minor in G, i.e., the largest cardinality of a set of coordinates shattered by the set of all d
cuts of G. Here, a partial cube minor is either a contraction of cuts or the restriction to one
of its sides, i.e., a specialization of the standard notion of graph minors [17]. Moreover, the
cube-minor of a partial cube G is a graph isomorphic to a hypercube that can be obtained
from G by a series of contractions and restrictions. Note that contractions can be seen as
simply ignoring a coordinate.
Proposition 2.1. dV C(Rk(x, y)) =
(
k + 1 if d(x, y) > k
d(x, y) if d(x, y)  k
Proof. By Lemma 1.2 it suffices to consider Rk( n,+n). From the definition of k-point
crossover it straightforwardly follows that Rk(x, y) = {+, }n, when k = n   1, since
there is a break point between any two coordinates. Now suppose k < n  1. If the break
points are consecutive, i.e., ij = j for 1  j  k, then Rk(x, y) induces {+, }k+1 on
the first k + 1 coordinates. The same holds if the break points are not consecutive and we
contract consecutive coordinates j and j + 1 that do not have a break point between them.
On the other hand, with k break points we can only “crossover” at most k + 1 coordinates,
whence dV C(Rk(x, y))  k + 1.
3 Oriented matroids and 2-point recombination sets
Oriented matroids [1] are an axiomatic abstraction of geometric and topological struc-
tures including convex polytopes, vector configurations, (pseudo)hyperplane arrangements,
point configurations in the Euclidean space, directed graphs, and linear programs. They re-
flect properties such as linear dependencies, facial relationship, convexity, duality, and have
bearing on solutions of associated optimization problems. Among several equivalent ax-
iomatizations of oriented matroids, the face or covector axioms best captures the geometric
flavour and thus is the most convenient one for our purposes.
Let E be a finite set. A sign vector X on E is a vector (Xe : e 2 E) with coordinates
Xe 2 {+, 0, }. The support of a sign vector X is the set X = {e 2 E |Xe 6= 0}. The
composition X Y of two sign vectors X and Y is defined coordinate-wisely as (X Y )e =
Xe, if Xe 6= 0, and (X   Y )e = Ye otherwise. Their difference set is D(X,Y ) = {e 2
E |Xe =  Ye}. We denote by  the product (partial) order on { , 0,+}E implied by the
standard ordering   < 0 < + of signs.
An oriented matroid M is ordered pair (E,F) of a finite set E and a set of covectors
F ✓ {+, , 0}E satisfying, for all X,Y 2 F , the following (face or covector) axioms:
(F0) 0 = (0, 0, . . . , 0) 2 F .
(F1)  X 2 F .
(F2) X   Y 2 F .
(F3) There is Z 2 F with Ze = 0 for e 2 D(X,Y ) and Zf = (X   Y )f for f 2
E \D(X,Y ).
M. Changat et al.: Transit sets of two-point crossover 5
−++−
+++−
−+++ −−+−
++++
+−+− −+−+
++−+ +−−−
+−−+
+−++ −−−+
−−−−
−+−−
Figure 1: The rhombododecahedral graph R2(----,++++) (top) with the binary labeling
corresponding to the isometric embedding into 4-dimensional hypercube. Below we show
its big face lattice generated using SageMath (www.sagemath.org).
Consider a subspace V ✓ R|E|, define, for every v 2 V , its sign vector s(v) coordinate-
wise by se(v) = sgn(ve) for all e 2 E, and denote by F the set of all sign vectors of V .
Oriented matroids obtained from a vector space in this manner are called representable or
linear.
The set C ⇢ F of cocircuits or vertices of M consists of the non-zero covectors that are
minimal with respect to the partial order . The set T ⇢ F of topes of M comprises the
covectors that are maximal with respect to . The cocircuits determine the set of covectors:
every covector X 2 F \{0} has a representation of the form X = V1  V2   . . . Vk, where
V1, V2, . . . Vk are cocircuits, and V1, V2, . . . Vk  X . Similarly, the topes determine the
oriented matroid: F = {X 2 {+, , 0}E | 8T 2 T : X   T 2 T }.
M = (E,F) is uniform of rank r if |X| = r + 1 for all cocircuits. The big face lattice
bF is a lattice obtained by adding the unique maximal element b1 to the partial order  on
F . The rank of a covector X is defined as its height in bF . The rank rk(M) of M is the
maximal rank of its covectors. The corank of M is |E|  rk(M).
As an example consider R2(x, y) with d(x, y) = 4. It can be verified that the elements
of R2(----,++++) are exactly the topes of the oriented matroid corresponding to the
Rhombododecahedron. It is shown together with its big face lattice in Figure 1. This
observation can be generalized with the help of the following result:
Proposition 3.1 ([13]). A set T ✓ {+, }X of VC-dimension d is the set of topes of a
uniform oriented matroid M on X if and only if T =  T and |T | = 2 d 1(|X|  1).
By Proposition 2.1, Equ.(1.1), and Theorem 3.1, this immediately implies
Theorem 3.2. For x, y 2 {+, }X , with d(x, y) = |X| = n the elements of Rk(x, y)
form the set of topes of a uniform oriented matroid M on X with VC-dimension dV C =
rk(M) = k + 1.
6 Art Discrete Appl. Math. 4 (2021) #P1.09
+++++
−−−−+−+−−−
++++− +−+++
+−−++
−−−+++−−−+
−−+++
−−++−
++−−+
++−−−−++−−
+++−− −+++−
+++−+ −++++
+−−−−−−+−− −−−+−
++−++
−−−−−
Figure 2: The transit graph R2(-----,+++++).
Since many of the known results on oriented matroids depend on the corank, we note
that Rk(x, y) has corank n  k   1.
One of the cornerstones of the theory of oriented matroids is the Topological Represen-
tation Theorem, which connects oriented matroids with pseudosphere arrangements, see
Appendix A for detailed definitions. Together with Theorem 3.2, it immediately implies
the following topological characterization of the recombination sets of k-point crossover:
Theorem 3.3. For x, y 2 {+, }X , with d(x, y) = |X| = n, the recombination set
Rk(x, y) can be topologically represented by a pseudosphere arrangement of dimension k,
where the minimal elements in the big face lattice correspond to the intersections of exactly
k pseudospheres, and there are 2
  n
k 1
 
such intersections.
The significance of this result is that it provides a representation of crossover opera-
tors in terms of topological objects. As an illustration of the usefulness of Theorem 3.3,
we now turn to a full characterization of the transit graphs of 2-point crossover opera-
tors. The smallest non-trivial examples are the graphs R2(----,++++) in Figure 1 and
R2(-----,+++++) in Figure 2.
Theorem 3.4. R2(a, b) with d(a, b) = t > 3 induces antipodal planar quandrangulation,
that is, a partial cube of diameter t with t2 t+2 vertices, 2t2 2t edges, t2 t quadrangles,
and all cuts of size 2t  2.
Proof. Let |V |, |E|, |Q| and |C| denote the number of vertices, edges, 4-faces, and edges
of a cut, respectively. From the definition of crossover operator, we can arbitrarily permute
coordinates, hence it follows that each cut has the same number of edges, this justifies that
we study |C|. From Theorem 3.2 it follows that vertices of R2(a, b) form the set of topes
of a uniform oriented matroid of rank 3 and corank t   3. As shown by [10] and in the
book by [1], rank 3 oriented matroids can be represented by pseudocircle arrangement on
S2. The corresponding tope graph is therefore planar. Hence R2(a, b) induces in particular
a planar antipodal partial cube. Corank t 3 implies that each intersection of pseudocurves
is the intersection of exactly two of them. Hence all faces of the dual – the tope graph – are
4-cycles, therefore R2(a, b) induces planar quadrangulation. Moreover, each intersection
M. Changat et al.: Transit sets of two-point crossover 7
Figure 3: Topological representation of rhombododecahedron (l.h.s.) in terms of its pseu-
docircle arrangement (doted curves) and the corresponding hyperplane arrangement (r.h.s.).
of two pseudocircles corresponds to cocircuit. In uniform oriented matroid of corank t  3
there are exactly 2
  t
t 2
 
cocircuits, which correspond to the 4-cycles in the dual graph.
Quadrangulations are maximal planar bipartite graphs – no edge can be added so that
graph remains planar and bipartite. Using Euler formula for planar graphs [20], we obtain
|E| = 2|V |   4. Equ.(1.1) furthermore, implies |E| = 2t2   2t and thus |C| = |E|/t =
2t  2.
As an example, Figure 3 shows the pseudocircle arrangement and the equivalent hyper-
plane arrangement of transit graph R2(----,++++) of Figure 1.
In order to get a better intuition on the structure of the 2-point crossover graphs we
derive their degree sequence.
Theorem 3.5. The degree sequence of R2(a, b) with t := d(a, b) > 3 equals
(t, t, 4, . . . , 4, 3, . . . , 3) with t2   3t vertices of degree 4 and 2t vertices of degree 3.
Proof. W.l.o.g., let a = 0 . . . 0 and b = 1 . . . 1. For any vertex c = x . . . xyx . . . x, x, y 2
{0, 1} we have that c 2 R2(a, b), hence deg(a) = deg(b) = t. Let c 2 R2(a, b) \ {a, b}.
Then we have two cases:
Case 1. c = xx . . . xxyy . . . yy and {x, y} = {0, 1}. Then c has at most four neighbors
in R2(a, b): c1 = yx . . . xxyy . . . yy, c2 = xx . . . xxyy . . . yx, c3 = xx . . . xyyy . . . yy
and c4 = xx . . . xxxy . . . yy. Since t > 3 it follows that c also has at least three neighbors
in R2(a, b).
