ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 24 (2024) #P3.07
https://doi.org/10.26493/1855-3974.2835.8f0
(Also available at http://amc-journal.eu)
On the minisymposium problem*
Peter Danziger †
Department of Mathematics, Toronto Metropolitan University, Toronto,
ON M5B 2K3, Canada
Eric Mendelsohn
Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada
Brett Stevens ‡
School of Mathematics and Statistics - Carleton University, Ottawa, ON K1S 5B6, Canada
Tommaso Traetta §
DICATAM, Università degli Studi di Brescia, Via Branze 43, 25123 Brescia, Italy
Received 26 February 2022, accepted 25 July 2023, published online 29 July 2024
Abstract
The generalized Oberwolfach problem asks for a factorization of the complete graph
Kv into prescribed 2-factors and at most a 1-factor. When all 2-factors are pairwise isomor-
phic and v is odd, we have the classic Oberwolfach problem, which was originally stated
as a seating problem: given v attendees at a conference with t circular tables such that the
ith table seats ai people and
∑t
i=1 ai = v, find a seating arrangement over the
v−1
2 days of
the conference, so that every person sits next to each other person exactly once.
In this paper we introduce the related minisymposium problem, which requires a solu-
tion to the generalized Oberwolfach problem on v vertices that contains a subsystem on m
vertices. That is, the decomposition restricted to the required m vertices is a solution to the
generalized Oberwolfach problem on m vertices. In the seating context above, the larger
conference contains a minisymposium of m participants, and we also require that pairs of
these m participants be seated next to each other for
⌊
m−1
2
⌋
of the days.
When the cycles are as long as possible, i.e. v, m and v−m, a flexible method of Hilton
and Johnson provides a solution. We use this result to provide further solutions when
v ≡ m ≡ 2 (mod 4) and all cycle lengths are even. In addition, we provide extensive
*We thank the anonymous referees for their many useful suggestions that helped strongly improve this paper.
†P. Danziger has received support from NSERC Discovery Grants RGPIN-2022-03816.
‡B. Stevens recieved support from NSERC Discovery Grant RGPIN-2017-06392.
§Corresponding author. T. Traetta has received support from GNSAGA of Istituto Nazionale di Alta Matem-
atica.
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Ars Math. Contemp. 24 (2024) #P3.07
results in the case where all cycle lengths are equal to k, solving all cases when m | v,
except possibly when k is odd and v is even.
Keywords: Minisymposium problem, (generalized) Oberwolfach problem, 2-factorizations, subsys-
tems.
Math. Subj. Class. (2020): 05C51, 05B30
1 Introduction
We assume that the reader is familiar with the fundamentals of graph theory and of design
theory and refer them to [41] and [14], respectively. In particular, a factor is a spanning
subgraph and an r-factor is a factor which is r-regular, so in a 1-factor every vertex has
degree one and a 2-factor is a disjoint union of cycles. Given a collection of factors, F , an
F-factorization of a graph G is a decomposition of the edges of G into subgraphs, each of
which is isomorphic to some F ∈ F . If F = {F} we speak of an F -factorization.
We use Kn to denote the complete graph on n vertices and K∗n to denote the graph
Kn when n is odd and Kn − I , where I is a 1-factor, when n is even. Similarly, Km,n
denotes the complete bipartite graph with parts of sizes m and n. If the parts are X and
Y , respectively, we may also speak of KX,Y . A 2-factor is called uniform if all of its con-
stituent cycles are of the same length; it is called Hamiltonian* if its cycles have longest
possible lengths given the requirements of the factorization. If a 2-factor, F , consists en-
tirely of cycles of a particular length, k say, we refer to an F -factor and an F -factorization
as a Ck-factor and Ck-factorization, respectively. Given a graph G, we denote by G[n]
the lexicographic product of G with the empty graph on n vertices. Specifically, the
vertex set of G[n] is V (G) × Zn (where Zn denotes the cyclic group of order n) and
(x, i)(y, j) ∈ E(G[n]) if and only if xy ∈ E(G), i, j ∈ Zn.
The well known Oberwolfach problem asks for a 2-factorization of K∗n into 2-factors
all of which are isomorphic to a given 2-factor F . A summary of results up until 2006 can
be found in [14, Section VI.12], in particular the case of uniform factors has been solved
[1, 2, 26].
Theorem 1.1 ([1, 2, 26]). Given integers v, k ≥ 3, there is a Ck-factorization of K∗v if and
only if k | v, except that there is no C3-factorization of K∗6 or K∗12.
The case when all cycles in F have even length has been completely solved in [6].
The case with exactly two cycles is solved in [38]. The case of the complete graph Kℵ,
where ℵ is any infinite cardinal has been completely solved in [15]. In the related Hamilton-
Waterloo problem two 2-factors F1 and F2 are specified and we are asked for a factorization
of K∗n into a given number of each of the factors. There has been much recent progress
in this problem, see [3, 5, 7, 10, 11, 12, 17, 27, 28, 29, 32, 39, 40]. More generally,
in the generalized Oberwolfach problem we are given a set of 2-factors F1, F2, . . . , Ft
of Kn and positive integers α1, α2, . . . , αt, where
∑
αi = ⌊n−12 ⌋, and are asked for a
factorization of K∗n which contains αi copies of the 2-factor Fi, see [6, 13, 21]. A major
recent development gives a non-constructive asymptotic existence result for the generalized
Oberwolfach problem [23].
E-mail addresses: danziger@torontomu.ca (Peter Danziger), mendelso@math.utoronto.ca (Eric
Mendelsohn), brett@math.carleton.ca (Brett Stevens), tommaso.traetta@unibs.it (Tommaso Traetta)
P. Danziger et al.: On the minisymposium problem 3
Other graphs have also been considered. In particular, Liu has shown the following for
the complete multipartite graph.
Theorem 1.2 ([30, 31]). Let k, t and u be positive integers with k ≥ 3. There exists a
Ck-factorization of Kt[u] if and only if k | tu, (t − 1)u is even, further k is even if t = 2,
and (k, t, u) ̸∈ {(3, 3, 2), (3, 6, 2), (3, 3, 6), (6, 2, 6)}.
