The International Congress on Hypergraphs, Graphs and Designs – HyGraDe 2017 This issue of ADAM – The Art of Discrete and Applied Mathematics offers a collection of papers presented at the International Congress on Hypergraphs, Graphs and Designs – HyGraDe 2017, which took place in Sant’Alessio Siculo, Sicily, Italy, June 20 – 24, 2017. HyGraDe 2017 was conceived with the idea of celebrating the 70th birthday of Mario Gionfriddo, a Sicilian mathematician who has devoted his long and successful career to the study of graphs, hypergraphs and designs. The conference HyGraDe 2017 was organized by Francesco Belardo (University of Naples Federico II) and Giovanni Lo Faro (University of Messina), both serving as chair of the Organizing Committee, Luca Giuzzi (University of Brescia), Enzo M. Li Marzi (Uni- versity of Messina), Lorenzo Milazzo (University of Catania), Salvatore Milici (University of Catania) and Antoinette Tripodi (University of Messina). The Scientific Committee con- sisted of Marco Buratti (University of Perugia), Giovanni Lo Faro, Guglielmo Lunardon (University of Naples Federico II) and Martin Milanič (University of Primorska). The conference brought together scientists working in different disciplines of Combina- torics. There were 85 participants from 13 different countries, 10 invited talks and 35 con- tributed talks touching on the latest developments in the corresponding research areas. The invited speakers were Richard Brualdi (University of Winsconsin), Marco Buratti (Univer- sity of Perugia), Charlie Colbourn (Arizona State University), Klavdjia Kutnar (University of Primorska), Josef Lauri (University of Malta), Curt Lindner (University of Auburn), Dragan Marušič (University of Primorska), Alex Rosa (McMaster University), Zsolt Tuza (Hungarian Academy of Sciences) and Vitaly Voloshin (Troy University). Mario Gion- friddo was the honoured participant at the congress. Among the invited speakers, six of them have been Mario’s co-authors in several scientific papers. The Conference Photo The Organizing Committee thanks all who contributed to the successful organization of this event. In particular the Organizing Committee is grateful to the sponsors of this conference: Universities of Catania, Messina and Naples Federico II, the Department of Mathematics and Informatics of Catania, the Department MIFT of Messina, the INDAM- GNSAGA and the non-profit association Combinatorics 2014. We also recognize the kind support of the University of Primorska, the Department of Mathematics and Applications “R. Caccioppoli” of Naples and the Accademia Peloritana dei Pericolanti. We are grateful to the Editors-in-Chief, Professors Tomo Pisanski and Dragan Marušič, for this issue of ADAM. We also thank all the colleagues who have participated in this initiative and the referees who have reviewed the papers. We would like to mention that a special issue of the Sicilian scientific journal Atti Accademia Peloritana dei Pericolanti – AAPP contains other contributions from the participants of HyGraDe 2017. The special issue of AAPP devoted to HyGraDe 2017 can be accessed at http://cab.unime.it/ journals/index.php/AAPP/issue/view/Vol96_Supplement2. Francesco Belardo Guest Editor ISSN 2590-9770 The Art of Discrete and Applied Mathematics 1 (2018) #P2.01 https://doi.org/10.26493/2590-9770.1235.c68 (Also available at http://adam-journal.eu) Circulant matrices and mathematical juggling⇤ Richard A. Brualdi Department of Mathematics, University of Wisconsin, Madison, WI 53706 Michael W. Schroeder Department of Mathematics, Marshall University, Huntington, WV Received 1 January 2018, accepted 25 April 2018, published online 26 July 2018 Abstract Circulants form a well-studied and important class of matrices, and they arise in many algebraic and combinatorial contexts, in particular as multiplication tables of cyclic groups and as special classes of latin squares. There is also a known connection between circulants and mathematical juggling. The purpose of this note is to expound on this connection de- veloping further some of its properties. We also formulate some problems and conjectures with some computational data supporting them. Keywords: Juggling, permutations, permanent, circulant matrices. Math. Subj. Class.: 05A05, 05E25, 15A15 1 Introduction Let n be a positive integer, and let t = (t1, t2, . . . , tn) be a sequence of n nonnegative integers. Then t is a juggling sequence of length n provided that 1 + t1, 2 + t2, . . . , n+ tn (1.1) are distinct modulo n, implying, in particular, that t1 + t2 + · · ·+ tn ⌘ 0 (mod n). Thus if (1.1) holds and balls are juggled where, at time i, there is at most one ball that lands in the juggler’s hand and is immediately tossed so that it lands in ti time units (1  i  n)1, then there are no collisions; that is, juggling balls with one hand according to these rules is possible (for a talented juggler!). The number of balls juggled equals (t1+t2+ · · ·+tn)/n. If we extend t to a two-way infinite sequence (ti : i 2 Z) where ti = ti mod n, then a ball ⇤We are indebted to a referee who helped improve our exposition. E-mail addresses: brualdi@math.wisc.edu (Richard A. Brualdi), schroederm@marshall.edu (Michael W. Schroeder) 1If ti = 0, then there is no ball to toss at time i. cb This work is licensed under http://creativecommons.org/licenses/by/3.0/ 2 Art Discrete Appl. Math. 1 (2018) #P2.01 caught at time i is tossed so that it lands at time i + ti. This defines certain orbits of the balls being juggled determined by the times at which a specified ball is caught and then tossed. The sequence t is a minimal juggling sequence provided that the integers ti have been reduced modulo n to 0, 1, . . . , n 1. In particular, ti = n (a ball is caught and tossed at time i to land in n time units) is equivalent to ti = 0 (no ball is caught and tossed at time i). For some references on mathematical juggling and related work, see e.g. [1, 4, 10]. We now briefly summarize the contents of this paper. In the next section we introduce many examples and discuss some basic properties of juggling sequences and show how they correspond to decompositions of all 1’s matrices. We also show how palindromic jug- gling sequences correspond to a special graph property. In Section 3, we elaborate on the connection between juggling sequences and circulant matrices as discussed in [3], and re- late juggling sequences to the permanent of circulants defined in terms of n indeterminates. In Section 4, we present some calculations concerning the coefficients of the distinct terms in the permanents of these circulants and discuss certain questions and conjectures. Finally, in Section 5 we discuss the existence of juggling sequences with additional properties. Part of the purpose of this paper is to draw attention to a number of directions, questions, and conjectures concerning juggling sequences and the permanent expansion of circulants. 2 Juggling sequences In this section we introduce some of the basic ideas of juggling sequences with many examples and, in the case of palindromic juggling sequences, establish a connection with matchings in complete graphs. A theorem of M. Hall, Jr. [8] for abelian groups when restricted to cyclic groups yields the following result concerning juggling sequences. Theorem 2.1. Let U = {u1, u2, . . . , un} be a multiset of n integers. Then there is at least one permutation ⇡ of {1, 2, . . . , n} such that u⇡ = (u⇡(1), u⇡(2), . . . , u⇡(n)) is a juggling sequence, that is, for which 1 + u⇡(1), 2 + u⇡(2), . . . , n+ u⇡(n) are distinct modulo n, if and only if u1 + u2 + · · ·+ un ⌘ 0 (mod n). (2.1) In this theorem there is no loss in generality in assuming that 0  u1, u2, . . . , un  n 1. In view of Theorem 2.1, we call a multiset U = {u1, u2, . . . , un} of n integers satisfy- ing (2.1) a juggleable set of size n. If u1, u2, . . . , un have been reduced modulo n, then we have a minimal juggleable set. It follows from Theorem 2.1 that U = {0, 1, 2, . . . , n 1} is a (minimal) juggleable set if and only if n is odd. Given U = {u1, u2, . . . , un}, whether or not U is a juggleable set is independent of which representatives of the equivalence classes modulo n determined by the ui have been chosen, in particular, whether or not the integers ui have been reduced modulo n. But if t = (t1, t2, . . . , tn) is a juggling sequence for the juggleable set U , the number of balls that are juggled depends on which representatives of the equivalence classes modulo n have been chosen, in particular, on whether or not the integers in U have been reduced modulo n. R. A. Brualdi and M. W. Schroeder: Circulant matrices and mathematical juggling 3 A juggling sequence (t1, t2, . . . , tn) is determined by a unique permutation of {1, 2, . . . , n} and conversely any permutation of {1, 2, . . . , n} determines a unique jug- gling sequence. Example 2.2. Let n = 7 and consider the permutation of {1, 2, 3, 4, 5, 6, 7} whose cycle decomposition is (1, 5, 6)(2, 4, 7, 3). (Thus in , 1 ! 