Case 2. c = x . . . xxyy . . . yyxx . . . x and {x, y} = {0, 1}. Then c has at most four
neighbors in R2(a, b): c1 = x . . . xxxy . . . yyxx . . . x, c2 = x . . . xyyy . . . yyxx . . . x,
c3 = x . . . xxyy . . . yxxx . . . x, and c4 = x . . . xxyy . . . yyyx . . . x. Since t > 3 it follows
that c also has at least three neighbors in R2(a, b).
Let x3 and x4 denote the number of vertices of degree 3 and 4 respectively. By the
handshaking lemma 2|E| =
P
v2V (G) deg(v). Therefore, it follows from arguments above
8 Art Discrete Appl. Math. 4 (2021) #P1.09
and Theorem 3.4 that
4t2   4t = 2t+
X
v2V (G)\{a,b}
deg(v)
4t2   6t = 3x3 + 4x4
Theorem 3.4 also implies that t2   t = x3 + x4. Solving this system of linear equations
yields x3 = 2t and x4 = t2   3t.
4 Concluding remarks
The recombination sets of 1-point crossover operators form isometric cycles in hypercube.
The partial cubes corresponding to k-point crossover operators have a VC-dimension of
k+1 unless they are smaller sub-hypercubes. We have considered here the uniform oriented
matroids that correspond to the k-point crossover operators and used this connection to
characterize the partial cubes of 2-point recombination sets. It remains an open question
for future research whether the connection with oriented matroids and their topological
representations can be utilized to better understand the structure of k-point recombination
graphs.
ORCID iDs
Manoj Changat https://orcid.org/0000-0001-7257-6031
Prasanth G. Narasimha-Shenoi https://orcid.org/0000-0002-5850-5410
Matjaž Kovše https://orcid.org/0000-0001-9473-7545
Shilpa Mohandas https://orcid.org/0000-0003-3378-2339
Abisha Ramachandran https://orcid.org/0000-0003-2778-5584
Peter F. Stadler https://orcid.org/0000-0002-5016-5191
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10 Art Discrete Appl. Math. 4 (2021) #P1.09
Appendix A: Pseudosphere arrangements
Consider the d-dimensional sphere Sd in Rd+1 and the corresponding (d+1)-dimensional
ball Bd+1 = {(x1, . . . , xd+1) 2 Rd+1 |x21 + . . .+ x2d+1  1}, whose boundary surface is
Sd.
A pseudosphere S ⇢ Sd is a tame embedded (d   1)-dimensional sphere. Its comple-
ment in Bd consist of exactly two regions, hence S can be oriented, by labeling one region
by S+e and the other by S e . A pseudosphere arrangement S = {Se | e 2 E} in the Eu-
clidean space Rd is a collection of (d 1)-dimensional pseudospheres on the d-dimensional
unit sphere Sd, where the intersection of any number of spheres is again a sphere and the
intersection of an arbitrary collection of closed sides is either a sphere or a ball, i.e., for all
R ⇢ E holds
(i) SR = Sd \i2R Si is empty or homeomorphic to a sphere.
(ii) If e 2 E and SR 6⇢ Se then SR \ Se is a pseudosphere in SR, SR \ S+e 6= ; and
SR \ S e 6= ;.
For a pseudosphere arrangement S , the position vector  (x) of a point x 2 Sd is defined
as  (x)e = 0 for x 2 Se,  (x)e = + for x 2 S+e and  (x)e =   for x 2 S e . The set of
all position vectors of S is denoted by  (S). A famous theorem due to [9] establishes an
correspondence between oriented matroids and pseudosphere arrangement.
Topological Representation Theorem ([2, 9]). Let M = (E,F) be an oriented matroid
of rank d. Then there exists a pseudosphere arrangement S in Sd such that  (S) = F .
Conversely, if S is a pseudosphere arrangement in Sd, then (E, (S)) is an oriented ma-
troid of rank d.
A pseudosphere arrangement naturally induces a cell complex on Sd, whose partial
order of faces corresponds precisely to the partial order  on covectors of the corresponding
oriented matroid. This fact served as motivation for the concept of covectors in the theory
of oriented matroids.
ISSN 2590-9770
The Art of Discrete and Applied Mathematics 4 (2021) #P1.10
https://doi.org/10.26493/2590-9770.1338.0b2
(Also available at http://adam-journal.eu)
On the equivalence between a conjecture of
Babai-Godsil and a conjecture of Xu concerning
the enumeration of Cayley graphs
Pablo Spiga
Pablo Spiga,Dipartimento di Matematica e Applicazioni, University of Milano-Bicocca
Via Cozzi 55, 20125 Milano, Italy
Received 20 November 2019, accepted 12 October 2020, published online 01 April 2021
Abstract
In this paper we show that two distinct conjectures, the first proposed by Babai and
Godsil in 1982 and the second proposed by Xu in 1998, concerning the asymptotic enu-
meration of Cayley graphs are in fact equivalent. This result follows from a more general
theorem concerning the asymptotic enumeration of a certain family of Cayley graphs.
Keywords: Regular representation, Cayley graph, automorphism group, asymptotic enumeration,
graphical regular representation, GRR, normal Cayley graph, Babai-Godsil conjecture, Xu conjec-
ture.
Math. Subj. Class.: 05C25, 05C30, 20B25, 20B15
1 Introduction
All digraphs and groups considered in this paper are finite. A digraph   is an ordered
pair (V,A) where the vertex-set V is a finite non-empty set and the arc-set A ✓ V ⇥ V
is a binary relation on V . The elements of V and A are called vertices and arcs of  ,
respectively. An automorphism of   is a permutation   of V with A  = A, that is,
(x , y ) 2 A for every (x, y) 2 A. Let R be a group and let S be a subset of R. The
Cayley digraph on R with connection set S (which we denote by  (R,S)) is the digraph
with vertex-set R and with (g, h) being an arc if and only if hg 1 2 S. The group R acts
regularly as a group of automorphisms of  (R,S) by right multiplication and hence R 
Aut( (R,S)). When R = Aut( (R,S)), the digraph   is called a DRR (for digraphical
regular representation). Babai and Godsil made the following conjecture.
E-mail address: pablo.spiga@unimib.it (Pablo Spiga)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Art Discrete Appl. Math. 4 (2021) #P1.10
Conjecture 1.1 ([6, Conjecture 3.13], [2]). Let R be a group of order r. The proportion of
subsets S of R such that  (R,S) is a DRR goes to 1 as r ! 1. More precisely,
lim
r!1
min
⇢
|{S ✓ R : Aut( (R,S)) = R}|
2r
: |R| = r
 
= 1.
This conjecture has been recently proved in [12].
This paper is the first step for proving yet another conjecture of Babai and Godsil con-
cerning the enumeration of Cayley graphs. A Cayley graph over R is a Cayley digraph
 (R,S) whose binary relation {(g, h) 2 R ⇥ R | gh 1 2 S} defining the arc-set of
 (R,S) is symmetric. (Incidentally, the binary relation {(g, h) 2 R ⇥ R | gh 1 2 S} is
riflexive if and only if S contains the identity element of G.) In terms of the connection set
S ✓ R,  (R,S) is a Cayley graph if and only if S = S 1, where S 1 := {s 1 | s 2 S}.
Given a subset S of R, we say that S is inverse-closed if S = S 1, that is,  (R,S) is
undirected, which in turn means that  (R,S) is a Cayley graph. When R = Aut( (R,S))
and S is inverse-closed, the graph   is called a GRR (for graphical regular representation).
While the number of Cayley digraphs on R is 2|R|, which is a number that depends
on the cardinality of R only, the number of undirected Cayley graphs on R is 2
|R|+|I(R)|
2
(see Lemma 2.2), where I(R) := {◆ 2 R | ◆2 = 1}, and hence depends on the algebraic
structure of R.
Although the difference between Cayley digraphs and Cayley graphs seems only minor
and to some extent only aesthetic, the behaviour between these two classes of combinatorial
objects with respect to their automorphisms can be dramatically different. For instance, it
was proved by Babai [1, Theorem 2.1] that, except for
Q8, C2 ⇥ C2, C2 ⇥ C2 ⇥ C2, C2 ⇥ C2 ⇥ C2 ⇥ C2 and C3 ⇥ C3,
every finite group R admits a DRR. Borrowing a phrase which I once heard from Tom
Tucker: “Besides some low level noise, every finite group admits a DRR”. The analogue
for GRRs is not the same. Indeed, it turns out that there are two (and only two) infinite
families of groups that do no admit GRRs. The first family consists of abelian groups of
exponent greater than two. If R is such a group and ◆ is the automorphism of R mapping
every element to its inverse, then every Cayley graph on R admits R o h◆i as a group of
automorphisms. Since R has exponent greater than 2, ◆ 6= 1 and hence no Cayley graph on
R is a GRR. The other family of groups that do not admit GRRs are the generalised dicyclic
groups, see [14, Definition 1.1] for a definition and also Definition 2.4 below. These two
families were discovered by Mark Watkins [19].
It was proved by Godsil [5] that abelian groups of exponent greater than 2 and gener-
alised dicyclic groups are the only two infinite families of groups that do not admit GRRs.
(A lot of papers have been published for determining those groups admitting a GRR, and
some of the most influential works along the way appeared in [7, 8, 9, 15, 16, 20].) The
stronger Conjecture 1.2 was made (at various times) by Babai, Godsil, Imrich and Lovász.