Originally the Oberwolfach problem was stated as a seating problem:
Given an odd number v of attendees at a conference with t circular tables such
that the ith table seats ai people and
∑t
i=1 ai = v, find a seating arrangement
over the v−12 days of the conference, so that every person sits next to each
other person exactly once.
In this paper we introduce the related minisymposium problem. In this case we require
a solution to the generalized Oberwolfach problem on v vertices such that its restriction to
a subset of m vertices constitutes a solution to the generalized Oberwolfach problem on
m vertices. Another way of considering the problem asks for a solution to the generalized
Oberwolfach problem on v vertices which contains a subsystem on m vertices. In the
seating context above, the larger conference contains a minisymposium of m participants,
and we also require that pairs of these m participants be seated next to each other for⌊
m−1
2
⌋
of the days. A similar problem has been considered, for example, in [9] for whist
tournaments.
Section 2 gives the formal definition of a minisymposium factorization and some nec-
essary conditions for its existence, as well as introduces some special cases. In Section 3
we show how to use a flexible theorem by Hilton and Johnson [25] to solve the case of
Hamiltonian* 2-factors (where the cycles are as long as possible). The same section con-
siders the case where all cycles are of even length and v ≡ m ≡ 2 (mod 4). Section 4
considers the uniform case, where all cycles have the same length. We completely solve
the case where all cycles are of length m when (v − 1)m is even. In Section 5 we discuss
and give some preliminary results on factorizations that contain more than one subsystem.
We provide some concluding remarks in the final section.
2 Preliminaries
We begin by giving a formal definition of a minisymposium factorization. The minisym-
posium problem is equivalent to the original Oberwolfach problem when v = m. Hence
we will generally assume that v > m.
Definition 2.1. Given positive integers v and m with v ≥ m, let
F =
{
Fi : i = 1, . . . ,
⌊
v − 1
2
⌋
−
⌊
m− 1
2
⌋}
,
be a collection of 2-factors on v vertices and let
G =
{
(Ti, Ui) : i = 1, . . . ,
⌊
m− 1
2
⌋}
,
where the Ti are 2-factors on m vertices and the Ui are 2-factors on v − m vertices. We
define a minisymposium factorization MSF(F ,G) as a factorization of K∗v into 2-factors
F ∈ F and Gi = Ti ∪Ui, where (Ti, Ui) ∈ G, such that T = {Ti : i = 1, . . . ,
⌊
m−1
2
⌋
} is
a factorization of a subgraph of K∗v isomorphic to K
∗
m.
4 Ars Math. Contemp. 24 (2024) #P3.07
Note that with the notation MSF(F ,G), we assume that the parameters v, m, T and
U = {Ui : i = 1, . . . ,
⌊
m−1
2
⌋
} are defined implicitly. We may also use the notation
MSF(F , (T ,U)), if we wish to explicitly refer to the factorizations T and U .
An MSF(F ,G) can be thought of as a 2-factorization of K∗v with a subsystem of size
m. When m = 1 or 2, this is just a factorization of K∗v into 2-factors in F , which is
equivalent to a solution of the generalized Oberwolfach problem, and so we will assume
m ≥ 3. Similarly, when m = v, this is a factorization into the Ti, so we assume that
m < v.
Removing the subsystem, we can talk about a 2-factorization of K∗v with a “hole” of
size m. However, care must be taken when either v or m is even as the placement of the
various 1-factors must be considered, as noted below.
We note that the size of F is⌊
v − 1
2
⌋
−
⌊
m− 1
2
⌋
=

v−m
2 v ≡ m (mod 2)
v−m+1
2 v + 1 ≡ m ≡ 0 (mod 2)
v−m−1
2 v ≡ m+ 1 ≡ 0 (mod 2).
• In the case when both v and m are even, we are considering a factorization of K∗v =
Kv − I , where I is a 1-factor, containing a factorization of a subgraph G− J of K∗v
where G ∼= Km and J is a 1-factor of G contained in I .
• When v is even and m is odd, we are considering a factorization of Kv − I , where
I is a 1-factor, containing a factorization of a subgraph G ∼= Km. Note that none of
the edges of I are contained in G.
• When v is odd and m is even, we are considering a factorization of K∗v = Kv
containing a factorization of G − J , where G ∼= Km and J is a 1-factor of G. We
note that the edges of J are not covered by factors in T , and hence must be covered
by factors in F .
• When both v and m are odd there is no 1-factor to consider.
Since in all cases G ∼= Km, we will henceforth refer to it as the Km. We note that none
of the edges of the Km are covered by F , except in the case of m even and v odd; in this
case, the edges of the 1-factor J of the Km are covered by F . We use this observation in
the proofs of the lemmas below, where for a given 2-factor F , we define ak(F ) to be the
number of cycles of length k in F .
Lemma 2.2. For a given v and m, if there is a minisymposium factorization, MSF(F ,G),
then for each factor F ∈ F using cF edges in the Km,
v −m ≥ −cF +
v∑
i=3
ai(F )
⌈
i
2
⌉
, (2.1)
m ≤ cF +
v∑
i=3
ai(F )
⌊
i
2
⌋
. (2.2)
In particular, if v is even, or m is odd, all of the cF = 0 and therefore,∑v
i=3 ai(F )
⌈
i
2
⌉∑v
i=3 ai(F )
⌊
i
2
⌋ ≤ v −m
m
. (2.3)
P. Danziger et al.: On the minisymposium problem 5
Proof. We first deal with the case when v is even, or m is odd. In this case none of the
edges of the Km appear in any F ∈ F . Therefore, for each F ∈ F , at most
⌊
i
2
⌋
vertices
of any cycle of length i in F are inside the Km, hence
m ≤
v∑
i=3
ai(F )
⌊
i
2
⌋
. (2.4)
Similarly, for any cycle of length i in F , at least ⌈ i2⌉ vertices of the cycle are not in the
Km, thus
v∑
i=3
ai(F )
⌈
i
2
⌉
≤ v −m. (2.5)
Thus Inequality (2.3) follows.
Now, if v is odd and m is even the m2 edges in the 1-factor J of the Km must be used
in factors from F . Suppose that F ∈ F uses cF of these edges. Each edge of the Km used
can increase the right hand side of Inequality (2.4) by no more than one and decrease the
left hand side of Inequality (2.5) by no more than one.