5 ! 6 ! 1 and 2 ! 4 ! 7 ! 3 ! 2). For each i = 1, 2, . . . , 7, define ti = (i) i mod 7, then t = (4, 2, 6, 3, 1, 2, 3) is a minimal juggling sequence. Reversing this procedure, let n = 9 and consider the juggling sequence t = (1, 5, 3, 4, 8, 3, 3, 6, 3). We obtain a permutation of {1, 2, 3, 4, 5, 6, 7, 8, 9} by calculating and re- ducing modulo 9: (1) = 1 + 1 = 2, (2) = 5 + 2 = 7, (3) = 3 + 3 = 6, (4) = 4 + 4 = 8, (5) = 8 + 5 = 4, (6) = 3 + 6 = 9, (7) = 3 + 7 = 1, (8) = 6 + 8 = 5, (9) = 3 + 9 = 3. Thus is the permutation with cycle decomposition (1, 2, 7)(3, 6, 9)(4, 8, 5). ⌃ Example 2.3. Let n = 3 and consider t = (4, 4, 1). Then to juggle according to t requires three balls and the balls determine three orbits of Z: · · · ! 1 ! 5 ! 9 ! 10 ! 14 ! 18 ! 19 ! · · · , · · · ! 2 ! 6 ! 7 ! 11 ! 15 ! 16 ! 20 ! · · · , · · · ! 3 ! 4 ! 8 ! 12 ! 13 ! 17 ! 21 ! · · · . (Here, for instance, 2 ! 6 represents the fact that at time unit 2, a ball is tossed so that it lands in 4 time units in the future, that is, at time unit 6; then the ball is tossed to land in 1 time unit in the future, that is at time unit 7.) Reducing t mod 3 to (1, 1, 1) results in only one ball and only one orbit: · · · ! 1 ! 2 ! 3 ! 4 ! 5 ! 6 ! · · · . Let Jm,n denote the m ⇥ n matrix of all 1’s. Juggling using the juggling sequence (4, 4, 1) gives a decomposition of the matrix J3,3 of all 1’s whereby any three consecutive matrices sum to J3,3. (The first subscript ‘3’ in J3,3 represents the number of balls juggled, the second ‘3’ represents the number of terms in the juggling sequence. The ordering of the rows is arbitrary.) This is indicated by · · · 4 4 1 4 4 1 4 4 1 1 1 1 1 1 1 1 1 1 · · · , giving J3,3 = 2 4 1 1 1 3 5+ 2 4 1 1 1 3 5+ 2 4 1 1 1 3 5 . Using the mod 3 reduction (1, 1, 1) of (4, 4, 1) gives the trivial decomposition J1,3 = ⇥ 1 1 1 ⇤ . ⌃ 4 Art Discrete Appl. Math. 1 (2018) #P2.01 Example 2.4. Let n = 5 and consider t = (3, 3, 4, 4, 1). Then juggling (with three balls) using this juggling sequence is indicated by · · · 3 3 4 4 1 3 3 4 4 1 3 3 4 4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 · · · , giving the decomposition J3,5 = 2 4 1 1 1 1 1 3 5+ 2 4 1 1 1 1 1 3 5+ 2 4 1 1 1 1 1 3 5 . The juggling sequence t = (2, 4, 2, 3, 4) corresponds to · · · 2 4 2 3 4 2 4 2 3 4 2 4 2 3 4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 · · · , and gives a different decomposition of J3,5. ⌃ We call a juggling sequence t = (t1, t2, . . . , tn) decomposable provided the per- mutation associated with t has at least two nontrivial cycles in its cycle decomposition. Equivalently, t is decomposable provided t = r + s where r = (r1, r2, . . . , rn), s = (s1, s2, . . . , sn) are juggling sequences such that {ri, si} = {0, ti} for i = 1, 2, . . . , n, and r, s 6= t. Any juggling sequence can be uniquely written as a sum of indecomposable juggling sequences arising from the unique cycle decomposition of the associated permu- tation. Example 2.5. With n = 9, t = (1, 5, 3, 4, 8, 3, 3, 6, 3) is a juggling sequence (4 balls). The corresponding decomposition is not that of J3,9 but, after permutation of columns, is J1,3 J1,3 ✓ 1 1 1 +  1 1 1 ◆ . ⌃ We summarize this discussion with the following theorem. Theorem 2.6. Let t = (t1, t2, . . . , tn) be a sequence of n integers. Then t is a (minimal) juggling sequence if and only if : {1, 2, . . . , n} ! {1, 2, . . . , n} defined by (i) ⌘ ti + i (mod n) is a permutation of {1, 2, . . . , n}. There is a one-to-one correspondence between minimal juggling sequences of length n and permutations of {1, 2, . . . , n}. Notice that Theorem 2.6 provides an algorithm to determine whether a sequence is a juggling sequence. Knutson (see [10]) showed how to generate all juggling sequences of length n with k balls (1  k  n) from the constant juggling sequence (k, k, . . . , k) of length n. There are two transformations used in the algorithm: R. A. Brualdi and M. W. Schroeder: Circulant matrices and mathematical juggling 5 I. Given a juggling sequence (k1, k2, . . . , kn), the cyclic shift (kn, k1, k2, . . . , kn1) is also a juggling sequence. II. Given a juggling sequence (k1, . . . , ki, . . . , kj , . . . , kn), then the swap (k1, . . . , (j i) + kj , . . . ,(j i) + ki, . . . , kn) is also a juggling sequence: i+ ((j i) + kj) = j + kj and j + ((j i) + ki) = i+ ki where the balls thrown at times i and j swap landing times. Theorem 2.7 ([10]). Any juggling sequence of length n with k balls can be generated from the constant juggling sequence (k, k, . . . , k) by cyclic shifts and swaps. We now consider a special property of juggling sequences that are palindromic. In the following argument, we use that a sequence (t0, t1, . . . , tn1) is a juggling sequence of length n if ti + i 6⌘ tj + j (mod n) for each i 2 {0, 1, . . . , n 1}, which is a direct consequence of the original definition. That is, if p is a juggling sequence of length n, then, modulo n, p+ (0, 1, 2, . . . , n 1) will be a permutation of {0, 1, 2, . . . , n 1}. Let n be odd and n = 2m+ 1. Let p = (pm, pm1, . . . , p1, p0, p1, . . . , pm1, pm) be a minimal palindromic juggling sequence. For each i 2 {1, 2, . . . ,m}, define xi = pi + i mod n and yi = pi i mod n. Since p is a juggling sequence, we have that, modulo n, (ym, . . . , y2, y1, p0, x1, x2, . . . , xm) = (pm m, . . . , p2 2, p1 1, p0, p1 + 1, p2 + 2, . . . , pm +m) = p+ (0, 1, . . . , n 1) (m,m, . . . ,m). Hence {p0, x1, . . . , xm, y1, . . . , ym} is a set of distinct values. Construct a digraph G(V,E) with vertex set V = {0, 1, 2, . . . , n 1} and edge set E = {e1, . . . , em}, where ei = (xi, yi) for each i 2 {1, 2, . . . ,m}. We define the length of edge ei to be yi xi mod n. Hence each ei has length n 2i; thus G is a directed near 1-factor whose set of edge lengths is {1, 3, 5, . . . , n 2}. Conversely, let V = {0, 1, 2, . . . , n 1} and suppose G(V,E) is a directed near 1- factor whose set of edge lengths is {1, 3, 5, . . . , n 2}. Then we may assume E = {e1, . . . , em}, where ei is the directed edge of length n 2i with ei = (xi, yi). Let p0 denote the vertex in G not incident to any edge, and for each i 2 {1, 2, . . . ,m}, let pi = xi i mod n. Then pi = yi + i mod n for each i 2 {1, 2, . . . ,m}. Define p = (pm, pm1, . . . , p1, p0, p1, . . . , pm1, pm). Then modulo n we have p+ (0, 1, . . . , n 1) = (ym, . . . , y2, y1, p0, x1, x2, . . . , xm) + (m,m, . . . ,m). Since all values in {p0, x1, . . . , xm, y1, . . . , ym} are distinct, p is a juggling sequence. These two operations which map between minimal palindromic juggling sequences of length n and directed near 1-factors on n vertices whose set of edge lengths is {1, 3, 5, . . . , n 2} are inverses of one another, which leads to the following theorem. Theorem 2.8. Let n be an odd positive integer. Then there is a one-to-one correspondence between minimal palindromic juggling sequences of length n and directed near 1-factors on the vertex set {0, 1, . . . , n 1} whose set of edge lengths is {1, 3, 5, . . . , n 2}. 