Conjecture 1.2 (see [2, Conjecture 2.1] and [6, Conjecture 3.13]). Let R be a group of
order r which is neither generalized dicyclic nor abelian of exponent greater than 2. The
proportion of inverse-closed subsets S of R such that  (R,S) is a GRR goes to 1 as
r ! 1. More precisely,
lim
r!1
min
⇢
|{S ✓ R : Aut( (R,S)) = R}|
2c(R)
: R admits a GRR and |R| = r
 
= 1.
P. Spiga: On the equivalence between a conjecture of Babai-Godsil and a conjecture of Xu 3
This conjecture is open at the moment and some of the techniques developed in [12]
for dealing with digraphs are not suited for dealing with undirected graphs.
The scope of this paper is twofold. Broadly speaking, we aim to start a long process
where we try to generalize and adapt the results obtained in [12] for eventually dealing
with undirected graphs and proving Conjecture 1.2. Given an inverse-closed subset S of
R, we let A := Aut( (R,S)). Now, the set S fails to give rise to a GRR essentially for
two different reasons.
1. There are non-identity group automorphisms of R leaving the set S invariant. This
case arises when NA(R) > R (this is the typical obstruction and we have encoun-
tered this obstruction already when we briefly discussed abelian groups of exponent
greater than 2).
2. The only group automorphism of R leaving the set S invariant is the identity and
there are some automorphisms of  (R,S) not lying in R. This case arises when
NA(R) = R and A > R : this obstruction is somehow mysterious and much harder
to analyze.
These two obstructions are clear (if not obvious) to readers familiar with the enumeration
problem of Cayley graphs [12] and in particular to readers familiar with [2]. Actually the
same obstructions arise in the enumeration problem of other types of Cayley graphs, for
instance in the asymptotic enumeration of DFRs [17] and GFRs [4, 18] and in the recent
solution of the GFR conjecture [18]. We start this process by dealing with the first natural
obstruction for the existence of GRRs.1
Theorem 1.3. Let R be a group of order r which is neither generalized dicyclic nor abelian
of exponent greater than 2. The proportion of inverse-closed subsets S of R such that
NAut( (R,S))(R) > R goes to 0 as r ! 1.
We observe that in Proposition 2.9 we have a quantified version of Theorem 1.3. More-
over, in Lemma 2.8 we have a more technical version of Theorem 1.3 which includes also
generalized dicyclic groups and abelian groups of exponent greater than 2. These two more
technical results are in our opinion needed to follow the footsteps of the argument in [12]
for the asymptotic enumeration of Cayley digraphs.
The second scope of this paper is to prove that a famous conjecture of Xu on the asymp-
totic enumeration of normal Cayley graphs is actually equivalent to Conjecture 1.2. A
Cayley (di)graph   on R is said to be a normal Cayley (di)graph on R if the regular rep-
resentation of R is normal in Aut( ), that is, R E Aut( ). Clearly, every DRR and every
GRR   on R is a normal Cayley (di)graph because R = Aut( ). Xu has conjectured that
almost all Cayley (di)graphs on R are normal Cayley (di)graphs on R. Indeed, Xu in [21]
has posed the following two conjectures.
Conjecture 1.4 (see [21]). The minimum, over all groups R of order r, of the proportion
of subsets S of R such that  (R,S) is a normal Cayley graph tends to 1 as r ! 1. More
precisely,
lim
r!1
min
⇢
|{S ✓ R : REAut( (R,S))}|
2r
: |R| = r
 
= 1.
1During the refereeing process of this paper a substantial step towards the second obstruction has been obtained
in [13]
4 Art Discrete Appl. Math. 4 (2021) #P1.10
Conjecture 1.5 (see [21]). The minimum, over all groups R of order r, of the proportion
of inverse-closed subsets S of R such that  (R,S) is a normal Cayley graph tends to 1 as
r ! 1. More precisely,
lim
r!1
min
⇢
|{S ✓ R : REAut( (R,S))}|
2c(R)
: |R| = r
 
= 1.
Conjecture 1.4 was shown to be true in [12] by proving the stronger Conjecture 1.1.
The veracity of Conjecture 1.5 when R is an abelian group and when R is a dicyclic group
was proved in [3, 14]. In this paper we show that Conjecture 1.2 and Conjecture 1.5 are
actually equivalent.
Theorem 1.6. Conjecture 1.2 holds true if and only if Conjecture 1.5 holds true.
2 Group automorphisms
Definition 2.1. Given a finite group R and x 2 R, we let o(x) denote the order of the
element x and we let I(R) := {x 2 R | o(x)  2} be the set of elements of R having
order at most 2. We let c(R) denote the fraction (|R|+ |I(R)|)/2, that is,
c(R) =
|R|+ |I(R)|
2
.
Given a subset X of R, we write I(X) := X\I(R). Finally, we denote by Z(R) the centre
of R.
Lemma 2.2. Let R be a finite group. The number of inverse-closed subsets S of R is 2c(R).
Proof. Given an arbitrary inverse-closed subset S of R, S \ I(R) is an arbitrary subset of
I(R) whereas in S \ (R \ I(R)) the elements come in pairs, where each element is paired
up to its inverse. Thus the number of inverse-closed subsets of R is
2|I(R)| · 2
|R\I(R)|
2 = 2c(R).
Definition 2.3. Let R be a finite group. Given an automorphism ' of R, we set
CR(') := {x 2 R | x' = x},
CR(')
inv := {x 2 R | x' = x 1}.
Observe that, when ' = idR is the identity automorphism of R, CR(')inv = I(R).
Given x 2 R, we denote by ◆x : R ! R the inner automorphism of R induced by
x, that is, m◆x = xmx 1, for every m 2 R. (Usually, the automorphism ◆x is defined
by m 7! m◆x = x 1mx, however for our application it is more convenient to define ◆x
by m 7! m◆x = xmx 1.) When A E R, we still denote by ◆x the restriction to A of the
automorphism ◆x, this makes the notation not too cumbersome to use and hopefully will
cause no confusion.
Finally, we let ◆ : R ! R be the permutation defined by x◆ = x 1, for every x 2 R.
In particular, when R is abelian, ◆ is an automorphism of R. Furthermore, ◆ = idR if and
only if R is an abelian group of exponent at most 2.
P. Spiga: On the equivalence between a conjecture of Babai-Godsil and a conjecture of Xu 5
Definition 2.4. Let A be an abelian group of even order and of exponent greater than 2,
and let y be an involution of A. The generalised dicyclic group Dic(A, y, x) is the group
hA, x | x2 = y, ax = a 1, 8a 2 Ai. A group is called generalised dicyclic if it is
isomorphic to some Dic(A, y, x). When A is cyclic, Dic(A, y, x) is called a dicyclic or
generalised quaternion group.
We let ◆̄A : Dic(A, y, x) ! Dic(A, y, x) be the mapping defined by (ax)◆̄A = ax 1
and a◆̄A = a, for every a 2 A. In particular, ◆̄A is an automorphism of Dic(A, y, x).
The role of the label “A” in ◆̄A seems unnecessary, however we use this label to stress
one important fact. An abstract group R might be isomorphic to Dic(A, y, x), for various
choices of A. Therefore, since the automorphism ◆̄A depends on A and since we might
have more than one choice of A, we prefer a notation that emphasizes this fact.
Lemma 2.5. Let R be a finite group and let ' be an automorphism of R with |R :
CR(')| = 2. Then one of the following holds:
1. 14 (|R|+ |I(R)|+ |CR(')|+ |CR(')
inv|)  c(R)  |R|32 ,
2. R is generalized dicyclic over the abelian group CR(') and ' = ◆̄CR('),
3. R is abelian of exponent greater than 2 and ' = ◆.
Proof. For simplicity, we let A := CR(') and we let o denote the left-hand side in (1).
Suppose that CR(')inv ✓ A. Then
CR(')
inv = CR(')
inv \A = {a 2 A | a' = a 1} = CA(')inv.
Since A = CR('), we have a' = a for every a 2 A and hence
CR(')
inv = CA(')
inv = CA(idA)
inv
.
Clearly, a 2 CA(idA)inv if and only if a = a 1, that is, a 2 I(A). Therefore
CR(')
inv = CA(')
inv = CA(idA)
inv = I(A).
Thus
o =
1
4
✓
|R|+ |I(R)|+ |R|
2
+ |I(A)|
◆
 1
4
✓
3
2
|R|+ 2|I(R)|
◆
=
|R|+ |I(R)|
2
  |R|
8
= c(R)  |R|
8
,
and (1) holds in this case.
Suppose that CR(')inv * A. In particular, there exists x 2 R \A with x' = x 1. As
|R : A| = 2, we have R = A [ Ax. For every a 2 A, since ' is an automorphism of R
fixing point-wise A and since xax 1 2 A, we deduce
xax
 1 = (xax 1)' = x'a'(x 1)' = x 1ax
6 Art Discrete Appl. Math. 4 (2021) #P1.10
and hence x2a = ax2, that is, x2 2 Z(hA, xi) = Z(R). As x2 2 A, we have x2 =
(x2)' = (x')2 = (x 1)2 = x 2, that is, x4 = 1. Summing up,
x
2 2 Z(R), x4 = 1. (2.1)
Now, let y 2 CR(')inv \ A. Then, y = ax, for some a 2 A. Moreover, y' = y 1 =
(ax) 1 = x 1a 1 and y' = (ax)' = a'x' = ax 1. Thus
x
 1
a
 1 = ax 1,
that is, xax 1 = a 1. Recall that ◆x : A ! A is the restriction to the normal subgroup A
of the inner automorphism of R determined by x, that is, a◆x = xax 1, for every a 2 A.
We have shown that CR(')inv \A = CA(◆x)invx. As CR(')inv \A = I(A), we get
CR(')
inv = I(A) [CA(◆x)invx (2.2)
and |CR(')inv| = |I(A)|+ |CA(◆x)inv|.