Theorem 2.3. For a given v and m, if there is a minisymposium factorization, MSF(F ,G),
then v ≥ 2m, unless v is odd and m is even, in which case v ≥ 2m− 1.
Proof. When v is even or m is odd, the left hand side of Inequality (2.3) is at least 1 and
therefore v ≥ 2m.
When v is odd and m is even,
∑
F∈F cF =
m
2 . Since v is odd, each F ∈ F must
contain at least one odd cycle, therefore
∑v
i=3 ai(F )⌈
i
2⌉ ≥ ⌈
v
2⌉ =
v+1
2 . Also note that
the number of factors F ∈ F is v−m+12 . Summing Inequality (2.1) over all of the F ∈ F
twice gives
(v −m+ 1)(v −m) ≥ 2
∑
F∈F
(
−cF +
v∑
i=3
ai(F )
⌈
i
2
⌉)
= −m+ 2
∑
F∈F
v∑
i=3
ai(F )
⌈
i
2
⌉
≥ −m+ 2
∑
F∈F
v + 1
2
= (v −m+ 1)v + 1
2
−m
Hence,
0 ≤ (v −m+ 1)(v −m)− (v −m+ 1)v + 1
2
+m
=
(v −m+ 1)(v − 2m− 1) + 2m
2
=
(v − (2m− 1))(v − (m+ 1))
2
.
6 Ars Math. Contemp. 24 (2024) #P3.07
When v = m + 1, the factors in U are required to be 2-regular graphs on a single vertex,
which is not possible, so v − (m + 1) > 0. Thus v − (2m − 1) ≥ 0 and the result
follows.
In the case where v is odd and m is even and v = 2m − 1, we have that U is a
factorization of Kv−m. So we can interchange both the roles of m and v −m, as well as
those of T and U . Thus, without loss of generality, we may assume that v ≥ 2m in all
cases.
There are two cases of initial special interest. Firstly, the case of uniform cycle lengths
(when all cycles in a factor are of the same length), which we consider in detail in Section 4.
Secondly, the case where the cycles are as long as possible, which in correspondence with
the definition of K∗n and the Hamiltonian-like nature of such factorizations we will call
Hamiltonian* factorizations. We formally define Hamiltonian* factorizations in Section 3.
There we show that a method of Hilton and Johnson completely settles their existence.
An MSF(F ,G) in which all of the factors in F , T and U are uniform with the same
cycle length k is called uniform and we refer to it as a UMSF(v,m, k). In this case we have
the following necessary conditions.
Theorem 2.4. If v > m and a UMSF(v,m, k) exists, then k ≥ 3, k | m and k | v.
Furthermore,
• if k is even, then v ≥ 2m;
• if k is odd, then v ≥ 2mkk−1 .
Proof. Since we are forming 2-factors with cycles of length k, we require k ≥ 3. The
divisibility conditions follow directly from the requirement for a factorization of K∗v and
K∗m into k-cycles. If k is even, then v is even, since it is a multiple of k, and Theorem 2.3
gives v ≥ 2m.
If k is odd, we note that for a Ck-factor F ∈ F , we have ai(F ) = vk when i = k and 0
otherwise, ⌊k2 ⌋ =
k−1
2 and ⌈
k
2 ⌉ =
k+1
2 . Thus, when v is even or m is odd, Inequality (2.3)
implies that v ≥ 2mkk−1 and the result follows.
This leaves the case when v and k are odd and m is even. We sum Inequality (2.1) over
all the F ∈ F to obtain∑
F∈F (v −m) ≥
∑
F∈F
(
−cF +
∑v
i=3 ai(F )
⌈
i
2
⌉)
1
2 (v −m+ 1)(v −m) ≥ −m/2 +
∑
F∈F
v
k
k+1
2
2k(v −m+ 1)(v −m) ≥ v(v −m+ 1)(k + 1)− 2mk.
Rearranging and expanding in v gives
(k − 1)v2 + (−3km+m+ k − 1)v + 2km2 ≥ 0. (2.6)
Let
f(v) = (k − 1)v2 + (−3km+m+ k − 1)v + 2km2.
By Theorem 2.3, v ≥ 2m− 1, but
f (2m− 1) = m(1 + k − 2m) < 0,
P. Danziger et al.: On the minisymposium problem 7
so v is at least as large as the larger of the two roots of f . Now
f
(
2mk
k − 1
− 2
)
= −2(m+ 1− k) < 0, and f
(
2mk
k − 1
− 1
)
= (k − 1)m > 0.
Thus f has its larger root between 2mkk−1 − 2 and
2mk
k−1 − 1.
It is left to check that v ̸= ⌊ 2mkk−1 −1⌋, ⌈
2mk
k−1 −1⌉. Recalling that k | v, if v = ⌊
2mk
k−1 −1⌋
or v = ⌈ 2mkk−1 − 1⌉, then there exist a rational number 0 ≤ ϵ < 1 and a positive integer a
such that
v =
2mk
k − 1
− 1± ϵ = ak.
Multiplying both sides by k−12k and rearranging we have that
m− ak − 1
2
= (1± ϵ)k − 1
2k
.
Since k is odd, the left side is an integer. However, 0 < (1 ± ϵ)k−12k < 1, a contradiction.
We conclude that v ≥ 2mkk−1 .
One interesting case in light of these necessary conditions is when m = k, i.e. a
UMSF(v, k, k). For these parameters, since k < v and k | v, we must have v ≥ 2k,
so the necessary conditions in Theorem 2.3 are satisfied. We consider these types of fac-
torizations in Section 4.
3 Hamiltonian* and bipartite factors
Considering non-uniform factors, an obvious case to consider is a Hamiltonian* minisym-
posium factorization, which is one in which the cycles have the longest possible lengths.
Specifically, the factors in F are all v-cycles, factors in T are all m-cycles and the factors in
U are all (v −m)-cycles. Such an MSF(F ,G) is denoted by HMSF(v,m). We sometimes
refer to the cycles in T and U as ‘short’ cycles. Because of the lengths of these cycles there
are no further necessary conditions beyond those of Theorem 2.3.
In a paper on the Oberwolfach problem, Hilton and Johnson prove the following theo-
rem on a flexible construction technique.