6 Art Discrete Appl. Math. 1 (2018) #P2.01 A similar construction gives a result for all positive even integers n. Theorem 2.9. Let n be a positive even integer. Then there is a one-to-one correspondence between minimal palindromic juggling sequences of length n and directed 1-factors on the vertex set {0, 1, . . . , n 1} whose set of edge lengths is {1, 3, 5, . . . , n 1}. Proof. Let n = 2m. The proof method is similar to that given for the argument to Theo- rem 2.8, so in what follows we give only the construction for the correspondence. Let {(xi, yi) | i 2 {1, 2, . . . ,m}} be a 1-factor on {0, 1, 2, . . . , n 1} with (xi, yi) having length 2i 1 for each i 2 {1, 2, . . . ,m}. For each i 2 {1, 2, . . . ,m}, let pi = xi m+ i mod n. Then pi = yi m i+ 1 mod n. So modulo n, (xm, . . . , x2, x1, y1, y2, . . . , ym) (0, 1, . . . , n 1) = (pm, . . . , p2, p1, p1, p2, · · · , pm). Therefore (pm, . . . , p2, p1, p1, p2, . . . , pm) is a minimal palindromic juggling sequence. Conversely, if (pm, . . . , p2, p1, p1, p2, . . . , pm) is a minimal palindromic juggling se- quence, then we may define xi = pi + m i mod n and yi = pi + m + i 1 mod n and have that {(xi, yi) | i 2 {1, 2, . . . ,m}} is the edge set of a directed 1-factor in which (xi, yi) has length 2i 1 for each i 2 {1, 2, . . . ,m}. Example 2.10. For n = 6, (2, 5, 2, 2, 5, 2) is the minimal palindromic juggling sequence corresponding to the directed 1-factor with edge set {(4, 5), (0, 3), (2, 1)}. Note the edges have lengths 1, 3, and 5, respectively. Similarly for n = 7, (2, 5, 3, 1, 3, 5, 2) is the minimal palindromic juggling sequence corresponding to the directed near 1-factor with unused vertex 1 and edge set {(4, 2), (0, 3), (5, 6)}. In this case, the edges have lengths 5, 3, and 1, respectively. ⌃ 3 Juggleable sets and circulants Let Pn be the set of minimal juggleable sets of size n. For U 2 Pn, let Jn(U) be the set of juggling sequences of length n with U as juggleable set. It follows from [2] that the number of minimal juggleable sets of size n is given by |Pn| = 1 n X d|n ✓ 2d 1 d ◆ ⇣ n d ⌘ (3.1) where is Euler’s totient function and the summation extends over all positive integers d dividing n. The number of minimal juggling sequences of length n is n!, since for each permutation (i1, i2, . . . , in) of {1, 2, . . . , n}, we have (i1 1) + (i2 2) + · · ·+ (in n) = nX i=1 i nX i=1 i = 0, and hence the multiset {i11, i22, . . . , inn} of integers taken modulo n, is a juggleable set. Let n, k, and ⌫ be positive integers. In [7] it is proved that the number of nonnegative integer solutions of u1 + u2 + · · ·+ un = k and nX i=1 iui ⌘ ⌫ (mod n) (3.2) R. A. Brualdi and M. W. Schroeder: Circulant matrices and mathematical juggling 7 equals the number of nonnegative integer solutions of v1 + v2 + · · ·+ vk = n and kX i=1 ivi ⌘ ⌫ (mod k). (3.3) Taking ⌫ = 0, we get the following duality result. Theorem 3.1. The number of minimal juggleable sets {u1, u2, . . . , un} with u1 + u2 + · · ·+un = k equals the number of minimal juggleable sets {v1, v2, . . . , vk} with v1+v2+ · · ·+ vk = n. The above discussion gives a characterization of the number of juggling sequences corresponding to each minimal juggleable set. Theorem 3.2. Let U = {u1, u2, . . . , un} be a minimal juggleable set. The number |J (U)| of juggling sequences with U as juggleable set equals the number of permuta- tions (j1, j2, . . . , jn) of {1, 2, . . . , n} such that ji ⌘ i + r (mod n) has ui solutions for each r = 0, 1, . . . , n 1. Another viewpoint (see [2]) is the following. Consider the n⇥ n circulant matrix C(x0, x1, . . . , xn1) = 2 666664 x0 x1 · · · xn2 xn1 xn1 x0 · · · xn3 xn2 ... ... . . . ... ... x2 x3 · · · x0 x1 x1 x2 · · · xn1 x0 3 777775 . (3.4) Thus C(x0, x1, . . . , xn1) = x0In + x1Pn + x2P 2 n + · · ·+ xn1Pn1n , where Pn is the n ⇥ n permutation matrix corresponding to the cyclic permutation (2, 3, . . . , n, 1) (thus P 0 n = Pn n = In). The book [6] contains a thorough discussion of circulants. Recall that the permanent of an n⇥ n matrix A = [aij : 0  i, j  n] is per(A) = X (i1,i2,...,in) a1i1a2i2 · · · an,in where the summation extends over all the permutations (i1, i2, . . . , in) of {1, 2, . . . , n}. Each term a1i1a2i2 · · · an,in in the permanent of C(x0, x1, . . . , xn1) is of the form x k0 0 x k1 1 · · ·x kn1 n1 where k0, k1, . . . , kn1 are integers such that 0  ki  n, (0  i  n 1) and k0 + k1 + · · ·+ kn1 = n, and ki is the number of integers r with 0  r  n 1 such that ir r ⌘ ki (mod n) and k0 · 0 + k1 · 1 + · · ·+ kn1 · (n 1) ⌘ 0 (mod n). 8 Art Discrete Appl. Math. 1 (2018) #P2.01 Thus the number of distinct terms in the permanent of the circulant C(x0, x1, . . . , xn1) equals the number |Pn| of juggleable sets of size n and thus is given by (3.1). Theorem 2.1 implies that the monomial x0x1 . . . xn1 is a term in per(A) if and only if 1 · 0 + 1 · 1 + · · ·+1 · (n1) ⌘ 0 (mod n); since 0+1+ · · ·+(n1) = n(n1)/2, x0x1 . . . xn1 is a term in per(A) if and only if n is odd. Now let n be even. Then a monomial of the form x k0 0 x k1 1 · · ·x kn1 n1 with kr = 2, ks = 0, and all other ki’s equal to 1, is a term in per(A) if and only if |r s| = n/2. In [9] it is shown that |Pn| equals the dimension of a certain symmetric space associated with a cyclic group of order n. See [12] for a comparison with the number of distinct terms occurring in the determinant. The following corollary is a direct consequence of Theorem 3.2 and the definitions of a circulant matrix and the permanent. Corollary 3.3. Two permutations j1, j2, . . . , jn and l1, l2, . . . , ln of {1, 2, . . . , n} give the same term in the permanent of C(x0, x1, . . . , xn1) if and only if |{i : ji ⌘ i+ r (mod n)}| = |{i : li ⌘ i+ r (mod n)}| for each r = 0, 1, . . . , n 1. If the common values are k0, k1, . . . , kn1, then the term in the permanent equals x k0 0 x k1 1 · · ·x kn1 n1 . Example 3.4. Table 1 gives the minimal juggleable sets of size n = 4 and their correspond- ing terms in per(C(x0, x1, . . . , xn1)), along with the juggling sequences corresponding Table 1: Minimal juggleable sets and juggling sequences for n = 4. Juggleable sets U {u0, u1, u2, u3} Corresponding term in the permanent Corresponding juggling sequences J4(U) Cardinalities |J4(U)| (coefficients) {0, 0, 0, 0} x40 (0, 0, 0, 0) 1 {1, 1, 1, 1} x41 (1, 1, 1, 1) 1 {2, 2, 2, 2} x42 (2, 2, 2, 2) 1 {3, 3, 3, 3} x43 (3, 3, 3, 3) 1 {0, 0, 2, 2} x20x22 (0, 2, 0, 2), (2, 0, 2, 0) 2 {1, 1, 3, 3} x21x23 (1, 3, 1, 3), (3, 1, 3, 1) 2 {0, 0, 1, 3} x20x1x3 (0, 0, 1, 3), (0, 1, 3, 0), (1, 3, 0, 0), (3, 0, 0, 1) 4 {0, 1, 1, 2} x0x21x2 (0, 1, 1, 2), (1, 1, 2, 0), (1, 2, 0, 1), (2, 0, 1, 1) 4 {1, 2, 2, 3} x1x22x3 (1, 2, 2, 3), (2, 2, 3, 1), (2, 3, 1, 2), (3, 1, 2, 2) 4 {0, 2, 3, 3} x0x2x23 (0, 2, 3, 3), (2, 3, 3, 0), (3, 3, 0, 2), (3, 0, 2, 3) 4 to each such pattern and their number. ⌃ R. A. Brualdi and M. W. Schroeder: Circulant matrices and mathematical juggling 9 As in Table 1 for n = 4, constant juggleable sets correspond to the n monomials terms x n 0 , xn1 , . . ., xnn1 of the permanent of the matrix C(x0, x1, . . . , xn1) each occuring with coefficient equal to 1. If {u1, u2, . . . , un} is a minimal juggleable set of size n, we define c(u1, u2, . . . , un) to be the number of juggling sequences of length n whose pattern is given by {u1, u2, . . . , un}. Therefore, c(u1, u2, . . . , un) equals the number of permutations (i1, i2, . . . , in) of {1, 2, . . . , n} such that ir r ⌘ ki (mod n) has ui solutions for i = 1, 2, . . . , n. The permanent of C(x0, x1, . . . , xn1) is then given by the homogeneous polynomial of degree n, X {u1,u2,...,un}2Pn c(u1, u2, . . . , un)x u1 0 x u2 1 · · ·xun1n , whose number of terms is given by (3.1). Thus from Table 1 we see that the permanent of C(x0, x1, x2, x3) equals 1x40x 0 1x 0 2x 0 3 + 1x 0 0x 4 1x 0 2x 0 3 + 1x 0 0x 0 1x 4 2x 0 3 + 1x 0 0x 0 1x 0 2x 4 3 + 2x 2 0x 0 1x 2 2x 0 3 + 2x00x 2 1x 0 2x 2 3 + 4x 2 0x 1 1x 0 2x 1 3 + 4x 1 0x 2 1x 1 2x 0 3 + 4x 0 0x 1 1x 2 2x 1 3 + 4x 1 0x 0 1x 1 2x 2 3. As the referee pointed out, c(u1, u2, . . . , un) is the number of ways to arrange the multiset consisting of u1 0’s, u2 1’s, . . . , un (n 1)’s into a juggling sequence. Some evaluation of these numbers can be found in sequence A006717 [11]. Theorem 3.5. If U = {u1, u2, . . . , un} is a minimal juggleable set of size n, then c(u1, u2, . . . , un) 1. (3.5) Equality holds in (3.5) if and only if U is a constant multiset. If n is a prime p and U is not a constant multiset, then p is a divisor of c(u1, u2, . . . , un). Proof. If U is a constant minimal juggleable set {k, k, . . . , k}, then xn k occurs as a term in the permanent of C(x0, x1, . . . , xn1) corresponding to the positions of the 1’s in P kn , that is, the positions (1, k + 1), (2, k + 2), . . . , (n, k + n) taken modulo n. If {u1, u2, . . . , un} is a non-constant juggleable set, there is a term in the permanent of C(x0, x1, . . . , xn1) equal to xu10 x u2 1 · · ·x un n1 not arising solely from the n positions (1, k+ 1), (2, k+ 2), . . . , (n, k + n) modulo n corresponding to the 1’s in the permutation matrices In, Pn, P 2n , . . . , P n1 n . The k ⇥ k principal submatrix C[i1, i2, . . . , ik | i1, i2, . . . , ik] = C(xi1 , xi2 , . . . , xik) of C determined by rows and columns i1, i2, . . . , ik is cyclically permutation equivalent (row and column indices are taken modulo n) to the submatrix C[i1+1, i2+1, . . . , ik+1 | i1 +1, i2 +1, . . . , ik +1] = C(xi1+1, xi2+1, . . . , xik+1) determined by rows and columns i1 + 1, i2 + 1, . . . , ik + 1 taken modulo n. Thus if we take a monomial in the permanent corresponding to a permutation j1, j2, . . . , jn, we get n 1 other equal monomials by sequentially adding 1 modulo n to each of j1, j2, . . . , jn and cyclically permuting: (j1, j2, . . . , jn) ! (jn + 1, j1 + 1, . . . , jn1 + 1) ! · · · (3.6) ! (j2 + (n 1), . . . , jn + (n 1), j1 + (n 1)). If U is a non-constant juggleable set, then not all these permutations can be equal. (If e.g. all of these n permutations are equal, then (j1, j2, . . . , jn) is a cyclic permutation 10 Art Discrete Appl. Math. 1 (2018) #P2.01 a, a+1, a+2, . . . , a+(n1) modulo n giving the monomial xn i with coefficient equal to 1.) This amounts to simultaneously permuting rows and columns of C(x0, x1, . . . , xn1) using the permutation matrix Pn and replacing the permutation (and its corresponding term in the permanent) with the image of (j1, j2, . . . , jn) under this action. The result is a term in the permanent with the same value; basically we have that the position (i, j) moves into the position (i + 1, j + 1) (indices taken mod n) under the action of Pn, so to position (i+ l, j + l) (indices taken mod n) under the action of P l. So the set of positions in those sets corresponding to powers of P have to be invariant under a cyclic shift by l in order to get another term in the permanent with the same value. If n is a prime this cannot happen unless the term is of the form xn i . Since there may be other terms of equal value in the permanent of C(x0, x1, . . . , xn1), we have that p | c(u1, u2, . . . , un). Corollary 3.6. If n is odd, the coefficient of c(1, 1, . . . , 1) of x0x1 · · ·xn1 in the perma- nent per(C(x0, x1, . . . , xn1)) is divisible by n. Proof. The corollary follows as in the proof of Theorem 3.5 since the term x0x1 · · ·xn1 comes from the juggleable set {0, 1, . . . , n 1} and whatever order gives a juggling se- quence, each of the (n 1) cyclic shifts is different, resulting in a contribution of n to the coefficient. 4 Coefficients in per(C(x0, x1, . . . , xn1)) We first consider the special case of n = 5. Example 4.1. Let n = 5. The formula (3.1) for the number of distinct terms in the perma- nent of C(x0, x1, x2, x3, x4) is 1 5 ✓ (5) + ✓ 9 5 ◆ (1) ◆ = 1 5 (4 + 126) = 26. There are five constant terms in the permanent each with coefficient 1 and there are twenty- one terms each with coefficient divisible by 5. So either we have two terms each with coefficient 10 and nineteen terms with coefficient 5, or we have one term with coefficient 15 and twenty terms with coefficient 5. The term x0x1x2x3x4 occurs in each of the following: 2 66664 x0 x1 x2 x3 x4 3 77775 , 2 66664 x0 x3 x1 x4 x2 3 77775 , 2 66664 x0 x2 x4 x1 x3 3 77775 . and thus, by cyclically simultaneously permuting rows and columns (changing the diagonal position in which x0 occurs by shifting along the main diagonal), appears in the permanent with coefficient at least 15 and therefore exactly 15. Note the positions occupied by the xi with i 6= 0 above: 2 66664 x1 x2 x3 x4 x1 x2 x3 x4 x1 x2 x3 x4 3 77775 . R. A. Brualdi and M. W. Schroeder: Circulant matrices and mathematical juggling 11 Each xi with i 6= 0 occupies all the positions in the submatrix obtained by striking out row 1 and column 1 that it occupies in C(x0, x1, x2, x3, x4). Thus this simple analysis gives per(C(x0, x1, x2, x3, x4)) = 4X i=0 x 5 i + 5(twenty other terms) + 15x0x1x2x3x4. ⌃ From calculations of per(C(x0, x1, . . . , xn1)) using Sage, we found the following information: • (n = 5): largest coefficient is 15 occuring uniquely for x0x1x2x3x4. Coefficients are 1, 5, 15. This confirms the calculations in Example 4.1. • (n = 6): largest coefficient is 24 occuring for the six terms of the form x 2 0x1x2x 0 3x4x5. Coefficients are 1, 2, 3, 6, 9, 12, 18, 24. • (n = 7): largest coefficient is 133 occuring uniquely for x0x1x2x3x4x5x6. Coefficients are 1, 7, 14, 21, 35, 42, 49, 133. • (n = 8): largest coefficient is 256 occuring for the 8 terms x 2 0x1x2x3x 0 4x5x6x7, x0x 2 1x2x3x4x 0 5x6x7, x0x1x 2 2x3x4x5x 0 6x7, x0x1x2x 2 3x4x5x6x 0 7, x 0 0x1x2x3x 2 4x5x6x7, x0x 0 1x2x3x4x 2 5x6x7, x0x1x 0 2x3x4x5x 2 6x7, x0x1x2x 0 3x4x5x6x 2 7. For instance, x20x1x2x3x04x5x6x7 occurs in the term 2 66666666664 x0 x2 x3 x6 x0 x5 x1 x7 3 77777777775 . There are 810 different terms that occur in per(C(x0, x1, . . . , x7)). The full set of coefficients in the permanent are {1, 2, 4, 6, 8, 12, 16, 20, 24, 32, 40, 48, 56, 64, 72, 80, 96, 128, 160, 256}. Note that the differences of consecutive coefficients in this list are: 1, 2, 2, 2, 4, 4, 4, 4, 8, 8, 8, 8, 8, 8, 8, 16, 32, 32, 96. Only the last is not a power of 2. 12 Art Discrete Appl. Math. 1 (2018) #P2.01 Conjecture 4.2. Our calculations have shown that for n = 4, the largest coefficient in per(C(x0, x1, . . . , xn1)) is 4 = 22 occuring 4 times, and for n = 8, the largest coeffi- cient is 256 = 28 occuring 8 times. We conjecture that if n is a power of 2, then the largest coefficient is also a power of 2 occuring for the terms of the form x20x1x2 · · · dxn/2 · · ·xn1, and cyclical translates of terms of this form (total number of different terms is n). Unfor- tunately, the occurrence of these terms does not seem to have a pattern. For instance, with n = 8, we have 2 66666666664 x0 x1 x2 x3 x4 x5 x6 x7 x7 x0 x1 x2 x3 x4 x5 x6 x6 x7 x0 x1 x2 x3 x4 x5 x5 x6 x7 x0 x1 x2 x3 x4 x4 x5 x6 x7 x0 x1 x2 x3 x3 x4 x5 x6 x7 x0 x1 x2 x2 x3 x4 x5 x6 x7 x0 x1 x1 x2 x3 x4 x5 x6 x7 x0 3 77777777775 (x20x1x2x3x5x6x7). This corresponds to the permutation of {1, 2, 3, 4, 5, 6, 7, 8} such that (1) = 1, (2) = 2, (3) = 8, (4) = 5, (5) = 7, (6) = 4, (7) = 6, (8) = 3. The coefficient of x20x1x2x3x5x6x7 is the number of permutations ⇡ of {1, 2, 3, 4, 5, 6, 7, 8}, such that ⇡(i) i ⌘ j (mod 8) has two solutions for j = 0, no solutions for j = 4, and one solution for j = 1, 2, 3, 5, 6, 7. A similar statement holds for all even n, and we seek the number of such solutions. Conjecture 4.3. Three conjectures/problems for n even (or perhaps just n a power of 2): (a) There exists a term x20x1x2 · · · dxn/2 · · ·xn1 in per(C(x0, x1, . . . , xn1)) arising from every choice of the two x0’s on the main diagonal. (b) In reference to (a), the largest number of terms occurs when the x0’s are chosen to be n/2 apart (cyclically, the same number of elements on the main diagonal between them). In the case of n = 8, there are 16 terms for a choice of x0’s which are 4 apart (5th x0 on the main diagonal minus 1st x0 on main diagonal) and 8 terms for all other choices of x0’s. (c) If n is a power of 2, the coefficient of x20x1x2 · · · dxn/2 · · ·xn1 is a power of 2. Problem 4.4. The matrix C(x0, x1, . . . , xn1) can be regarded as a special latin square. The coefficient of x0x1 · · ·xn1 in per(C(x0, x1, . . . , xn1)) equals the number of trans- versals of this latin square. In [5] it is shown that if n is odd and sufficiently large, the coefficient of x0x1 · · ·xn1 in per(C(x0, x1, . . . , xn1)) is greater than (3.246)n. In the cases of n = 5 and n = 7, the number of latin square transversals of C(x0, x1, . . . , xn1) equals 15 = 5 ⇥ 3 and 133 = 7 ⇥ 19, respectively. Since a latin square transversal is mapped into a latin square transversal by multiplying C(x0, x1, . . . , xn1) by the full cycle permutation matrix Pn, it follows that for odd n, the number of latin square transversals, that is, c(1, 1, . . . , 1) is divisible by n. See also Theorem 3.5 and Corollary 3.6. R. A. Brualdi and M. W. Schroeder: Circulant matrices and mathematical juggling 13 If n is odd, the term x0x1 · · ·xn1 occurs in per(C(x0, x1, . . . , xn1) with a nonzero coefficient. A conjecture would be that this term has the largest coefficient. Thinking of the xi as n different colors giving n! multicolored transversals, the conjecture is saying that the number of multicolored transversals with all colors different is greater than the num- ber of multicolored transversals of any other prescribed color type (so at least two colors the same). This coefficient is equal to the number of transversals of C(x0, x1, . . . , xn1) considered as a latin square, so finding this exactly is probably not attainable (see [5]). Remark 4.5. Concerning Problem 4.4 and the juggleable set {1, 2, . . . , n} with n odd, corresponding to the term x0x1 · · ·xn1 in per(C(x0, x1, . . . , xn1)). A permutation (i1, i2, . . . , in) of this juggleable set is a juggling sequence