Suppose that |CA(◆x)inv|  3|A|/4. Thus, by (2.2), we have
o =
1
4
✓
3
2
|R|+ |I(R)|+ |I(A)|+ |CA(◆x)inv|
◆
 1
4
✓
3
2
|R|+ 2|I(R)|+ 3|A|
4
◆
=
1
4
✓
3
2
|R|+ 2|I(R)|+ 3|R|
8
◆
=
|R|+ |I(R)|
2
  |R|
32
= c(R)  |R|
32
,
and (1) holds in this case.
Suppose that |CA(◆x)inv| > 3|A|/4, that is, the automorphism ◆x of A inverts more
than 3/4 of its elements. By a result of Miller [11], A is abelian. Since A is abelian, it
is easy to verify that CA(◆x)inv is a subgroup of A. As |CA(◆x)inv| > 3|A|/4, we get
CA(◆x)inv = A and ◆x acts on A inverting each of its elements. From (2.2), we have
CR(')
inv = I(A) [Ax. (2.3)
If I(R) ✓ I(A), then no element in Ax is an involution and hence x has order 4 from (2.1).
When A has exponent greater than 2, we deduce R ⇠= Dic(A, x2, x) is a generalized di-
cyclic group over A, ' = ◆̄A and (2) holds in this case. When A has exponent at most 2,
we have I(A) = A and ' = ◆. Hence I(R) = A, R is an abelian group of exponent greater
than 2 and (3) holds in this case. Therefore, we may suppose I(R) * I(A).
Let x0 2 I(R) \ A. Then, x0 = ax, for some a 2 A. Then 1 = x02 = (ax)2 =
axax = a(xax 1)x2 = aa 1x2 = x2 and hence x2 = 1. Now, for every b 2 A, we
have (bx)2 = bxbx = b(xbx 1) = bb 1 = 1. This shows I(R) \ A = Ax. Therefore,
P. Spiga: On the equivalence between a conjecture of Babai-Godsil and a conjecture of Xu 7
I(R) = I(A) [Ax and hence I(R) = CR(')inv from (2.3). We deduce
o =
1
4
✓
3|R|
4
+ |I(R)|+ |CR(')inv|
◆
=
1
4
✓
3
2
|R|+ 2|I(R)|
◆
=
|R|+ |I(R)|
2
  |R|
8
= c(R)  |R|
8
,
and (1) holds in this case.
Lemma 2.6. Let R be a finite group and let ' be an automorphism of R with |R :
CR(')| = 3. Then one of the following holds:
1. 14 (|R|+ |I(R)|+ |CR(')|+ |CR(')
inv|)  c(R)  |R|96 ,
2. R is abelian of exponent greater than 2 and ' = ◆.
Proof. For simplicity, we let A := CR(') and we let o denote the left-hand side in (1). As
|R : A| = 3, we may write R = A [Ax [Ax0, for some x, x0 2 R.
Suppose that CR(')inv ✓ A [Ay, for some y 2 {x, x0}. Then
CR(')
inv = (CR(')
inv\A)[(CR(')inv\Ay) = I(A)[(CR(')inv\Ay) ✓ I(A)[Ay,
because ' fixes each element of A. Thus |CR(')inv|  |I(R)|+ |A| and
o  1
4
✓
4
3
|R|+ |I(R)|+ |I(R)|+ |A|
◆
 1
4
✓
4
3
|R|+ 2|I(R)|+ |R|
3
◆
=
|R|+ |I(R)|
2
  |R|
12
= c(R)  |R|
12
,
and (1) holds in this case.
Therefore we may suppose that CR(')inv \ Ax 6= ; and CR(')inv \ Ax0 6= ;. In
particular, replacing x and x0 if necessary, we may suppose that x, x0 2 CR(')inv, that is,
x
' = x 1 and x0' = x0 1.
CASE: AER.
As R/A is cyclic of order 3, we may assume that x0 = x 1 and that x has odd order. For
every a 2 A, we have xax 1 2 A and hence
xax
 1 = (xax 1)' = x'a'(x 1)' = x 1ax,
that is, x2a = ax2. Therefore x2 2 Z(hx,Ai) = Z(R). As x has odd order, we deduce
x 2 Z(R). From this it is easy to deduce that
CR(')
inv = I(A) [ I(A)x [ I(A)x 1. (2.4)
8 Art Discrete Appl. Math. 4 (2021) #P1.10
Assume that |I(A)|  3|A|/4. Thus, by (2.4), we have
o =
1
4
✓
4
3
|R|+ |I(R)|+ |I(A)|+ |I(A)|+ |I(A)|
◆
 1
4
✓
4
3
|R|+ 2|I(R)|+ 2|I(A)|
◆
 1
4
✓
4
3
|R|+ 2|I(R)|+ 23|A|
4
◆
=
1
4
✓
4
3
|R|+ 2|I(R)|+ |R|
2
◆
=
1
4
✓
11
6
|R|+ 2|I(R)|
◆
=
|R|+ |I(R)|
2
  |R|
24
= c(R)  |R|
24
,
and (1) holds in this case.
Assume that |I(A)| > 3|A|/4. By [11], A is abelian. Thus I(A) is a subgroup of A
with |I(A)| > 3|A|/4. It follows that A is an elementary abelian 2-group. As x 2 Z(R),
we deduce that R is abelian and ' = ◆; thus (2) holds in this case.
CASE: A IS NOT NORMAL IN R.
Let K be the core of A in R. Observe that the group R acts on the right cosets of A in
R. As |R : A| = 3, this action gives rise to a transitive permutation representation of R
inside the symmetric group of degree 3. The kernel of this permutation representation is,
by definition, K and hence R/K is isomorphic to a subgroup of the symmetric group of
degree 3. Therefore |R : K|  3! = 6. Since by hypothesis A is not normal in R, we
deduce that K is a proper subgroup of A. As |R : A| = 3 and |R : K|  6, we get that
|R : K| = 6 and that R/K is isomorphic to the dihedral group of order 6.
Suppose that CR(')inv \ Ky = ;, for some y 2 R \ A. As R \ A is the union of
four K-cosets and as CR(')inv \Ky = ;, we deduce |CR(')inv \ (R \ A)|  3|K|. As
CR(')inv \A = I(A), we get |CR(')inv| = |CR(')inv \A|+ |CR(')inv \ (R \A)| 
|I(A)|+ 3|K| and hence
o  1
4
✓
4
3
|R|+ |I(R)|+ |I(A)|+ 3|K|
◆
 1
4
✓
4
3
|R|+ 2|I(R)|+ 3 |R|
6
◆
=
1
4
✓
11
6
|R|+ 2|I(R)|
◆
=
|R|+ |I(R)|
2
  |R|
24
= c(R)  |R|
24
,
and (1) holds in this case. Thus we may suppose CR(')inv\Ky 6= ;, for every y 2 R\A.
Let x1, x2, x3, x4 2 R \ A with R = A [ Kx1 [ Kx2 [ Kx3 [ Kx4 and with
x1, x2, x3, x4 2 CR(')inv. As usual we denote by ◆xi : K ! K the automorphism
of K defined by k◆xi = xikx 1i , for every k 2 K. For each i 2 {1, . . . , 4}, let y 2
CR(')inv\Kxi. Then y = kxi, for some k 2 K and hence x 1i k 1 = (kxi) 1 = y 1 =
y
' = (kxi)' = k'x
'
i
= kx 1
i
, that is, xikx 1i = k
 1 and k 2 CK(◆xi)inv. This shows
CR(')
inv = I(A) [CK(◆x1)invx1 [CK(◆x2)invx2 [CK(◆x3)invx3 [CK(◆x4)invx4.
(2.5)
Suppose that |CK(◆xi)inv|  3|K|/4, for some i 2 {1, 2, 3, 4}. Then
|CR(')inv|  |I(A)|+ 3|K|+
3|K|
4
= |I(A)|+ 15|K|
4
= |I(A)|+ 5|R|
8
.
P. Spiga: On the equivalence between a conjecture of Babai-Godsil and a conjecture of Xu 9
Thus
o  1
4
✓
4
3
|R|+ |I(R)|+ |I(A)|+ 5|R|
8
◆
 1
4
✓
47
24
|R|+ 2|I(R)|
◆
=
|R|+ |I(R)|
2
  |R|
96
= c(R)  |R|
96
,
and (1) holds in this case. Therefore, we may suppose that |CK(◆xi)inv| > 3|K|/4, for
each i 2 {1, 2, 3, 4}. The work of Miller [11] shows that K is abelian and that, for ev-
ery i 2 {1, 2, 3, 4}, xi acts by conjugation on K by inverting each of its elements. In
particular, (2.5) becomes
CR(')
inv = I(A) [ (R \A). (2.6)
As R/K is isomorphic to the dihedral group of order 6, we deduce that there exist
i, j, k 2 {1, 2, 3, 4} with xixj 2 Kxk. From the previous paragraph, xi, xj and xk act by
conjugation on K by inverting each of its elements. Therefore, for every y 2 K, we have
y
 1 = yxk = yxixj = (yxi)xj = (y 1)xj = y,
that is, y2 = 1. This yields that K is an elementary abelian 2-group and hence K ✓ I(A).
Eq (2.6) gives CR(')inv ◆ K [ (R \ A) and hence |CR(')inv|   |K| + |R \ A| =
5|R|/6 > 3|R|/4. Again, from the work of Miller [11], we deduce that R is abelian and
' = ◆, and (2) holds in this case.
Before proving the main step towards the proof of Theorem 1.3, we need a preliminary
observation.