Theorem 3.1 ([25]). Let m and n be integers, 1 ≤ m < n. Let (s1, . . . , st), si ∈ 1, 2, 1 ≤
i ≤ t, be a composition of n−1. Let Km be edge coloured with t colours c1, . . . , ct. Let fi
be the number of edges coloured ci and Km(ci) be the ith colour class. This colouring can
be extended to an edge-colouring of Kn in which the colour class Kn(ci) is an si-factor,
1 ≤ i ≤ t, and when si = 2, Kn(ci) contains exactly one more cycle than Km(ci) if and
only if for all i = 1, 2, . . . , t:
fi ≥ si(m− n/2),
sin is even,
∆(Km(ci)) ≤ si.
This theorem is sufficient to provide a solution to the Hamiltonian* minisymposium
factorization.
8 Ars Math. Contemp. 24 (2024) #P3.07
Corollary 3.2. An HMSF(v,m) exists if and only if m ≥ 3, v ≥ 2m−1 in case m is even,
and v ≥ 2m in case m is odd.
Proof. The given conditions are necessary by Theorem 2.3. To prove sufficiency, we will
define an edge colouring of the Km from a decomposition of the Km into Hamiltonian
cycles and possibly a single 1-factor using Theorem 1.1. If m is odd, this defines edge
colours ci, 1 ≤ i ≤ (m− 1)/2. If v is also odd, extend this to a (v − 1)/2-edge colouring
of the Km by including (v −m)/2 empty colour classes ci, (m+ 1)/2 ≤ i ≤ (v − 1)/2.
Let si = 2 for all 1 ≤ i ≤ (v − 1)/2. If v is even, extend the colouring to a v/2-edge
colouring of the Km by adding empty colour classes. Let si = 2 for all 1 ≤ i < v/2 and
sv/2 = 1. In both cases, it can be verified that Theorem 3.1 now gives an HMSF(v,m) as
desired.
If m is even, define m/2 edge colour classes of the Km from a decomposition into
Hamiltonian cycles and one 1-factor. Let c1 be the colour class of the 1-factor. If v is also
even, extend this to a v/2-edge colouring by adding empty colour classes. Let s1 = 1 and
si = 2 for 2 ≤ i ≤ v/2. If v is odd, extend this to a (v − 1)/2-edge colouring by adding
empty colour classes. Let si = 2 for 1 ≤ i ≤ v/2. In both cases, it can be verified that
Theorem 3.1 now gives an HMSF(v,m) as desired.
Theorem 3.1 is more than just an existence result; a recursive procedure can be extracted
from the proof to algorithmically build the edge decompositions. We have a more direct
construction of all HMSF(v,m) which uses difference methods and decompositions of
Cayley graphs [16].
Theorem 3.1 can be used much more generally to build minisymposium factorizations.
Essentially it shows that it is possible to extend any 2-factorization of the Km to one of Kv ,
provided that the necessary conditions hold, where the additional 2-factors are Hamiltonian,
with an additional 1-factor when v is even.
When all of the cycles of the factors in F , U and T are bipartite (i.e. contain only even
cycles), we apply the Theorem of Häggkvist [24] (given below) to HMSF(v,m) to give us
a solution to the minisymposium problem when v ≡ m ≡ 2 (mod 4).
Theorem 3.3 ([24]). If F is a bipartite 2-regular graph of order 2n, then there is a factor-
ization of Cn[2] into 2 isomorphic copies of F .
We note that in the case where the factors are bipartite, so all cycle lengths are even, v
and m are both even and Theorem 2.3 gives v ≥ 2m.
Theorem 3.4. If v ≡ m ≡ 2 (mod 4), and F = {Fi : 1 ≤ i ≤ (v −m)/2}, U = {Uj :
1 ≤ j ≤ (m−2)/2} and T = {Tj : 1 ≤ j ≤ (m−2)/2} are sets of bipartite factors with
Fi = Fi+1, Uj = Uj+1 and Tj = Tj+1 for every odd i and j, then an MSF(F , (T ,U))
exists if and only v ≥ 2m.
Proof. We note that if v ≡ m ≡ 2 (mod 4), the number of the Fi is (v − m)/2 and the
number of the Uj and the Tj is (m− 2)/2, so the number of the Fi, Uj and Tj are all even.
We take an HMSF(v/2,m/2), which exists by Corollary 3.2, with factors F ′i of order v/2,
U ′i of order (v − m)/2 and T ′i of order m/2. We blow up each vertex by 2 and apply
Theorem 3.3 to factor each F ′i [2] into 2 copies of Fi, each U
′
j [2] into 2 copies of Uj and
each T ′j [2] into 2 copies of Tj .
P. Danziger et al.: On the minisymposium problem 9
If v ≡ 0 (mod 4) or m ≡ 0 (mod 4), then v/2 or m/2 would be even and the
HMSF(v/2,m/2) would contain 1-factors either in Kv/2 or the Km/2. When a 1-factor is
blown up as done in Theorem 3.3, it results in a C4-factor, which prevents constructing the
desired MSF unless F , T , and U already contain this kind of factor.
An immediate consequence of Theorem 3.4 is the following relating to uniform factors.
Corollary 3.5. If k > 3, k ≡ 2 (mod 4), v ≡ m ≡ k (mod 2k), then a UMSF(v,m, k)
exists if and only if v ≥ 2m.
4 Uniform factors
In this section we consider the case of uniform factors, i.e. when all cycles are of the same
length, k. We recall from Theorem 2.4 that in order for a UMSF(v,m, k) to exist, we
require that k ≥ 3, which we will assume throughout this section. We also require k | m
and k | v. Additionally, if k is even, then v ≥ 2m and if k is odd, then v ≥ 2mkk−1 .
Corollary 3.5 gives a powerful result in the case when k ≡ 2 (mod 4) and v ≡ m ≡ k
(mod 2k). The case where k = 3 has been considered in [34, 35, 36] when v and m are
both odd, and [18, 19, 20, 22, 37] when they are both even. However, the case when m and
v have opposite parities appears to be completely open. We summarize these results in the
following theorem.
Theorem 4.1 ([20, 35]). If v ≡ m (mod 2), there exists a UMSF(v,m, 3) if and only if
v ≥ 3m, v ≡ m ≡ 0 (mod 3), and if v,m are even, then v,m > 12.