giving the term x0x1 · · ·xn1 in per(C(x0, x1, . . . , xn1)) provided 1 + i1, 2 + i2, . . . , n+ in are distinct modulo n. If this is the case, then any cyclic permutation of (i1, i2, . . . , in) is also a juggling sequence (since subtracting 1 modulo n from distinct integers modulo n gives distinct integers mod- ulo n, thereby giving n terms in per(C(x0, x1, . . . , xn1)) equal to x0x1 · · ·xn1. The difficulty in calculating the coefficient of x0x1 · · ·xn1 in per(C(x0, x1, . . . , xn1)) is knowing how many permutations i1, i2, . . . , in of the set {1, 2, . . . , n} have the property that 1 + i1, 2 + i2, . . . , n + in are distinct modulo n. So one might consider the additive group Z(n)n = Zn ⇥ Zn ⇥ · · ·⇥ Zn (n copies of Zn) and the mapping T : Z(n) n ! Z(n) n given by T (i1, i2, . . . , in) = (1 + i1, 2 + i2, . . . , n+ in) = (1, 2, . . . , n) + (i1, i2, . . . , in) mod n. Unfortunately, this mapping is not a homomorphism and so does not seem useful. But it does seem that for a juggleable set {u1, u2, . . . , un} with at least one repeat, that is, the number of permutations (u1, u2, . . . , un) of this pattern such that 1+u1, 2+u2, . . . , n+un are distinct modulo n is smaller than when there is no repeat in {u1, u2, . . . , un}. But it seems difficult to make a comparison. Remark 4.6. Assume n is odd. Then x10x11x12 · · ·x1n1 occurs in per(C(x0, x1, . . . , xn1)) with a nonzero coefficient. We can think of this term as generating other terms that occur in per(C(x0, x1, . . . , xn1)) as follows: We increase or decrease (by 1) some of the exponents of this term to get x 1+a0 0 x 1+a1 1 x 1+a2 2 · · ·x 1+an1 n1 where each ai 2 {1, 0,1}, and n1X i=0 ai = 0 (4.1) and, in order that the result is a term in per(C(x0, x1, . . . , xn1)), we must have n1X i=0 iai ⌘ 0 (mod n). (4.2) 14 Art Discrete Appl. Math. 1 (2018) #P2.01 (By (4.2), P n1 i0 (1 + ai) = 0 and P n1 i=0 i(ai + 1) ⌘ 0 (mod n) and thus gives a term in this permanent.) We can do a similar operation on the resulting term but then we need to be sure that the resulting exponents are always between 0 and n. Continuing like this we can generate all terms that occur in this permanent. So in this operation we increase s 1 exponents by 1 and decrease s exponents by 1, so adding (a0, a1, a2, . . . , an1), subject to the condition (4.2), to the vector of exponents in a term in our permanent. One line of investigation is to try to determine when this operation increases/decreases the coefficient of the corresponding terms in our permanent. In particular, when with one application starting with the term x10x11x12 · · ·x1n1, does the coefficient decrease? Note that in one application, we must reduce two exponents to 0 in order that we satisfy (4.2); in general there must be at least four changes in exponents. See the following example. Example 4.7. Let n = 9. We start with the term x0x1x2x3x4x5x6x7x8. We can change exponents by using the vector (0, 0, 1,1, 0,1, 1, 0, 0). Since 1 · 2 1 · 3 + (1) · 5 + 1 · 6 = 0 ⌘ 0 (mod 9), x0x1x 2 2x4x 2 6x7x8 is a term in our permanent. ⌃ Problem 4.8. If n is even, then we can also ask for the term(s) with the largest coefficient. If n = 4, there are four terms that appear with the largest coefficient of 4, namely x 2 0x1x3, x0x 2 1x2, x1x 2 2x3, x0x2x 2 3. A conjecture might be: If n is even then the terms in per(C(x0, x1, . . . , xn1)) that occur with the largest coefficient are the terms with the property that xi occurs with exponent 2, xi+n/2 (subscript mod n) occurs with exponent 0, and all other xi appear with exponent 1. Remark 4.9. We have that there is a nonzero term in per(C(x0, x1, . . . , xn1)) with ex- actly two nonzero exponents (so binomials) if and only if n is not a prime. The reason is as follows: Suppose xa i x b j occurs with a nonzero coefficient where 0  j < i  n 1 and i 6= j, and a, b 1, and a+ b = n (and so a, b  n 1). Then by Hall’s theorem ai+ bj = ai+ (n a)j ⌘ 0 (mod n), that is, a(i j) ⌘ 0 (mod n). If n is a prime p, this is a contradiction since p - a and p - (i j). If n is not a prime, say n = uv where 1 < u, v < n1. Then we may choose a = u, and i and j so that ij = v, and get a term xa i x (na) j with a nonzero coefficient. In investigating binomials in per(C(x0, x1, . . . , xn1)) it is sufficient to consider bi- nomials of the form xa0xbk where 1  k  n 1. Thus we consider the terms of per(x0In + xkP kn ) different from xn0 and xnk . This permanent is easily computed: per(x0In + xkP k n ) = dX t=0 ✓ d t ◆ x t n d 0 x (dt)nd k where d = gcd(n, k). Thus the largest coefficient of a binomial is d d 2 . More generally, let H ✓ {0, 1, . . . , n 1}. If we set xj = 0 if j 62 H , then the permanent of the resulting matrix CH(x0, x1, . . . , xn1) gives all the terms that occur in R. A. Brualdi and M. W. Schroeder: Circulant matrices and mathematical juggling 15 per(C(x0, x1, . . . , xn1)) and their coefficients in which the only xi that can occur are those with i 2 H . By also setting xi = 1 for i 2 H , the permanent equals the number of terms in per(CH(x0, x1, . . . , xn1)). Remark 4.10. Now consider terms in per(C(x0, x1, . . . , xn1)) where there are exactly three nonzero exponents (so in the juggling context, three different heights in throwing the balls). These terms are then trinomials. Which trinomial has the largest coefficient among all trinomials that occur in per(C(x0, x1, . . . , xn1))? The conjecture is that the maximum coefficient occurs when the exponents are as equal as possible; in particular if n = 3k, then the trinomial will largest coefficient is conjectured to be xk0xkkx k 2k and its cyclic permutations. In investigating trinomials in per(C(x0, x1, . . . , xn1)) it suffices to consider terms of the form xa0xbrxcs where 0 < r < s < n and a + b + c = n, that is it suffices to consider the trinomials in per(x0In + xrP r n + xsP s n ). The conjecture is that the largest coefficient of a trinomial in this permanent occurs when the exponents are as equal as possible and the powers of Pn, i.e. the subscripts of the x’s are as equally spaced as possible (in the cyclic sense). If n = 3k, then after permutations x0In + xkP kn + x2kP 2k n becomes a direct sum of k 3⇥ 3 matrices of the form x0I3 + xkP3 + x2kP 2 3 . 5 Juggling sequences with additional properties Let U = {u1, u2, . . . , un} be a minimal juggleable set, and let u⌧(1), u⌧(2), . . . , u⌧(n) be a juggling sequence corresponding to U . Thus ⌧ is a permutation of {1, 2, . . . , n} and it is natural to ask about the existence of such permutations ⌧ with additional properties, equiv- alently, extensions of Theorem 2.1 by imposing additional restrictions on the permutation ⌧ . Juggling sequences correspond to transversals in the circulant C(x0, x1, . . . , xn1) and thus we seeks transversals of C(x0, x1, . . . , xn1) whose pattern has additional properties. Two natural permutations to consider are involutions and centrosymmetric permutations of of {1, 2, . . . , n}. Involutions are permutations of {1, 2, . . . , n} where for all i and j, (i) = j implies (j) = i, and these correspond to transversals of C(x0, x1, . . . , xn1) whose positions have a symmetric matrix pattern, that is, transversal patterns invariant under a reflection about the main diagonal. A permutation is centrosymmetric pro- vided that for all i, (i) + (n + 1 i) = n + 1 and these correspond to transversals of C(x0, x1, . . . , xn1) whose positions have a centrosymmetric matrix pattern, that is, transversal patterns invariant under a 180 degree rotation. There are permutations that are both symmetric and centrosymmetric. Example 5.1. Let n = 4 and let = (2, 1, 4, 3). As a permutation matrix, equals 2 664 1 1 1 1 3 775 which is invariant under a reflection about the diagonal and a rotation of 180 degrees. Thus is both an involution (invariant under a reflection about the main diagonal) and a cen- trosymmetric permutation (invariant under a 180 degree rotation). Notice that is also 16 Art Discrete Appl. Math. 1 (2018) #P2.01 invariant under reflection about the antidiagonal running from the lower left to the upper right, and this holds in general for permutations that are both symmetric and centrosym- metric. ⌃ Let U = {u1, u2, . . . , un} be a multiset where ui 2 {0, 1, . . . , n1} for 0  i  n1. We say that U is balanced mod n provided that its nonzero elements can be paired as {a, b} so that a+b ⌘ 0 (mod n). Thus if n is even, 0 and n/2 each occur an even, possibly zero, number of times, and if n is odd, 0 occurs an