Lemma 2.7. Let ' be an automorphism of a finite group R and let
 :=
|CR(')inv|
|R| .
If 12 <  < 1, then there exists a positive integer q   2 with  =
q+1
2q . In particular, if
2
3 < , then  =
3
4 and there exists an abelian subgroup A of R such that |R : A| = |A :
CA(x)| = 2 for every x 2 R \A.
Proof. The first assertion follows at once from the classification of Liebeck and MacHale [10,
Structure Theorem, page 61] of the finite groups admitting an automorphism inverting more
than half of the elements. (Actually, the first statement of this lemma can also be found in
the third paragraph of the introductory section in [10].)
Suppose now that 23 < . Then, from the first statement, there exists q 2 N with q   2
and  = q+12q . Now,
2
3 <
q+1
2q only when q = 2; hence  =
3
4 . We now invoke once
again the work of Liebeck and MacHale. In [10, Structure Theorem, page 61], the finite
groups admitting an automorphism inverting more than half of the elements are partitioned
into three types. Namely, Type I*, Type II* and Type III*. It is readily seen that none
of the groups in Type II* or Type III* admits an automorphism ' with |CR(')
inv|
|R| =
3
4 .
Therefore, R is of Type I*, which means that there exists an abelian subgroup A with
|R : A| = |A : CA(x)|, for every x 2 R \A.
Lemma 2.8. Let R be a finite group and let ' be a non-identity automorphism of R. Then,
one of the following holds
10 Art Discrete Appl. Math. 4 (2021) #P1.10
1. the number of '-invariant inverse-closed subsets of R is at most 2c(R) 
|R|
96 ,
2. CR(') is abelian of exponent greater than 2 and has index 2 in R, R is a generalized
dicyclic group over CR(') and ' = ◆̄CR('),
3. R is abelian of exponent greater than 2 and ' = ◆.
Proof. In the first part of the proof, we establish the result when ' has order p, where p is
a prime number.
Recall that ◆ : R ! R is the permutation of R defined by x◆ = x 1, for every x 2 R.
Let H := h◆,'i  Sym(R). Clearly, the number of '-invariant inverse-closed subsets of
R is 2o, where o is the number of H-orbits, that is, o is the number of orbits of H in its
action on R. From the orbit-counting lemma, we have
o =
1
|H|
X
h2H
|FixR(h)|, (2.7)
where FixR(h) := {x 2 R | xh = x} is the fixed-point set of h in its action on R.
For every x 2 R, we have x◆' = (x 1)' = (x') 1 = x'◆ and hence ◆' = '◆.
Therefore H is an abelian group. Moreover, FixR(◆) = I(R) and FixR('`) = CR(') for
every ` 2 {1, . . . , p  1}.
Suppose R is abelian of exponent at most 2. As R has exponent at most 2, ◆ is the
identity permutation and hence H = h'i is cyclic of prime order p. From (2.7) and from
the fact that |CR(')|  |R|/2, we obtain
o =
1
p
(|R|+(p 1)|CR(')|) 
1
p
✓
|R|+ (p  1) |R|
2
◆
=
(p+ 1)|R|
2p
 3|R|
4
= |R|  |R|
4
and part (1) of the lemma holds in this case because c(R) = (|R| + |I(R)|)/2 = |R| and
thus
o = |R|  |R|
4
= c(R)  |R|
4
.
In particular, for the rest of the argument we suppose that R has exponent greater than
2. Thus H has order 2p.
CASE 1: p is odd.
As H is abelian of order 2p, we deduce that H is cyclic and FixR(◆'`) = CR('`) \
FixR(◆) = CR(') \ I(R), for every ` 2 {1, . . . , p   1}. Now, (2.7) yields (in the sec-
ond inequality we are using the fact that ' is not the identity automorphism and hence
|CR(')|  |R|/2)
o =
1
2p
(|R|+ |I(R)|+ (p  1)|CR(')|+ (p  1)|CR(') \ I(R)|)
 1
2p
(|R|+ |I(R)|+ (p  1)|CR(')|+ (p  1)|I(R)|)
=
|R|+ |I(R)|
2
  |R|
2
+
1
2p
(|R|+ (p  1)|CR(')|)
 c(R)  |R|
2
+
1
2p
✓
|R|+ (p  1) |R|
2
◆
= c(R)  |R|
✓
1
2
  p+ 1
4p
◆
= c(R)  |R|p  1
4p
 c(R)  |R|
6
.
P. Spiga: On the equivalence between a conjecture of Babai-Godsil and a conjecture of Xu 11
CASE 2: p = 2.
If ' = ◆, then R is an abelian group of exponent greater than 2 and we obtain that part (3)
holds in this case. Therefore, we may suppose that ' 6= ◆. As H = h', ◆i is abelian of
order 2p, we deduce that H = {idR, ◆,', ◆'} is elementary abelian of order 4. Moreover,
FixR(◆) = I(R), FixR(') = CR(') and FixR(◆') := CR(')inv. Thus
o =
1
4
 
|R|+ |I(R)|+ |CR(')|+ |CR(')inv|
 
.
From Lemmas 2.5 and 2.6, we may suppose that |R : CR(')|   4.
Miller [11] has shown that a non-identity automorphism of a non-abelian group inverts
at most 3|R|/4 elements. Therefore, |CR(')inv|  3|R|/4. Observe that the same in-
equality holds when R is abelian because CR(')inv is a proper subgroup of R and hence
|CR(')inv|  |R|/2  3|R|/4. In particular, if |R : CR(')|   5, then we deduce
o =
1
4
✓
|R|+ |I(R)|+ |R|
5
+
3|R|
4
◆
=
1
4
✓
39
20
|R|+ |I(R)|
◆
 |R|+ |I(R)|
2
  |R|
80
= c(R)  |R|
80
.
For the rest of the argument we may suppose that |R : CR(')|  4 and hence |R :
CR(')| = 4. Therefore
o =
1
4
✓
5|R|
4
+ |I(R)|+ |CR(')inv|
◆
. (2.8)
Assume |CR(')inv|  2|R|/3. Then, from (2.8), we get
o =
1
4
✓
5|R|
4
+ |I(R)|+ 2|R|
3
◆
=
1
4
✓
23
12
|R|+ |I(R)|
◆
 |R|+ |I(R)|
2
  |R|
48
= c(R)  |R|
48
.
Therefore, we may assume that |CR(')inv| > 2|R|/3.
As 2|R|/3 < |CR(')inv|  3|R|/4, from Lemma 2.7 we deduce that
|CR(')inv| =
3|R|
4
and that R contains an abelian subgroup A with |R : A| = |A : CA(x)| = 2, for every
x 2 R \A.
Suppose that A is not '-invariant. Since ' has order p = 2, A \ A' has index 4 in R
and is '-invariant. Observe that R/(A \ A') is an elementary abelian 2-group of order 4.
Let T be the index 2 subgroup of R containing A \A' and with A 6= T 6= A'. We have
CR(')
inv = (CR(')
inv \A) [ (CR(')inv \A') [ (CR(')inv \ T ).
12 Art Discrete Appl. Math. 4 (2021) #P1.10
Let a 2 CR(')inv \ A. Then a 1 = a' 2 A' \ A and hence CR(')inv \ A =
CA\A'(')inv and (similarly) CR(')inv \A' = CA\A'(')inv. Therefore
CR(')
inv = CA\A'(')
inv [ (CR(')inv \ T ).
We deduce
|CR(')inv| = |CA\A'(')inv|+ |CR(')inv \ (T \ (A \A'))|
 |A \A'|+ (|T |  |A \A'|)
= |T | = |R|
2
;
however this contradicts |CR(')inv| = 3|R|/4. Thus A is '-invariant.
CASE 2.1: ' inverts each element in A, that is, a' = a 1, for every a 2 A.
As |CR(')inv| = 3|R|/4 > |R|/2 = |A|, there exists x 2 R \ A with x' = x 1. It
follows that CR(')inv = A [CA(x)x and hence
|CR(')inv| = |A|+
|A|
2
. (2.9)
A computation gives CR(') = I(A) [ {ax | a 2 A, a2 = x 2}. Let a, b 2 A with the
property that a2 = x 2 = b2. Then (ab 1)2 = a2b 2 = x 2x2 = 1. This shows that
either {ax | a 2 A, a2 = x 2} is the empty set or {ax | a 2 A, a2 = x 2} = {bāx |
b 2 I(A)}, where ā 2 A is a fixed element with ā2 = x 2. In particular, |CR(')| 2
{|I(A)|, 2|I(A)|}. As |R : CR(')| = 4, we deduce that either |A : I(A)| = 2 and
{ax | a 2 A, a2 = x 2} = ;, or |A : I(A)| = 4 and {ax | a 2 A, a2 = x 2} 6= ;. In the
first case, from (2.9), we have
|CR(')inv| = |A|+ |A|/2 = |A|+ |I(A)| 
|R|
2
+ |I(R)|.
Thus
o  1
4
✓
5
4
|R|+ |I(R)|+ |R|
2
+ |I(R)|
◆
 1
4
✓
7
4
|R|+ 2|I(R)|
◆
=
|R|+ |I(R)|
2
  |R|
16
= c(R)  |R|
16
.
In the second case, from (2.9), we have
|CR(')inv| = |A|+ |A|/2 = |A|+ 2|I(A)|
=
|R|
2
+ |I(A)|+ |I(A)| = |R|
2
+
|R|
8
+ |I(A)|
 5|R|
8
+ |I(R)|.
Thus
o  1
4
✓
5
4
|R|+ |I(R)|+ 5|R|
8
+ |I(R)|
◆
=
1
4
✓
15
8
|R|+ 2|I(R)|
◆
=
|R|+ |I(R)|
2
  |R|
32
= c(R)  |R|
32
.