We will find the following results useful. A corollary of a result in [4] yields the fol-
lowing.
Theorem 4.2 ([4]). If G is a Hamiltonian decomposable graph, then G[n] is also Hamil-
tonian decomposable. In particular, Cm[n] has a Cmn-factorization for every m ≥ 3.
Piotrowski [33] has shown the following result for m ≥ 4. The case m = 3 is covered
by Theorem 1.2.
Theorem 4.3 ([33]). There exists a Cm-factorization of Cm[n], except if n = 2 and m is
odd, or when (m,n) = (3, 6).
Piotrowski [33] has also shown the following result.
Theorem 4.4 ([33]). Let F be a bipartite 2-regular graph of order 2n. The complete
bipartite graph K2[n] has an F -factorization if and only if n is even, except when n = 6
and F consists of two 6-cycles.
We now give some recursive constructions for uniform minisymposium factorizations.
Theorem 4.5. Let m ≥ k ≥ 3 and t ≥ 2 be integers. If (t − 1)m is even and k | m,
then there is a UMSF(mt,m, k), except that there is no UMSF(6t, 6, 3) UMSF(12t, 12, 3),
UMSF(12, 6, 6), or UMSF(2m,m, k) when k is odd.
Proof. The non-existence of a UMSF(6t, 6, 3) and UMSF(12t, 12, 3) are covered by The-
orem 4.1. Since a UMSF(2m,m, k) is equivalent to a Ck-factorization of the complete
bipartite graph K2[m], it clearly does not exist when the cycle length k is odd, or when
k = m = 6 by Theorem 4.4. In all remaining cases, the following conditions simultane-
ously hold:
10 Ars Math. Contemp. 24 (2024) #P3.07
1. (m, k) ̸∈ {(6, 3), (12, 3)},
2. (t,m, k) ̸= (2, 6, 6),
3. if k is odd, then t > 2.
The assumptions of Theorems 1.1 and 1.2 are then satisfied. Hence there is a Ck-factorization
of Kt[m] and a Ck-factorization of K∗m, which we use to fill in the parts of size m in Kt[m].
This completes the proof.
Considering the necessary conditions in Theorem 2.4, we get the following corollaries.
Corollary 4.6. Suppose that either k is even or v is odd, and m | v. Then there exists a
UMSF(v,m, k) if and only if k | m, v ≥ 2m when k is even, and v ≥ 3m when k is odd,
except that UMSF(v, 6, 3), UMSF(v, 12, 3) and UMSF(12, 6, 6) do not exist.
Proof. Taking v = mt, Theorem 2.4 gives the necessary conditions k | m and v ≥ 2m
when k is even. When k is odd, the necessary condition from Theorem 2.4 is v ≥ 2mkk−1 , but
since m | v, this implies k ≥ 3m. Given the conditions of k and v, the sufficiency comes
from Theorem 4.5.
We note that if m | v this corollary completely solves all cases except when k is odd
and v is even. One case of particular interest is when k = m, in this case m | v is necessary.
Corollary 4.7. Let m(t− 1) be even. Then a UMSF(tm,m,m) exists if and only if t ≥ 2
when m is even, t ≥ 3 when m is odd, and (t,m) ̸= (2, 6).
The previous results all require m | v, however the next theorem allows us to recur-
sively construct solutions to cases where m does not divide v.
Theorem 4.8. Assume there is a UMSF(v,m, k) and let t ≥ 1. Then there exists a
UMSF(vtk,mtk, ℓ), with ℓ ∈ {k, kt}, in each of the following cases:
(1) v and m have the same parity;
(2) v and t are even, ℓ = tk, and m and k are both odd, except possibly when (k, t) =
(3, 2).
Proof. Letting w ∈ {m, v}, we factorize K∗wtk into Γw = K∗w[tk] and Γw = K∗wtk − Γw.
Note that Γw is the vertex disjoint union of
(1) w copies of K∗tk when w is odd, or
(2) w/2 copies of K∗2tk when w is even.
Without loss of generality, we can assume that Γm ⊆ Γv and Γm ⊆ Γv , except when v is
odd and m is even. In this case the components of Γm are copies of K∗2tk, while those of
Γv are isomorphic to K∗tk, therefore Γm ⊆ Γv cannot hold. We proceed by constructing
(a) a Cℓ-factorization of Γv containing a Cℓ-factorization of Γm, and
(b) a Cℓ-factorization of Γv containing a Cℓ-factorization of Γm,
P. Danziger et al.: On the minisymposium problem 11
which together will provide the desired UMSF (vtk,mtk, ℓ).
We blow up each vertex of the UMSF(v,m, k) by tk, to obtain a Ck[tk]-factorization
of Γv containing a Ck[tk]-factorization of Γm. To construct (a) it is therefore enough to
factorize Ck[tk] into Cℓ-factors, ℓ ∈ {k, tk}. By Theorem 4.3 there is a Ck-factorization
of Ck[tk], except when (t, k) = (2, 3). In this case, the desired UMSF(6v, 6m, 3) exists
by Theorem 4.1. Considering that Ck[tk] = Ck[t][k], by Theorem 4.2 there exists a Ckt-
factorization of Ck[t] which we blow up by k to obtain Ckt[k]-factorization of Ck[tk]. By
Theorem 4.3, each Ckt[k]-factor can be further decomposed into Ckt-factors yielding a
Ckt-factorization of Ck[tk].
It is left to construct (b). If m and v have the same parity, the components of Γm and
Γv are pairwise isomorphic: they are copies of K∗tk or K
∗
2tk. It is then enough to build a
Cℓ-factorization of K∗tk and K
∗
2tk for ℓ ∈ {k, kt}. They exist by Theorem 1.1 except when
ℓ = k = 3 and one of the following two conditions hold,
1. mv is odd and t ∈ {2, 4}, or
2. m and v are even, and t ∈ {1, 2}.
In each of these cases, the existence of the desired UMSF (3vt, 3mt, 3) is guaranteed by
Theorem 4.1.
If v and t are even, ℓ = tk, and both m and k are odd, the components of Γm are
isomorphic to K∗tk, while those of Γv are isomorphic to K
∗
2tk. Since we can factorize
K∗2tk into K2[tk] and two copies of K
∗
tk, it is enough to decompose both K2[tk] and K
∗
tk
into Ctk-factors. These factorizations exist by Theorem 4.4 and Theorem 1.1, respectively,
except possibly when (k, t) = (3, 2).