odd number of times. If U is balanced mod n, then it is an immediate consequence of Theorem 2.1 that U is a juggleable set with each xi with i 6= 0 occurring with an even, possibly zero, exponent in per(C(x0, x1, . . . , xn1)). Example 5.2. Let n = 8 and let U = {0, 0, 1, 7, 1, 7, 4, 4}. Then U is balanced mod 8 and hence is a juggleable set. In C(x0, x1, . . . , x7) below we have realizations 1 2 3 4 5 6 7 8 1 x0 x1 x2 x3 x4 x5 x6 x7 2 x7 x0 x1 x2 x3 x4 x5 x6 3 x6 x7 x0 x1 x2 x3 x4 x5 4 x5 x6 x7 x0 x1 x2 x3 x4 5 x4 x5 x6 x7 x0 x1 x2 x3 6 x3 x4 x5 x6 x7 x0 x1 x2 7 x2 x3 x4 x5 x6 x7 x0 x1 8 x1 x2 x3 x4 x5 x6 x7 x0 . corresponding to the term x20x21x24x27 in per(C(x0, x1, . . . , x7)), achieved in the permanent per(C(x0, x1, . . . , x7)) by an involution (dark gray) and by a centrosymmetric permutation (light gray). ⌃ Example 5.3. Let n = 6 and consider the multiset U = {2, 2, 2, 4, 4, 4} balanced mod 6 with the pairing {2, 4}, {2, 4}, {2, 4}. In both case we seek a corresponding transversal in 2 6666664 x0 x1 x2 x3 x4 x5 x5 x0 x1 x2 x3 x4 x4 x5 x0 x1 x2 x3 x3 x4 x5 x0 x1 x2 x2 x3 x4 x5 x0 x1 x1 x2 x3 x4 x5 x0 3 7777775 , consisting of three x2’s and three x4’s. We have indicated such a realization in the cen- trosymmetric case, but it is straightforward to check that it cannot be attained by an invo- lution. ⌃ We have done a substantial amount of calculation with the following consequences: (i) For n  19 a prime, all balanced mod n multisets can be achieved by a transversal with a symmetric pattern. When n = 15, there are 16 balanced mod 15 multisets that cannot be achieved by a transversal with a symmetric pattern, e.g. the multiset {0, 6, 6, 6, 6, 6, 6, 6, 9, 9, 9, 9, 9, 9, 9} cannot be so achieved. On the other hand, for n = 18, there are 48 620 balanced mod 18 multisets satisfying (2.1) and only 36 195 can be achieved with a symmetric pattern. R. A. Brualdi and M. W. Schroeder: Circulant matrices and mathematical juggling 17 (ii) For odd n  21, all balanced mod n multisets can be achieved by a transversal with a centrosymmetric pattern. As a consequence of the data obtained we make two conjectures: Conjecture 5.4. If n is a prime, then every balanced mod n multiset can be achieved by a transversal with a symmetric pattern. Conjecture 5.5. If n is odd, then a balanced mod n multiset can be achieved by a transver- sal with a centrosymmetric pattern. If n is even, then the unachievable balanced mod n multisets only have terms with the same parity. References [1] E. Banaian, S. Butler, C. Cox, J. Davis, J. Landgraf and S. Ponce, Counting prime juggling patterns, Graphs Combin. 32 (2016), 1675–1688, doi:10.1007/s00373-016-1711-1. [2] R. A. Brualdi and M. Newman, An enumeration problem for a congruence equation, J. Res. Nat. Bur. Standards Sect. B 74B (1970), 37–40, doi:10.6028/jres.074b.003. [3] S. Butler and R. Graham, private communication. [4] S. Butler and R. Graham, Enumerating (multiplex) juggling sequences, Ann. Comb. 13 (2010), 413–424, doi:10.1007/s00026-009-0040-y. [5] N. J. Cavenagh and I. M. Wanless, On the number of transversals in Cayley tables of cyclic groups, Discrete Appl. Math. 158 (2010), 136–146, doi:10.1016/j.dam.2009.09.006. [6] P. J. Davis, Circulant Matrices, Pure and Applied Mathematics, John Wiley & Sons, New York, 1979. [7] M. L. Fredman, A symmetry relationship for a class of partitions, J. Comb. Theory Ser. A 18 (1975), 199–202, doi:10.1016/0097-3165(75)90008-4. [8] M. Hall, Jr., A combinatorial problem on abelian groups, Proc. Amer. Math. Soc. 3 (1952), 584–587, doi:10.2307/2032592. [9] D. I. Panyushev, Fredman’s reciprocity, invariants of abelian groups, and the permanent of the Cayley table, J. Algebraic Combin. 33 (2011), 111–125, doi:10.1007/s10801-010-0236-6. [10] B. Polster, The Mathematics of Juggling, Springer-Verlag, New York, 2003. [11] N. J. A. Sloane (ed.), The On-Line Encyclopedia of Integer Sequences, published electronically at https://oeis.org. [12] H. Thomas, The number of terms in the permanent and the determinant of a generic circulant matrix, J. Algebraic Combin. 20 (2004), 55–60, doi:10.1023/b:jaco.0000047292.01630.a6. ISSN 2590-9770 The Art of Discrete and Applied Mathematics 1 (2018) #P2.02 https://doi.org/10.26493/2590-9770.1236.ce4 (Also available at http://adam-journal.eu) On silver and golden optical orthogonal codes ⇤ Marco Buratti Dipartimento di Matematica e Informatica, Università di Perugia, via Vanvitelli 1 Received 1 January 2018, accepted 24 June 2018, published online 3 August 2018 Abstract It is several years that no theoretical construction for optimal (v, k, 1) optical orthog- onal codes (OOCs) with new parameters has been discovered. In particular, the literature almost completely lacks optimal (v, k, 1)-OOCs with k > 3 which are not regular. In this paper we will show how some elementary difference multisets allow to obtain three new classes of optimal but not regular (3p, 4, 1)-, (5p, 5, 1)-, and (2p, 4, 1)-OOCs which are de- scribable in terms of Pell and Fibonacci numbers. The OOCs of the first two classes (resp. third class) will be called silver (resp. golden) since they exist provided that the square of a silver element (resp. golden element) of Zp is a primitive square of Zp. Keywords: Silver and golden ratio, Pell and Fibonacci numbers, difference packing, optimal optical orthogonal code, strong difference family, difference multiset. Math. Subj. Class.: 05B10, 94B25 1 Introduction The real numbers 1 + p 2 (the silver ratio), 1+ p 5 2 (the golden ratio) and their marvelous properties are very well known. Disregarding their geometrical meaning (see, e.g., [17]), they can be defined in the same algebraic way in any finite field Fq of an appropriate order q. By the Law of Quadratic Reciprocity (see, e.g., [20]), it is well known that 2 is a non- zero square in Fq if and only if q ⌘ 1 or 7 (mod 8) and that 5 is a non-zero square in Fq if and only if q ⌘ 1 or 4 (mod 5). Thus, for a prime p ⌘ 1 or 7 (mod 8), we naturally define the silver elements of Zp as the two elements 1+x and 1x of Zp where x and x are the square roots of 2 modulo p. Also, for a prime p ⌘ 1 or 4 (mod 5), we naturally define the golden elements of Zp as the two elements 21(1 + x) and 21(1 x) of Zp where x and x are the square roots of 5 modulo p. We recall that the Pell sequence is the integer sequence {Pn} defined by P0 = 0, P1 = 1 and by the recursive formula Pn = 2Pn1 + Pn2 for n 2, and that the ⇤This work has been performed under the auspices of the G.N.S.A.G.A. of the C.N.R. (National Research Council) of Italy. E-mail address: buratti@dmi.unipg.it (Marco Buratti) cb This work is licensed under http://creativecommons.org/licenses/by/3.0/ 2 Art Discrete Appl. Math. 1 (2018) #P2.02 Fibonacci sequence is the integer sequence {Fn} defined by F0 = 0, F1 = 1 and by the recursive formula Fn = Fn1 + Fn2 for n 2. Two good textbooks on Pell and Fibonacci numbers are [19] and [18], respectively. As in the real field, if ✓ is a silver element of Fq then we have ✓ n = Pn✓ + Pn1 8n 1 (1.1) Also, if is a golden element of Fq then we have n = Fn+ Fn1 8n 1 (1.2) An optical orthogonal code of length v, weight k, auto-correlation a and cross- correlation c – briefly, a (v, k,a,c)-OOC – can be seen as a set of k-subsets (codeword- sets) of Zv such that (i) any two distinct translates of a codeword-set share at most a elements (auto-correlation property); (ii) any two translates of two distinct codeword-sets share at most c elements (cross-correlation property). This topic, introduced by Chung, Salehi and Wei [12], has been well studied for a long time in view of its many applications (see, e.g., [13]). In particular, a (v, k, 1, 1)-OOC – briefly, a (v, k, 1)-OOC – can be viewed as a set of k-subsets of Zv (codeword-sets) such that no element of Zv \ {0} can be represented as a difference of two elements of a codeword-set in more than one way. Such an OOC is said to be optimal when its size (that is the number of its codeword-sets) reaches the upper Johnson bound b v1k(k1)c. There is a huge literature on optical orthogonal codes but, as far as this author is aware, in the last seven years no theoretical construction for a class of optimal (v, k, 1)-OOCs with new parameters has been discovered. In this paper we find three classes of optimal OOCs with new parameters: an optimal (3p, 4, 1)-OOC and an optimal (5p, 5, 1)-OOC for each prime p ⌘ 7 (mod 8) such that the silver elements of Zp are generators of Z⇤p/{1,1} (both these codes will be called silver); an optimal (2p, 4, 1)-OOC for each prime p ⌘ 11 or 29 (mod 30) such that the golden elements of Zp are generators of Z⇤p/{1,1} (this code will be called golden). The strategy to get our silver/golden OOCs is to use some elementary difference mul- tisets (which are strong difference families with only one block) but, in the end, we will show that all these codes can be presented in terms of Pell/Fibonacci numbers. 