P. Spiga: On the equivalence between a conjecture of Babai-Godsil and a conjecture of Xu 13
CASE 2.2: ' does not invert each element in R \A.
Observe that CA(')inv is a subgroup of A because A is abelian. In particular, |CR(')inv\
A|  |A|/2 = |R|/4. As |CR(')inv| = 3|R|/4, we deduce that
• ' inverts each element in R \A and
• |CR(')inv \A| = |R|/4.
Fix x 2 R \ A. In particular, for every a 2 A, we have x 1a 1 = (ax) 1 = (ax)' =
a
'
x
' = a'x 1 and hence a' = x 1a 1x. From this it follows
CR(')
inv = CA(x) [Ax and CR(') = CA(◆x)inv [ I(R \A),
where I(R \A) := {m 2 R \A | m2 = 1}.
Suppose that I(R \ A) = ;. Then CR(') = CA(◆x)inv and hence |A : CA(◆x)inv| =
2 because |R : CR(')| = 4. As |A : CA(x)| = 2, we deduce that |A : CA(x) \
CA(◆x)inv|  4. Clearly, CA(x) \ CA(◆x)inv ✓ I(A) and hence |A : I(A)|  4. We
deduce
|CR(')| = |CA(◆x)inv| = |CA(◆x)inv \ (A \CA(x))|+ |CA(◆x)inv \CA(x)|
 |A|
4
+ |I(A)|  |R|
8
+ |I(R)|.
Thus
o =
1
4
 
|R|+ |I(R)|+ |CR(')|+ |CR(')inv|
 
=
1
4
✓
7|R|
4
+ |I(R)|+ |CR(')|
◆
 1
4
✓
7|R|
4
+ |I(R)|+ |R|
8
+ |I(R)|
◆
=
1
4
✓
15|R|
8
+ 2|I(R)|
◆
 |R|+ |I(R)|
2
  |R|
32
= c(R)  |R|
32
.
Suppose that I(R \ A) 6= ;. In particular, we may suppose that x 2 I(R \ A), that is,
x
2 = 1. From this it follows that
CR(') = CA(◆x)
inv [CA(◆x)invx,
CR(')
inv = CA(x) [Ax,
I(R) = I(A) [CA(◆x)invx.
As |CR(')| = |R|/4, we deduce |CA(◆x)inv| = |A|/4. Assume that |I(R)|   |CR(')|,
that is, |I(A)|   |CA(◆x)inv|. Thus
o =
1
4
 
|R|+ |I(R)|+ |CR(')|+ |CR(')inv|
 
=
1
4
✓
7|R|
4
+ 2|I(R)|
◆
 |R|+ |I(R)|
2
  |R|
16
= c(R)  |R|
16
.
Assume that |I(R)| < |CR(')|, that is, |I(A)| < |CA(◆x)inv|. Observe now
CA(x) \ I(A) = CA(◆x)inv \ I(A) = CA(x) \CA(◆x)inv.
14 Art Discrete Appl. Math. 4 (2021) #P1.10
As |I(R)| < |CR(')|, from these equalities we deduce I(A) = CA(x) \ CA(◆x)inv and
that CA(x) 6= CA(◆x)inv. Moreover,
|I(A)| = |A|
8
, |CA(x)| =
|A|
2
, |CA(◆x)inv| =
|A|
4
.
In particular, c(R) = (|R|+ |I(R)|)/2 = 19|R|/32. Thus
o =
1
4
|R|
✓
1 +
3
16
+
1
4
+
3
4
◆
=
35|R|
64
=
19|R|
32
  3|R|
64
= c(R)  3|R|
64
.
The proof of the lemma is now completed when ' has prime order.
Suppose now that o(') is not a prime number. Let p be the largest prime divisor of
o(') and let  := 'o(')/p. As  is a non-identity automorphism of R of prime order, we
are in the position to apply Lemma 2.8 to the group R and to the automorphism  . Let 2o
be the number of orbits of h'i on R.
If part (1) of Lemma 2.8 holds for  , then part (1) of Lemma 2.8 holds for ' because
every '-invariant subset of R is also  -invariant.
Assume then that part (3) of Lemma 2.8 holds for  . Then R is abelian of exponent
greater than 2 and  = ◆. Hence p = o( ) = 2. As p is the largest prime divisor of d, we
deduce that d is a power of 2. As o(')   4 and 'd/2 = ◆, the action of h'i on R has orbits
of cardinality 1 on CR('), of cardinality at least 2 on CR(◆) \CR('), and of cardinality
at least 4 on R \CR(◆). It follows that the number of subsets of R which are '-invariant is
at most
2|CR(')| · 2
|CR(◆)\CR(')|
2 · 2
|R\CR(◆)|
4 = 2
|R|+|CR(◆)|
4 +
|CR(')|
2 .
Observe that every '-invariant subset of R is also inverse-closed because 'd/2 = ◆. Thus
2o  2
|R|+|CR(◆)|
4 +
|CR(')|
2 .
Observe, also, that CR(◆) = I(R). As c(R) = (|R|+ |I(R)|)/2, by rewriting the previous
equation, we deduce
2o  2c(R) 
|R|+|I(R)| 2|CR(')|
4 . (2.10)
If |I(R)|  2|CR(')|   0, then (2.10) yields
o  2c(R) 
|R|
4 .
Thus part (1) holds for R and '. If |I(R)|   2|CR(')| < 0, then |I(R)| < 2|CR(')|.
However, as CR(')  CR( ) = I(R), we deduce I(R) = CR(') and hence (2.10)
yields
o  2c(R) 
|R| |I(R)|
4 .
Since R has exponent greater than 2, we have |R|  |I(R)|  |R|/2 and hence
o  2c(R) 
|R|
8 .
Thus part (1) holds for R and '.
Assume then that part (2) of Lemma 2.8 holds for  . Then CR( ) is abelian of expo-
nent greater than 2 and has index 2 in R, R is a generalized dicyclic group over CR( )
P. Spiga: On the equivalence between a conjecture of Babai-Godsil and a conjecture of Xu 15
and  = ◆̄CR( ). Hence p = o( ) = 2. As p is the largest prime divisor of d, we deduce
that d is a power of 2. As o(')   4 and 'd/2 =  = ◆̄CR( ), the action of h'i on R has
orbits of cardinality 1 on CR('), of cardinality at least 2 on CR(◆̄CR( )) \CR('), and of
cardinality at least 4 on R \CR(◆̄CR( )). As CR( ) = CR(◆̄CR( )), the action of h'i on
R has orbits of cardinality 1 on CR('), of cardinality at least 2 on CR( )\CR('), and of
cardinality at least 4 on R \CR( ). Since the number of inverse-closed subsets of CR(')
is c(CR(')) = (|CR(')| + |I(CR('))|)/2, it follows that the number of inverse-closed
subsets of R which are '-invariant is at most
2
|CR(')|+|I(CR('))|
2 · 2
|CR( )\CR(')|
2 · 2
|R\CR( )|
4 = 2
|R|+|CR( )|
4 +
|I(CR('))|
2 .
As |CR( )| = |R|/2, we have
2o  2
3|R|
8 +
|I(CR('))|
2 .
As c(R) = (|R|+ |I(R)|)/2, by rewriting the previous equation, we deduce
2o  2c(R) 
|R|+4|I(R)| 4|I(CR('))|
8 . (2.11)
As |I(R)|  |I(CR('))|   0, from (2.11) e deduce
2o  2c(R) 
|R|
8 .
In particular, part (1) holds for R and '.
Proposition 2.9. Let R be a finite group and suppose that R is not an abelian group of
exponent greater than 2 and that R is not a generalized dicyclic group. Then the set
{S ✓ R | S = S 1, R 6= NAut( (R,S))(R)}
has cardinality at most 2c(R) |R|/96+(log2 |R|)
2
.
As R  NAut( (R,S))(R), the condition R 6= NAut( (R,S))(R) is equivalent to the
fact that R is a proper subgroup of NAut( (R,S))(R).
Proof. Let, for the time being, R be any finite group. For every ' 2 Aut(R) with ' 6= idR
and for every S ✓ R with S = S 1 and S' = S, we have ' 2 NAut( (R,S))(R) \ R and
' fixes the identity vertex of  (R,S). Conversely, for every S ✓ R with S = S 1 and
for every ' 2 NAut( (R,S))(R) \ R with ' fixing the identity vertex of  (R,S), we have
' 2 Aut(R) with ' 6= idR. Therefore,
{S ✓ R | S = S 1, R 6= NAut( (R,S))(R)} =
[
'2Aut(R)
' 6=idR
{S ✓ R | S = S 1, S' = S}
and
|{S ✓ R | S = S 1, R 6= NAut( (R,S))(R)}| =
X
'2Aut(R)
' 6=idR
|{S ✓ R | S = S 1, S' = S}|.
(2.12)
16 Art Discrete Appl. Math. 4 (2021) #P1.10
Suppose now that R is not an abelian group of exponent greater than 2 and that R is not a
generalized dicyclic group.
Since a chain of subgroups of R has length at most log2(|R|), R has a generating
set of cardinality at most blog2(|R|)c  log2(|R|). Any automorphism of R is uniquely
determined by its action on the elements of a generating set for R. Therefore
|Aut(R)|  |R|blog2(|R|)c  2(log2(|R|))
2
. (2.13)
Let ' 2 Aut(R) with ' 6= idR. We now apply Lemma 2.8 to the group R and to
the non-identity automorphism ' of R. As R is neither abelian of exponent greater than
2 nor generalized dicyclic, parts (2) and (3) of Lemma 2.8 do not hold. Hence, part (1) of
Lemma 2.8 holds, that is,
|{S ✓ R | S = S 1, S' = S}|  2c(R) 
|R|
96 . (2.14)
Now the proof follows from (2.12), (2.13) and (2.14).