We may now use the result on triples (Theorem 4.1) to obtain the following.
Corollary 4.9. Let v ≡ m ≡ 0, 3 (mod 6), with v ≥ 3m and m ̸∈ {0, 6, 12}. Then there
exists a UMSF(3tv, 3tm, 3t) for all t > 0.
Additionally, we may use Theorem 3.5 to obtain the following result.
Corollary 4.10. Let 3 < k, k ≡ 2 (mod 4), v ≡ m ≡ k (mod 2k) and v ≥ 2m. Then
there exists a UMSF(vtk,mtk, k) and a UMSF(vtk,mtk, tk) for all t > 0.
We note that the above result can be used to obtain UMSF’s with cycle length, subsys-
tem size or number of vertices congruent to 0 (mod 4) by taking t even. However, in all
cases, the number of vertices and subsystem size will be divisible by k2.
5 Multiple subsystems
A natural question to ask is if a system can have multiple subsystems. In general, it seems
likely to be hard to navigate through the lattice of subsystems and all the possible ways the
subsystems can be distributed across the main system. However, when the subsystems are
disjoint, have small common intersections or are nested, the problem is more tractable. We
give some preliminary results in the next three subsections.
12 Ars Math. Contemp. 24 (2024) #P3.07
5.1 Disjoint subsystems
In the uniform case, the flexibility of Theorem 1.2 allows us to create a large number of
disjoint subsystems. We refer to a factorization of K∗v into k-cycles with subsystems on
disjoint vertex sets of sizes mj for 1 ≤ j ≤ n as a UMSF(v, {mj}, k).
Lemma 5.1. Let k | mj for 1 ≤ j ≤ n. Let m be an integer such that there is a
UMSF(m,mj , k) for each 1 ≤ j ≤ n. Then there exists a UMSF(ms, {mj}, k) for all
s ≥ max{2, n} if k is even, and for all s ≥ max{3, n} such that (s − 1)m is even if k is
odd, except when (k, s,m) = (6, 2, 6).
Proof. Theorem 1.2 guarantees the existence of a Ck-factorization of Ks[m]. For each
1 ≤ j ≤ n, place a copy of the ingredient UMSF(m,mj , k) on the jth part of size m
of Ks[m], and any Ck-factorization of K∗m on each of the remaining parts. The definite
exception (k, s,m) = (6, 2, 6) follows from the non-existence of a UMSF(12, 6, 6) (see
Theorem 4.5).
As with the uniform factorizations containing a single subsystem in this paper, the
easiest case is when mj | m for all 1 ≤ j ≤ n and either k is even or m is odd.
Corollary 5.2. Let m = lcm{mj : j = 1, . . . , n}, and assume the following conditions
are all satisfied:
(1) k | mj for all j ∈ {1, . . . , n} and m | v;
(2) k(m− 1) is even;
(3) if k = 3, then mj ̸∈ {6, 12} for all j;
(4) if (k,m) = (6, 12), then mj ̸= 6 for all j;
(5) (v,m, k) ̸= (12, 6, 6);
(6) v/m ≥ max{2, n} if k is even;
(7) v/m ≥ max{3, n} and v is odd if k is odd.
Then there exists a UMSF(v, {mj}, k).
Proof. Since each mj is a divisor of m and conditions (1)–(4) hold, we can apply either
Theorem 1.1 or Corollary 4.6, as needed, to ensure the existence of a UMSF(m,mj , k) for
every j ∈ {1, . . . , n}. These are the ingredient designs needed in Lemma 5.1, which can
be applied in view of conditions (5)–(7).
We note that for any fixed multiset of mj , this corollary constructs UMSF(v, {mj}, k)
for all but a finite number of v permitted by the necessary conditions when mj | v and
either k is even or m is odd.
P. Danziger et al.: On the minisymposium problem 13
5.2 Scattered subsystems
The proof of Lemma 5.1 builds systems whose factors intersect either all of the subsystems
or none of them. A balancing of the sizes of these intersections could be an interesting
property. For instance, we could ask for systems whose factors do not intersect more than
one subsystem. In other words, we ask for a Ck-factorization F of K∗v that contains n
subsystems of sizes m1,m2, . . . ,mn, such that no two factors of any of the subsystems are
contained in the same factor of F . We denote such a factorization by UMSF(v, [mj ], k)
and say that the subsystems are scattered.
Partial results in this direction can be easily obtained by making use of cycle frames.
We recall that a k-cycle frame (k-CF) of Ks[m] is a decomposition of Ks[m] into holey
Ck-factors; a holey Ck-factor is a vertex-disjoint union of k-cycles covering all vertices
Ks[m] except those belonging to one part. The following result, proven in [8], provides
necessary and sufficient conditions for the existence of k-cycle frames.
Theorem 5.3 ([8]). Let m ≥ 2 and k, s ≥ 3. There exists a k-cycle frame of Ks[m] if and
only if m is even, m(s− 1) ≡ 0 (mod k), k is even when s = 3, and (k,m, s) ̸= (6, 6, 3).
By making use of Theorem 5.3, we obtain the following.
Lemma 5.4. Let u ∈ {1, 2}. If there exists a UMSF(2m+ u,mj , k) for each 1 ≤ j ≤ n,
then there exists a UMSF(2ms+ u, [mj ], k) whenever 2s ≡ 2 (mod k) and s ≥ n.
Proof. Let n ≥ 1, 2s ≡ 2 (mod k) and s ≥ n. It follows that 2m(s − 1) ≡ 0 (mod k),
k = 4 when s = 3, and (k, 2m, s) ̸= (6, 6, 3). Therefore, Theorem 5.3 guarantees the
existence of a k-cycle frame F of Ks[2m]. Let Pi denote the i-th part of Ks[2m], for
1 ≤ i ≤ s. Also, let
F = {Fij : 1 ≤ i ≤ s, 1 ≤ j ≤ m},
where the Fijs are the holey Ck-factors of F missing Pi, for 1 ≤ i ≤ s. By assumption,
there is a UMSF(2m + u,mj , k) on Pi ∪ {∞1,∞u}, say Hi = {Hij : 1 ≤ j ≤ m}. It
follows that F∗ = {Fij ∪Hij : 1 ≤ i ≤ s, 1 ≤ j ≤ m} is a Ck-factorization of K2ms+u
with scattered subsystems of sizes m1,m2, . . . ,mn. Indeed, the factors of the subsystems
belong to the Hijs, each of which belongs to exactly one factor of F∗.