2 Difference packings via strong difference families Given a k-multisubset, in particular a k-subset, B = {b1, . . . , bk} of an additive group G, we call list of differences from B the multiset B of all possible differences bi bj with (i, j) an ordered pair of distinct elements of {1, . . . , k}. One calls (G, k, 1) difference packing any set D of k-subsets of G (blocks) with the property that its list of differences, namely the multiset sum D := ] B2D B, M. Buratti: On silver and golden optical orthogonal codes 3 does not have repeated elements. It is evident that the size of D cannot exceed b |G|1k(k1)c. For this reason, one says that D is optimal when its size reaches this value. The difference leave of a (G, k, 1) difference packing D is defined to be the set of all elements of G not appearing in D. Thus D is optimal provided that its difference leave has size less or equal to k(k 1). The difference packing is a relative difference family [5] if its difference leave is a subgroup H of G. In this case someone also speaks of a r-regular difference packing if H has order r (see, e.g., [24]). Note that a (Zv, k, 1) difference packing is nothing but a (v, k, 1)-OOC. The problem of factoring a group into subsets and its variants [23] could play a crucial role in the construction of difference packings. Also, the construction of a |G|-regular (G⇥Fq, k, 1) difference packing can be facilitated by a suitable strong difference family in G, a concept formally introduced in [6] but implicitly used for a long time. A t-(G, k, µ) strong difference family is a t-multiset S of k-multisubsets (blocks) of a group G such that S covers all elements of G exactly µ times. The parameter µ is called the index of the strong difference family and a trivial counting shows that it is necessarily equal to k(k1)t|G| . Of course it is possible to consider, more generally, strong difference families whose blocks have variable sizes [7]. In order to explain why strong difference families might be good to construct relative difference families and more generally OOCs, we have to introduce some notation and terminology. Denote by F⇤q the multiplicative group of the field Fq . Given a subset B of a direct product G⇥ Fq and given c 2 F⇤q , denote by (1, c) ·B the subset of G⇥ Fq obtained from B by multiplying the second coordinates of all its elements by c and leaving invariant their first coordinates. If B is a set of subsets of G ⇥ Fq and g 2 G, we denote by gB the list of the second coordinates of all elements of B whose first coordinate is g so that one can write B = [ g2G {g}⇥gB. Let us say that two subsets C and of F⇤q are companions if the list C · := {c | c 2 C; 2 } does not have repeated elements. In this case it is evident that the size of C cannot exceed b q1|| c. Thus we say that C is an optimal companion of when its size reaches this value. In particular, we say that C is a perfect or near-perfect companion of when its size is exactly equal to q1|| or q2 || , respectively. In these last two cases we have C · = F⇤q or C · = F⇤q \ {x} for some x 2 F⇤q and one says that C · is a factorization of F⇤q in the former case and that 1xC · is a near factorization of F ⇤ q in the latter (see [23]). The next proposition is very elementary. Proposition 2.1. Let B = {B1, . . . , Bt} be a set of k-subsets of G⇥Fq such that all gB are sets admitting a common companion C. Then D := {(1, c) · B | c 2 C,B 2 B} is a (G⇥ Fq, k, 1) difference packing. The proof is straightforward; indeed, by assumption, C · gB does not have repeated elements, hence D = S g2G{g}⇥ (C ·gB) is also without repeated elements. Now we show that the above proposition cannot give optimal optical orthogonal codes for arbitrarily high values of q unless the projection of B on G is a strong difference family. Proposition 2.2. Let D be a difference packing as in Proposition 2.1 and set µ = k(k1)t|G| . Then, for q > k(k 1)µ, D is optimal if and only if the following conditions hold: 4 Art Discrete Appl. Math. 1 (2018) #P2.02 (i) The projection of B on G is a t-(G, k, µ) strong difference family; (ii) C is an optimal companion of gB for every g 2 G; (iii) the remainder of the Euclidean division of q by µ does not reach µt . Proof. (=)): The size of D is |C| · t, therefore we have |C| · t = b |G|q1k(k1)c because D is optimal. This gives |C| k(k 1) in view of the hypothesis q > k(k 1)µ. For each g 2 G, let Lg(B) be the complement of C ·gB in Fq . We have |Lg(B)| = q|C|·|gB| for each g 2 G. This implies that |Lg(B)| = |Lh(B)|+|C|·(|hB||gB|) for any pair of elements g and h of G. Thus, if |gB| < |hB| we would have |Lg(B)| > |C| and then Lg(B) would have size greater than k(k1) in view of the previous paragraph. This is clearly absurd since {g} ⇥ Lg(B) is contained in the difference leave of D whose size is at most k(k 1). We conclude that |gB| is a constant, i.e., the projection of B on G is a strong difference family with t blocks of size k. This implies that its index is k(k1)t|G| which is equal to µ. Thus |gB| = µ for every g 2 G. Now assume that C is not optimal. In this case the size of Lg(B) would be a constant at least equal to µ, hence the difference leave of D, which is clearly given by S g2G{g}⇥ Lg(B), would have size greater than µ|G| = tk(k 1), therefore greater than k(k 1) contradicting the optimality of D. If r is the remainder of the Euclidean division of q by µ, then the difference leave of D has size r · |G|. Thus, since D is optimal, we must have r · |G|  k(k 1) which means r < µ t . ((=): Straightforward. Note that condition (iii) is certainly satisfied when t = 1, namely when B is a singleton {B}. In this case one says that the projection of B on G, say ⇡(B), is a (G, k, µ) difference multiset (also called a difference cover in [3]) rather than to say that {⇡(B)} is a 1-(G, k, µ) strong difference family. The above proposition suggests the following strategy for getting families of optimal difference packings. Start with a t-(G, k, µ) strong difference family S which will be used as “skeleton” of the desired optimal difference packing. Then take a prime power q = µn + r with 1  r  µt and try to “lift” S to a suitable t-set B of k-subsets of G ⇥ Fq in such a way that all gB admit a common optimal companion C. For r = 1 this strategy has been used (sometimes implicitly) in many papers to construct relative difference families and, in particular, regular OOCs. The elder constructions are surveyed in [2]. More recent constructions can be found in [8, 9, 11, 14, 15, 21, 22, 25]. Here one often tries to have each gB a complete system of representatives for the cosets of the subgroup of F⇤q of index µ, namely the group Cµ of non-zero µ-th powers of Fq . Indeed in this case a common companion of each gB is clearly given by Cµ itself. On the other hand, as far as this author is aware, the above strategy has been never applied with r > 1 probably because the existence of a common optimal but not perfect companion of all the set gB seems to be almost a miracle. Indeed, the probability that even a single set ⇢ F⇤q admits an optimal companion C diminishes dramatically if || is not a divisor of q1. Consider, for instance, that Theorem 2.8 and Theorem 2.9 in [4] imply that for q ⌘ 1 (mod 3) the number of 3-subsets of F⇤q admitting a perfect companion is at least equal to q( q13 ) 3, while for q ⌘ 2 (mod 3) the number of 3-subsets of F⇤q admitting a near-perfect companion is less or equal to q · (q 1) with the Euler totient function. M. Buratti: On silver and golden optical orthogonal codes 5 This probably explains why, at the moment, we have only a few known classes of optimal but not regular (v, k, 1)-OOCs with k > 3 (see [1, 4, 10]). Anyway in this paper we manage to find three new classes of optimal but not regu- lar OOCs adopting the strategy described above with S equal to one the following very elementary strong difference families: (a) the (Z3, 4, 4) difference multiset {0, 0, 1, 1} for getting an optimal (3p, 4, 1)-OOC with p = 8n+ 7 a prime whose silver elements are generators of Z⇤p/{1,1}; (b) the (Z5, 5, 4) difference multiset {0, 1, 1, 4, 4} for getting an optimal (5p, 5, 1)-OOC with p a prime as above; (c) the (Z2, 4, 6) difference multiset {0, 1, 1, 1} for getting an optimal (2p, 4, 1)-OOC with p = 30n+ 11 or p = 30n+ 29 a prime whose golden elements are generators of Z⇤p/{1,1}. 