3 Proofs of Theorems 1.3 and 1.6
Proof of Theorem 1.3. Let R be a finite group of order r which is neither generalized di-
cyclic nor abelian of exponent greater than 2. By Lemma 2.2 and Proposition 2.9, we
have
lim
r!1
|{S ✓ R | S = S 1, R 6= NAut( (R,S))(R)}|
|{S ✓ R | S = S 1}|  limr!1 2
  r96+(log2(r))
2
= 0.
Proof of Theorem 1.6. Let R be a finite group. It was shown in [3, 14] that Xu conjecture
holds true when R is a generalized dicyclic group or when R is an abelian group of expo-
nent greater than 2. In particular, for the rest of the proof we may assume that R is neither
a generalized dicyclic group nor an abelian group of exponent greater than 2.
Let us denote by N (R) := {S ✓ R | S = S 1, R E Aut( (R,S))}, C(R) := {S ✓
R | S = S 1, R = Aut( (R,S))} and T (R) := {S ✓ R | S = S 1}. If Conjecture 1.2
holds true, then
lim
|R|!1
|C(R)|
|T (R)| = 1
and hence
lim
|R|!1
|N (R)|
|T (R)| = 1,
because C(R) ✓ N (R), that is, Conjecture 1.5 holds true. Conversely, suppose that Con-
jecture 1.5 holds true, that is, lim|R|!1 |N (R)|/|T (R)| = 1. Now,
N (R) = C(R) [ {S ✓ R | S = S 1, REAut( (R,S)), R < Aut( (R,S))}
✓ C(R) [ {S ✓ R | S = S 1, R < NAut( (R,S))(R)}
P. Spiga: On the equivalence between a conjecture of Babai-Godsil and a conjecture of Xu 17
and hence, by Theorem 1.3, we have
1 = lim
|R|!1
|N (R)|
|T (R)|
 lim
|R|!1
|C(R)|
|T (R)| + lim|R|!1
|{S ✓ R | S = S 1, R < NAut( (R,S))(R)}|
|T (R)|
= lim
|R|!1
|C(R)|
|T (R)| ,
that is, Theorem 1.2 holds true.
ORCID iDs
Pablo Spiga https://orcid.org/0000-0002-0157-7405
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doi:10.1016/s0012-365x(97)00152-0, graph theory (Lake Bled, 1995).
Noncommutative Lattices
by Jonathan E. Leech
J. E. Leech, Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond,
volume 4 of Famnit Lectures, Slovenian Discrete and Applied Mathematics Society and
University of Primorska Press, Koper, 2021.
ISBN 978-961-95273-0-6 (printed edn.), ISBN 978-961-293-027-1 (electronic edn.)
Weakenings of lattices, where the meet and join operations may fail to be commutative,
attracted from time to time the attention of various communities of scholars, including
ordered algebra theorists (for their connection with preordered sets) and semigroup theo-
rists (who viewed them as structurally enriched bands possessing a dual multiplication).
Recently, noncommutative generalisations of lattices and related structures have seen a
growth in interest, with new ideas and applications emerging. The adjective “noncommu-
tative” is used here in the inclusive sense of “not-necessarily-commutative”. Much of this
recent activity derives in some way from the initiation, thirty years ago, by Jonathan Leech,
of a research program into structures based on Pascual Jordan’s notion of a noncommuta-
tive lattice. Indeed, the research began by studying multiplicative bands of idempotents in
rings, and realising that under certain conditions such bands would also be closed under an
“upward multiplication”. In particular, for multiplicative bands that were left regular, any
maximal such band in a ring was also closed under the circle operation (or quadratic join)
x y = x+y xy. And any band closed under both operations satisfied certain absorption
identities, e.g., e(e f) = e = e  (ef). These observations indicated the presence of struc-
turally strengthened bands with a roughly lattice-like structure. These algebras are called
skew lattices and are defined as algebras (S;^,_) of type (2, 2), where both operations ^
and _ are associative and satisfy the four absorption identities x^(x_y) = x = (y_x)^x
and their dual. Absorption causes both operations to be idempotent. In the case of maximal
left regular bands in rings, ^ and _ are given as e ^ f = ef and e _ f = e+ f   ef .
Parallel to this was an expanding role of results related to structures that were weakened
or modified forms of (generalised) Boolean algebras. This was especially important in the
study of a second class of motivating examples, algebras of partial functions between pairs
of sets, A and B. Here, the skew operations are defined as follows:
f ^ g = g|G\F ; f _ g = f [ g|G F ,
where F,G ✓ A denote the actual functional domains of the partial functions f and g
respectively. A relative complement, defined by f \ g = f|F G, can be added to the alge-
braic structure, making definable the complement in the Boolean interval {g : g ✓ f} of all
functions approximating a given f . These algebras of partial functions provided examples
of so-called skew Boolean algebras and related structures, much as subsets of a given set
led to basic examples of Boolean algebras and distributive lattices. The pioneering papers
by Leech on skew lattices and skew Boolean algebras have attracted the attention of math-
ematicians from around the world, and in the last thirty years many interesting papers have
been published on the subject. As a result of these developments, skew lattices have grown
into a theory worth studying for its own sake. The 2020/21 monograph Noncommutative
Lattices: Skew Lattices, Skew Boolean Algebras and Beyond by Jonathan E. Leech pro-
vides an excellent, organised and comprehensive account of much that has been published
on the subject up through 2017. The book is mainly written for algebraists and mathemati-
cians, but readers interested in applications to logic and computer science may also find it
useful. The core of this monograph is the first four chapters. More specialised topics are
studied in the last three chapters. The content of the monograph will be explained in more
detail in the remaining part of this review.
In the first chapter of the book the author recalls various basic facts about bands (idem-
potent semigroups) that are pertinent to the rest of the monograph. In particular, he em-
phasises that a knowledge of regular bands, and their left and right-sided cases, is crucial
to understanding much that will be said about skew lattices. The author of this review has
particularly appreciated Section 1.3, where a noncommutative lattice is defined as a double
band satisfying a specified set among eight possible absorption identities. The comparison
of these absorption laws naturally brings the reader into the definitions of quasilattices,
paralattices, antilattices and skew lattices.
The basic theory for skew lattices is developed in Chapter 2. Of particular importance
in Section 2.1 are the two core structural results for skew lattices, analogues of the Clifford-
McLean Theorem and the Kimura Factorization Theorem, given originally for bands and
regular bands, respectively. There are two basic subvarieties of skew lattices: lattices (full
commutation) and anti-lattices (no non trivial commutation). The Clifford-McLean The-
orem for skew lattices thus states that every skew lattice is a lattice of anti-lattices. More
precisely, Green’s relation D on a skew lattice S, defined by xD y iff x ^ y ^ x = x
and y ^ x ^ y = y, is a congruence making S/D the maximal lattice image of S, and all
congruence classes of D are maximal anti-lattices in S (Theorem 2.1.3). The Kimura Fac-
torisation Theorem for skew lattices interestingly states that every skew lattice S factorises
as the fibered product of its maximal right-handed image and its maximal left-handed im-
age (Theorem 2.1.5).
An element that join commutes with all elements in a skew lattice also meet commutes
with all elements (and conversely). In general two elements commuting under one opera-
tion need not commute under the other operation. A skew lattice S is called symmetric if
this does not happen. All skew lattices in rings (using multiplication and the circle opera-
tion) are symmetric. In Section 2.2 many results on symmetric skew lattices are presented.
We mention here that, if S is a symmetric skew lattice for which S/D is countable, then
S has a lattice section (Theorem 2.2.7), i.e., a sublattice T of S having nonempty inter-
section with each D-class of S, in which case, T ⇠= S/D. A characterisation of having a
left-handed section and a right-handed section is also given in Theorem 2.2.8.
In Section 2.3 normal skew lattices are studied. In a normal skew lattice the lower set
{x : x  e} is a lattice, for every e. Of special interest are distributive, symmetric, normal
skew lattices characterised in Theorem 2.3.4 by the identities a^ (b_ c) = (a^ b)_ (a^ c)
and (a _ b) ^ c = (a ^ c) _ (b ^ c). This strengthened form of distributivity is called
strong distributivity. Thanks to Theorem 2.3.6, every normal skew lattice of idempotents
in a ring is strongly distributive. In this case the operations are given by e ^ f = ef again,
but e _ f = (e + f   ef)2 = e + f + fe   efe   fef , the cubic join. Of course when
e + f   ef is idempotent, both outcomes agree. Strongly distributive skew lattices are
also of interest due to their connections to skew Boolean algebras, the subject of Chapter 4.
A skew lattice can be embedded into (the skew lattice reduct of) a skew Boolean algebra
precisely when it is strongly distributive.
In Section 2.4 and 2.5 a detailed study of the natural partial order  on a skew lattice
is provided. This study is based on the behaviour of primitive skew lattices, consisting of
exactly two D-classes. Primitive skew lattices have a simple description given in terms of
cosets (Theorem 2.4.1). In Section 2.6 the decompositions of (mostly symmetric) normal
skew lattices are studied. For instance, the Reduction Theorem 2.6.9 implies that every
symmetric normal skew lattice can be embedded in the product of its maximal lattice image
and its maximal distributive image. In Theorem 2.6.12 and its corollaries the variety of
strongly distributive skew lattices is shown to be generated by a special primitive skew
lattice 5, a noncommutative 5-element variant of the lattice 2 for which the latter is its
maximal lattice image. A similar result holds for the variety of symmetric, normal skew
lattices.