In the UMSF(2ms+u, [mj ], k) constructed in the proof of Lemma 5.4, two subsystems
may intersect in 0, 1, or 2 vertices, which are necessarily in the set {∞1,∞u}.
Theorem 4.5 provides sufficient conditions for the existence of a UMSF(v,m, k) if m
is a divisor of v. From that, we easily obtain the following corollary.
Corollary 5.5. Let u ∈ {1, 2}, k ≥ 3, and let k | mj | (2m + u) for each 1 ≤ j ≤ n.
Then there exists a UMSF(2ms+ u, [mj ], k) whenever the following conditions hold:
(1) mj is even or (2m+ u)/mj is odd,
(2) 2n ≤ 2s and 2s ≡ 2 (mod k),
except when (mj , k) ∈ {(6, 3), (12, 3)}, and except possibly when (2m + u,mj , k) =
(12, 6, 6), or k is odd and 2m+ u = 2mj , for some j ∈ {1, . . . , n}.
Note that for values of the triple (2m + u,mj , k) determining a possible exception in
Corollary 5.5 it is possible for a UMSF(2ms + u, [mj ], k) to exist. However, our method
cannot construct them because the UMSF(2m+u,mj , k) to use in the construction does not
exist. It is possible that other construction methods would build a UMSF(2ms+u, [mj ], k).
14 Ars Math. Contemp. 24 (2024) #P3.07
5.3 Nested subsystems
A scenario complementary to the susbsystems being all disjoint is when the subsystems
are completely nested, on vertex sets M1 ⊃ M2 ⊃ · · · ⊃ Mn−1. We modify our notation
slightly for this section to make it less cumbersome in this specific context.
Definition 5.6. Let v = m0 > m1 > · · · > mn−1 > mn = 0 be non-negative integers.
For 1 ≤ i ≤ ⌊m0−12 ⌋ and 0 ≤ j < n, let Ui,j be a 2-regular graph of order
|V (Ui,j)| =

mj −mj+1, if 1 ≤ i ≤ ⌊mj+1−12 ⌋,
mj , if ⌊mj+1+12 ⌋ ≤ i ≤ ⌊
mj−1
2 ⌋,
0, if ⌊mj+12 ⌋ ≤ i ≤ ⌊
m0−1
2 ⌋.
A nested minisymposium factorization nMSF({Ui,j : 1 ≤ i ≤ ⌊m0−12 ⌋, 0 ≤ j < n}) is a
2-factorization F = {Fi : 1 ≤ i ≤ ⌊m0−12 ⌋} of K
∗
v such that
• V (K∗v ) = M0 ⊃ M1 ⊃ · · · ⊃ Mn−1 ⊃ Mn = ∅ are nested sets with |Mj | = mj ;
• each 2-factor Fi =
⋃n−1
j=0 Fi,j , where
V (Fi,j) =

Mj \Mj+1 if 1 ≤ i ≤ ⌊mj+1−12 ⌋
Mj if ⌊mj+1+12 ⌋ ≤ i ≤ ⌊
mj−1
2 ⌋
∅ otherwise,
and each Fi,j is isomorphic to Ui,j ;
• for every 0 ≤ ℓ < n, {
⋃n−1
j=ℓ Fi,j : 1 ≤ i ≤ ⌊
mℓ−1
2 ⌋} is a 2-factorization of a graph
isomorphic to K∗mℓ .
In other words, the factorization F of K∗v restricted to vertex set Mℓ factorizes a graph
isomorphic to K∗mℓ into 2-factors whose structure is determined by the Ui,js.
Our construction of nested minisymposium factorizations is most tidily expressed by
defining holey factorizations.
Definition 5.7. Given positive integers v and m with v ≥ m, let
U =
{
Ui : 1 ≤ i ≤
⌊
v − 1
2
⌋}
,
be a collection of 2-regular graphs on v−m vertices for 1 ≤ i ≤ ⌊m−12 ⌋, and on v vertices
for ⌊m+12 ⌋ ≤ i ≤ ⌊
v−1
2 ⌋. A holey factorization HF(U) is a decomposition F = {Fi : 1 ≤
i ≤
⌊
v−1
2
⌋
} of K∗v −G (i.e., K∗v minus the edges of G) where each Fi ∼= Ui and G ∼= K∗m.
If v is even, then there is a 1-factor, Iv , on the vertices of K∗v whose edges are not
present in K∗v − G. If v and m are both even there is a 1-factor, Jm, on the vertices of G
which is a subgraph of Iv . If v is even and m odd, then no edges of Iv are induced on the
vertices of G. If v is odd and m is even, then there is a 1-factor Jm on the vertices of G
whose edges are present in K∗v −G.
By removing the 2-factors of a subsystem or “filling the hole” with them (making the
Jm in the hole coincide with the Im of the subsystem as required by the parities of v and
m) we have an equivalence between the existence of minisymposium factorizations and
holey factorizations.
P. Danziger et al.: On the minisymposium problem 15
Theorem 5.8. Let T = {Ti : 1 ≤ i ≤ ⌊m−12 ⌋} be a 2-factorization of K
∗
m and
U =
{
Ui : 1 ≤ i ≤
⌊
v − 1
2
⌋}
,
be a collection of 2-regular graphs on v−m vertices for 1 ≤ i ≤ ⌊m−12 ⌋ and on v vertices
for ⌊m+12 ⌋ ≤ i ≤ ⌊
v−1
2 ⌋. Then a HF(U) exists if and only if a
MSF({Ui : ⌊m+12 ⌋ ≤ i ≤ ⌊
v−1
2 ⌋}, {(Ti, Ui) : 1 ≤ i ≤ ⌊
m−1
2 ⌋})
exists.
Because in a nested minisymposium factorization the holes are nested and emptying or
filling them does not affect the edges outside the hole, this equivalence extends to nested
minisymposium factorizations and shows that they can be constructed exactly when the
various holey factorizations of K∗mj with holes of size mj+1 exist.