Chapter 3 is devoted to the study of quasilattices, paralattices, and especially refined
quasilattices. The variety of refined quasilattices contains the variety of skew lattices and
it is defined as intersection of the variety of quasilattices and of the variety of paralattices.
Particular attention is given to their congruence lattices and to related topics such as Green’s
equivalences and simple algebras. Since all skew lattices are refined quasilattices, the study
in this chapter has implications for skew lattices.
Skew Boolean algebras are studied in Chapters 4 and 7. In Section 4.1 skew Boolean
algebras are formally defined as structural enhancements of strongly distributive skew lat-
tices. Skew Boolean algebras are shown to be subdirect products of primitive skew Boolean
algebras; moreover, every skew Boolean algebra can be embedded into a power of 5, a 5-
element primitive algebra (Corollaries 4.1.6 and 4.1.7). In Section 4.2, special attention is
given to classifying finite algebras, and in particular, to classifying finitely generated (and
thus finite) free skew Boolean algebras. The main results are Theorems 4.2.2 and 4.2.6,
with the latter describing the structure of finitely generated free algebras. In Section 4.4
skew Boolean algebras with finite intersections are introduced, that is, algebras for which
the natural partial order has meets that are called intersections. All skew Boolean alge-
bras S for which S/D is finite have intersections as do, more generally, all complete skew
Boolean algebras. In Chapter 7 the role of skew Boolean algebras in universal algebra is ex-
amined, in particular in the study of what might be termed “generalised Boolean phenom-
ena”, a topic of interest in universal algebra, with connections to discriminator varieties,
iBCK-algebras and more recently, Church algebras. The reviewer thinks that the character-
isation of one-pointed discriminator varieties in terms of right-handed skew Boolean alge-
bras with intersections is one of the most beautiful results of the theory. Chapter 6 (Skew
Lattices in Rings) is also devoted to the skew Boolean algebras of idempotents in rings, and
in particular, the case where the idempotents in a ring are closed under multiplication and
thus naturally form a skew Boolean algebra.
We conclude the review of this excellent monograph with the belief that it will be the
main reference on the subject of noncommutative lattices for many years.
Antonino Salibra https://orcid.org/0000-0001-6552-2561
Department of Environmental Sciences, Informatics and Statistics,
Università Ca’ Foscari Venezia, Via Torino 155, 30173 Venezia, Italia
E-mail address: salibra@unive.it
Minisymposium Announcement and Call for Papers –
Chemical Graph Theory
This is a call for submission of papers for a special issue of the journal The Art of Discrete
and Applied Mathematics (ADAM), on topics in Chemical Graph Theory.
Additionally, we are announcing a related 15-speaker minisymposium on Chemical Graph
Theory at CanaDAM 2021 (Canadian Discrete and Applied Mathematics) Conference. The
CanaDAM 2021 Conference is taking place online from May 25 – May 28, 2021. Further
information on CanaDAM 2021 can be found at
https://2021.canadam.math.ca/
About the minisymposium: This minisymposium in chemical graph theory explores var-
ious applications of graph theory to chemistry. A molecule can be described as a graph,
where vertices represent atoms and edges represent chemical bonds: benzenoids and fuller-
enes are two examples of such graph classes. Properties of those graphs, such as perfect
matchings and graph spectra, can be used to model characteristics of molecules, including
stability, reactivity, and electronic structure. Other related topics in chemical graph theory
include enumeration of graphs classes and algorithms for their enumeration.
About the journal: The Art of Discrete and Applied Mathematics (ADAM) is a modern,
dynamic, platinum open access, electronic journal that publishes high-quality articles in
contemporary discrete and applied mathematics (including pure and applied graph theory
and combinatorics), with no costs to authors or readers.
Papers should be submitted by 31 December 2021 via the ADAM website. When sub-
mitting a paper, please choose the option “Chemical Graph Theory Issue of ADAM” so
that it is directed to the correct editors. Papers that are accepted will appear online soon
after acceptance, and papers that are not processed in time for the special issue may still be
accepted and published in a subsequent regular issue of ADAM.
Nino Bašić and Elizabeth Hartung
Guest Editors
Jonathan E. Leech: Noncommutative Lattices:
Skew Lattices, Skew Boolean Algebras and Beyond
About the book: The extended study of non-commutative lattices was begun in 1949
by Ernst Pascual Jordan, a theoretical and mathematical physicist and co-worker of Max
Born and Werner Karl Heisenberg. Jordan introduced noncommutative lattices as algebraic
structures potentially suitable to encompass the logic of the quantum world. The modern
theory of noncommutative lattices began forty years later with Jonathan Leech’s 1989 paper
“Skew lattices in rings.” Recently, noncommutative generalizations of lattices and related
structures have seen an upsurge in interest, with new ideas and applications emerging,
from quasilattices to skew Heyting algebras. Much of this activity is derived in some way
from the initiation of Jonathan Leech’s program of research in this area. The present book
consists of seven chapters, mainly covering skew lattices, quasilattices and paralattices,
skew lattices of idempotents in rings and skew Boolean algebras. As such, it is the first
research monograph covering major results due to this renewed study of noncommutative
lattices. It will serve as a valuable graduate textbook on the subject, as well as a handy
reference to researchers of noncommutative algebras.
About the author: Jonathan Leech graduated from the University of Hawaii and earned
a PhD at the University of California, Los Angeles. He has taught mathematics at the
University of Tennessee, later at Missouri Western State University and finally at Westmont
College in Santa Barbara, California. He has been a Visiting Professor at Case Western
Reserve University, the Universidad de Granada in Spain and Universidade Mackenzie
in Brazil, and a scholar in residence at both the University of Sidney and the University
of Tasmania in Australia. Throughout his academic career Professor Leech has studied
algebraic structures related to semigroups, with much of his emphasis being on the theory
of noncommutative lattices, and of skew lattices in particular. He laid the foundations
of the modern theory of noncommutative lattices in a number of (co)authored seminal
publications. His work has inspired many mathematicians around the world to pursue
research in this area.
J. E. Leech, Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and
Beyond, volume 4 of Famnit Lectures, Slovenian Discrete and Applied Mathematics
Society and University of Primorska Press, Koper, 2021, 284 pp., ISBN 978-961-
95273-0-6.
The paperback edition of the book was published on March 5, 2021 by SDAMS,
the Slovenian Discrete and Applied Mathematics Society. The cost of the book is
20.00 EUR + shipping. Society members have discount of 5.00 EUR. Orders should
be sent to info@sdams.si. An invoice will be sent upon receipt of the order. The
book will be shipped after payment is received.
Call for papers for the Wilfried Imrich 80 issue of ADAM
On May 25, 2021, Wilfried Imrich turned 80. Wilfried’s impact on the development
of the Slovenian graph theory school is utmost important and lasting. As a small token of
appreciation, we are opening a special issue of ADAM dedicated to Wilfried to be edited
by Tanja Dravec, Marko Jakovac, Sandi Klavžar, and Janez Žerovnik. You are invited
to submit a paper related to Wilfried’s work by October 1, 2021. All accepted papers
will be published on-line as soon as possible, the issue will be completed in 2022. When
submitting a paper, please choose the option ”The Wilfried Imrich 80 Issue of ADAM” so
that it is directed to the correct editors.
Tanja Dravec, Marko Jakovac, Sandi Klavžar, Iztok Peterin, Janez Žerovnik
Guest editors
Petra Šparl Award 2022: Call for Nominations
The Petra Šparl Award was established in 2017 to recognise in each even-numbered year
the best paper published in the previous five years by a young woman mathematician in
one of the two journals Ars Mathematica Contemporanea (AMC) and The Art of Discrete
and Applied Mathematics (ADAM). It was named after Dr Petra Šparl, a talented woman
mathematician who died mid-career in 2016.
The award consists of a certificate with the recipient’s name, and invitations to give a
lecture at the Mathematics Colloquium at the University of Primorska, and lectures at the
University of Maribor and University of Ljubljana. The first award was made in 2018 to
Dr Monika Pilśniak (AGH University, Poland) for a paper on the distinguishing index of
graphs, and then two awards were made for 2020, to Dr Simona Bonvicini (Università di
Modena e Reggio Emilia, Italy) for her contributions to a paper giving solutions to some
Hamilton-Waterloo problems, and Dr Klavdija Kutnar (University of Primorska, Slovenia),
for her contributions to a paper on odd automorphisms in vertex-transitive graphs.
The Petra Šparl Award Committee is now calling for nominations for the next award.
Eligibility: Each nominee must be a woman author or co-author of a paper published in
either AMC or ADAM in the calendar years 2017 to 2021, who was at most 40 years old at
the time of the paper’s first submission.
Nomination Format: Each nomination should specify the following:
(a) the name, birth-date and affiliation of the candidate;
(b) the title and other bibliographic details of the paper for which the award is recom-
mended;
(c) reasons why the candidate’s contribution to the paper is worthy of the award, in at
most 500 words; and
(d) names and email addresses of one or two referees who could be consulted with regard
to the quality of the paper.
Procedure: Nominations should be submitted by email to any one of the three members
of the Petra Šparl Award Committee (see below), by 31 October 2021.
Award Committee:
• Marston Conder, m.conder@auckland.ac.nz
• Asia Ivić Weiss, weiss@yorku.ca
• Aleksander Malnič, aleksander.malnic@guest.arnes.si
Marston Conder, Asia Ivić Weiss and Aleksander Malnič
Members of the 2022 Petra Šparl Award Committee