Theorem 5.9. Let v = m0 > m1 > · · · > mn−1 > mn = 0 be positive integers. For
1 ≤ i ≤ ⌊m0−12 ⌋ and 0 ≤ j < n, let Ui,j be a 2-regular graph of order
|V (Ui,j)| =

mj −mj+1, if 1 ≤ i ≤ ⌊mj+1−12 ⌋,
mj , if ⌊mj+1+12 ⌋ ≤ i ≤ ⌊
mj−1
2 ⌋,
0, if ⌊mj+12 ⌋ ≤ i ≤ ⌊
m0−1
2 ⌋.
A nested minisymposium factorization nMSF({Ui,j : 1 ≤ i ≤ ⌊m0−12 ⌋, 0 ≤ j < n})
exists if and only if a HF({Ui,j : 1 ≤ i ≤ ⌊mj−12 ⌋}) exists for each 0 ≤ j < n.
Proof. The forward direction is proved simply by restricting the system to Mj and re-
moving the subsystem on Mj+1. The converse is proved by a recursive construction
starting with j = n − 1: in this case, a HF({Ui,n−1 : 1 ≤ i ≤ ⌊mn−1−12 ⌋}) is an
nMSF({Ui,n−1 : 1 ≤ i ≤ ⌊mn−1−12 ⌋}), say Fn−1.
At stage j < n − 1, use Theorem 5.8 to construct an nMSF({Ui,ℓ : 1 ≤ i ≤
⌊mj−12 ⌋, j ≤ ℓ < n}), say Fj , by filling the hole in the HF({Ui,j : 1 ≤ i ≤ ⌊
mj−1
2 ⌋})
with the nMSF({Ui,ℓ : 1 ≤ i ≤ ⌊mj+1−12 ⌋, j + 1 ≤ ℓ < n}), denoted by Fj+1, built at
stage j + 1.
Between the extremes of disjoint and nested subsystems, there are factorizations with
multiple subsystems with arbitrary intersections. Some structured instances of this much
more general problem may be amenable to solution but we leave this to future work.
6 Conclusions and further work
We have introduced the minisymposium problem: a subsystem variant of the generalized
Oberwolfach problem. This variant asks for a solution to a generalized Oberwolfach prob-
lem that contains a subsystem of a given size. When v, the number of vertices, is even,
it is traditional in 2-factor decomposition problems to ask for decompositions of Kv − I
where I is a 1-factor. When the number of vertices in the system and the subsystem are
both even, then we require that the 1-factor in the subsystem be a subgraph of the 1-factor
in the full system. Therefore when the parities of the system and the subsystem agree, the
16 Ars Math. Contemp. 24 (2024) #P3.07
problem becomes more tractable. When the parities are opposite, either the 1-factor of the
full system must avoid the subsystem, or the edges of the 1-factor in the subsystem must
be in 2-factors of the whole system.
Clearly, this is a very broad statement and we identify some particularly interesting
cases, Hamiltonian* and uniform. In the Hamiltonian* minisymposium problem there are
as few cycles as possible and in the uniform minisymposium problem all cycles are of the
same length. We have shown that the work of Hilton and Johnson provides a complete
solution for the Hamiltonian* minisymposium problem in Corollary 3.2. In the case when
v ≡ m ≡ 2 (mod 4), Theorem 3.4 uses this Hamiltonian* construction to provide a
wide range of solutions when the resulting factors are all bipartite. In particular, a uniform
factorization with k ≡ 2 (mod 4), v ≥ 2m and v ≡ m ≡ k (mod 2k) always exists.
Corollary 4.10 can be used to extend this to uniform factorizations where k ≡ 0 (mod 4)
or v ≡ m ≡ 0 (mod 2k).
In Section 4 we considered the uniform case. We have solved a large part of the spec-
trum. In particular, when k is even or v is odd, Corollary 4.6 gives all cases when m | v
and Corollary 4.7 completely solves all cases when k = m has the same parity as v. Theo-
rem 4.8 gives a powerful recursive construction which is applicable in cases where m does
not divide v. By applying it to the case when k = 3, we obtain uniform factorizations
with cycle lengths divisible by 3. The case when m is odd and v is even seems to be the
hardest. Even in the simplest case when k = 3, which has been well studied otherwise
[18, 19, 20, 22, 34, 35, 36, 37], the case with v and m having opposite parities has not been
previously considered and remains open.
While the Hamiltonian* problem is solved and we have made significant inroads into
the uniform case, the general problem remains wide open. We expect that when m and
v have the same parity solutions will be easier to find. When m and v have opposite
parity we expect that v odd with m even is more tractable than the reverse. Considering
2-factorizations where a solution to the Oberwolfach problem is known might be a good
starting point. A natural case to consider is the case when all factors are isomorphic i.e.
Fi ∼= Tj ∪ Uj for all i and j. Uniform factorizations are an example of this, but other
variations are possible, for example, requiring all factors to be isomorphic to Cv−m ∪Cm.
Indeed, Theorem 3.4 solves all these cases when the factors are bipartite and v ≡ m ≡ 2
(mod 4), but this broader variant remains open.
More complex variants can also be considered. We have briefly considered systems
with multiple subsystems. When these subsystems are completely nested the problem es-
sentially reduces to the existence of the necessary ingredients as described in Theorem 5.9.
Let {mj} be a multiset of subsystem sizes. When the subsystems are pairwise disjoint, v is
divisible by each mj and either v is odd or at least one subsystem is even, then Lemma 5.1
and Corollary 5.2 use Theorem 1.2 to construct a UMSF(v, {mj}, k) for all but a finite
number of admissible v. Even in the seemingly simple case when the subsystems are all
disjoint the problem remains generally open even for the uniform case. Further partial re-
sults are obtained when the subsystems are scattered, that is, when no two minisymposia
have meetings taking place on the same day. Cycle frames in Theorem 5.3 allow us to
construct uniform factorizations as described in Lemma 5.4 and Theorem 5.5. The more
general case when the intersections of multiple subsystems are arbitrary is completely open.
P. Danziger et al.: On the minisymposium problem 17
ORCID iDs
Tommaso Traetta https://orcid.org/0000-0001-8141-0535
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