3 On the silver (3p, 4, 1) and (5p, 5, 1) optical orthogonal codes Note that the silver elements of Zp are precisely the solutions of the congruence x2 2x 1 ⌘ 0 (mod p), i.e., the elements ✓ of Zp such that ✓ + 1 = ✓(✓ 1). This property is crucial for getting the following construction. Theorem 3.1. Let p = 8n + 7 be a prime and let ✓ be a silver element of Zp. If ✓ is a generator of Z⇤p/{1,1}, then there exists an optimal (3p, 4, 1)-OOC and an optimal (5p, 5, 1)-OOC. Proof. By the Chinese Remainder Theorem, Z3p and Z5p are isomorphic to Z3 ⇥ Zp and Z5⇥Zp, respectively. So it is enough to show that, under the given assumption, there exists an optimal (Z3 ⇥ Zp, 4, 1) difference packing and an optimal (Z5 ⇥ Zp, 5, 1) difference packing. The assumption on ✓ implies that {✓i | 0  i  4n + 2} is a complete system of representatives for the cosets of {1,1} in Z⇤p so that we have Z⇤p = {1,1} · {1, ✓, ✓2, . . . , ✓4n+1, ✓4n+2} and then {✓2i | 0  i  2n} · {±✓,±✓2} = Z⇤p \ {1,1}. Thus we can claim that C := {✓2i | 0  i  2n} is an optimal companion of {±✓,±✓2}. (3.1) Let us lift the (Z3, 4, 4) difference multiset {0, 0, 1, 1} to the following 4-subset of Z3⇥Zp B = {(0, ✓), (0,✓), (1, ✓2), (1,✓2)}. The difference table of B (see Table 1) shows that we can write: 0B = {±2✓,±2✓2}; 1B = 2B = {±✓(✓ 1),±✓(✓ + 1)}. (3.2) Then, recalling that ✓ + 1 = ✓(✓ 1), we have: 0B = 2{±✓,±✓2}; 1B = 2B = (✓ 1){±✓,±✓2}. We conclude, by (3.1), that C is an optimal companion of gB for every g 2 Z3 and then D = {(1, c) · B | c 2 C} is the desired optimal (Z3 ⇥ Zp, 4, 1) difference packing by Proposition 2.2. 6 Art Discrete Appl. Math. 1 (2018) #P2.02 Table 1: The difference table of B = {(0, ✓), (0,✓), (1, ✓2), (1,✓2)}. (0, ✓) (0,✓) (1, ✓2) (1,✓2) (0, ✓) • (0, 2✓) (2, ✓ ✓2) (2, ✓ + ✓2) (0,✓) (0,2✓) • (2,✓ ✓2) (2,✓ + ✓2) (1, ✓2) (1, ✓2 ✓) (1, ✓2 + ✓) • (0, 2✓2) (1,✓2) (1,✓2 ✓) (1,✓2 + ✓) (0,2✓2) • Now, let us lift the (Z5, 5, 4) difference multiset {0, 1, 1, 4, 4} to the following 5-subset of Z5 Zp B = {(0, 0), (1, ✓), (1,✓), (4, ✓2), (4,✓2)}. Table 2 is its difference table. Table 2: The difference table of B = {(0, 0), (1, ✓), (1,✓), (4, ✓2), (4,✓2)}. (0, 0) (1, ✓) (1,✓) (4, ✓2) (4,✓2) (0, 0) • (4,✓) (4, ✓) (1,✓2) (1, ✓2) (1, ✓) (1, ✓) • (0, 2✓) (2, ✓ ✓2) (2, ✓ + ✓2) (1,✓) (1,✓) (0,2✓) • (2,✓ ✓2) (2,✓ + ✓2) (4, ✓2) (4, ✓2) (3, ✓2 ✓) (3, ✓2 + ✓) • (0, 2✓2) (4,✓2) (4,✓2) (3,✓2 ✓) (3,✓2 + ✓) (0,2✓2) • Recalling again that ✓ + 1 = ✓(✓ 1), we can write 0B = 2{±✓,±✓2}, 1B = 4B = {±✓,±✓2}, 2B = 3B = (✓ 1){±✓,±✓2} so that, by (3.1), C is an optimal companion of gB for each g 2 Z5. We conclude that D = {(1, c) · B | c 2 C} is the desired optimal (Z5 ⇥ Zp, 5, 1) difference packing by Proposition 2.2. The assertion follows. The optimal OOCs arising from the above result will be called silver. We remark that the assumption on ✓ is equivalent to ask that ✓2, that is 2✓ + 1, is a primitive square of Zp and that this assumption does not depend on the chosen silver element; indeed the product of the two silver elements is 1, hence they have the same orders in Z⇤p/{1,1}. We also note that the difference leaves of the constructed pakings are {0}⇥ {0, 2,2} [ {1, 2}⇥ {0, ✓ 1, 1 ✓} for the (Z3 ⇥ Zp, 4, 1) difference packing and {0}⇥ {0, 2,2} [ {1, 4}⇥ {0, 1,1} [ {2, 3}⇥ {0, ✓ 1, 1 ✓} M. Buratti: On silver and golden optical orthogonal codes 7 for the (Z5 ⇥ Zp, 5, 1) difference packing. Among the 2399 primes p congruent to 7 modulo 8 and not exceeding 100 000 we have checked that ✓ is not a generator of Z⇤p/{1,1} in “only” 599 cases. Thus, roughly speaking, it seems that the two constructions succeed three times out of four. Remark 3.2. Using formula (1.1), the optimal difference packings constructed in Theo- rem 3.1 can be more explicitly written in terms of Pell numbers. They are of the form D = {Bi | 0  i  2n} with Bi = {(0, P2i+1✓ + P2i), (0,P2i+1✓ P2i), (1, P2i+2✓ + P2i+1), (1,P2i+2✓ P2i+1)} when D is a (Z3 ⇥ Zp, 4, 1) difference packing and with Bi = {(0, 0), (1, P2i+1✓ + P2i), (1,P2i+1✓ P2i), (4, P2i+2✓ + P2i+1), (4,P2i+2✓ P2i+1)} when D is a (Z5 ⇥ Zp, 5, 1) difference packing. By way of illustration we explicitly construct a silver (141, 4, 1)-OOC. We have 141 = 3p with p = 47 = 8n + 7 prime, n = 5. A silver element of Zp is ✓ = 8; indeed we have 8 + 1 ⌘ 82 8 (mod 47). Here the group Z⇤p/{1,1} has prime order 23, hence ✓ is certainly a generator of it and Theorem 3.1 can be applied. The reduction modulo p of the Pell sequence up to its 22-nd term is (0, 1, 2, 5, 12, 29, 23, 28, 32, 45, 28, 7, 42, 44, 36, 22, 33, 41, 21, 36, 46, 34, 20). Thus, applying Remark 3.2, the blocks of a (Z3 ⇥ Z47, 4, 1) difference packing are the following: {(0, ✓), (0,✓), (1, 2✓ + 1), (1,2✓ 1)} {(0, 5✓ + 2), (0,5✓ 2), (1, 12✓ + 5), (1,12✓ 5)} {(0, 29✓ + 12), (0,29✓ 12), (1, 23✓ + 29), (1,23✓ 29)} {(0, 28✓ + 23), (0,28✓ 23), (1, 32✓ + 28), (1,32✓ 28)} {(0, 45✓ + 32), (0,45✓ 32), (1, 28✓ + 45), (1,28✓ 45)} {(0, 7✓ + 28), (0,7✓ 28), (1, 42✓ + 7), (1,42✓ 7)} {(0, 44✓ + 42), (0,44✓ 42), (1, 36✓ + 44), (1,36✓ 44)} {(0, 22✓ + 36), (0,22✓ 36), (1, 33✓ + 22), (1,33✓ 22)} {(0, 41✓ + 33), (0,41✓ 33), (1, 21✓ + 41), (1,21✓ 41)} {(0, 36✓ + 21), (0,36✓ 21), (1, 46✓ + 36), (1,46✓ 36)} {(0, 34✓ + 46), (0,34✓ 46), (1, 20✓ + 34), (1,20✓ 34)} The isomorphism f : (x, y) 2 Z3 ⇥ Z47 ! 48y 47x 2 Z141 turns the above blocks into the following eleven codeword-sets forming the desired silver (141, 4, 1)-OOC with difference leave {0, 7, 40, 45, 47, 94, 96, 101, 134}: {102, 39, 64, 124}, {42, 99, 7, 40}, {9, 132, 25, 22}, {12, 129, 49, 139}, {63, 78, 34, 13}, {84, 57, 61, 127}, {18, 123, 97, 91}, {24, 117, 4, 43}, {126, 15, 115, 73}, {27, 114, 28, 19}, {36, 105, 100, 88}. 8 Art Discrete Appl. Math. 1 (2018) #P2.02 As far as this author is aware, the above optimal OOC is new but the same cannot be said for its parameters. Indeed it was proved in [16] that there exists a perfect (v, 4, 1) difference family for all v ⌘ 1 (mod 12) not exceeding 10 000 except v = 25 and v = 37. Also, according to Remark 1.4 in [1], any perfect (v, 4, 1) difference family can be also seen as an optimal (w, k, 1)-OOC for all w’s between v and v + k(k 1) included. Thus we have the existence of an optimal (v, 4, 1)-OOC for all v’s not exceeding 10 012 except v = 25 (indeed an optimal (v, 4, 1)-OOC with 26  v  48 is known to exist anyway). 4 On the golden (2p, 4, 1) optical orthogonal codes Note that the golden elements of Zp are precisely the solutions of the congruence x2x 1 ⌘ 0 (mod p), i.e., the elements of Zp such that + 1 = 2. This property is crucial for getting the following construction. Theorem 4.1. Let p ⌘ 11 or 29 (mod 30) be a prime and let be a golden element of Zp. If is a generator of Z⇤p/{1,1}, then there exists an optimal (2p, 4, 1)-OOC. Proof. We have to show that, under the given assumption, there exists an optimal (Z2 ⇥ Zp, 4, 1) difference packing. Indeed Z2⇥Zp is isomorphic to Z2p by the Chinese Remain- der Theorem. We can write p = 6n+ 5 for a suitable n, hence p12 = 3n+ 2, and the assumption on implies that we have Z⇤p = {1,1} · {1,,2, . . . ,3n,3n+1}. It is then clear that C := {3i1 | 1  i  n} is an optimal companion of {±1,±,±2}. (4.1) Indeed we have C · {±1,±,±2} = Z⇤p \ {±1,±}. Let us lift the (Z2, 4, 6) difference multiset {0, 1, 1, 1} to the 4-subset B of Z2 Zp B = {(0, 0), (1, 1), (1,), (1,2}. Looking at the difference table of B (see Table 3) we see that we have 0B = {±( 1),±( 1),±(+ 1)( 1)}; 1B = {±1,±,±2}. Thus, recalling that + 1 = 2, we can write 0B = ( 1){±1,±,±2}, 1B = {±1,±,±2} so that, by (4.1), C in an optimal companion of gB for each g 2 Z2. We conclude that D = {(1, c) · B | c 2 C} is the desired optimal (Z5 ⇥ Zp, 5, 1) difference packing by Proposition 2.2. The optimal OOCs arising from the above result will be called golden. We remark that the assumption on is equivalent to ask that 2, that is + 1, is a primitive square of Zp and it does not depend on the chosen golden element; indeed the product of the two golden elements is 1, hence their orders in Z⇤p/{1,1} are the same. We also note that the difference leave of the constructed difference packing is {0}⇥ {0, 1,1, 1, 1 } [ {1}⇥ {0, 1,1,,}. M. Buratti: On silver and golden optical orthogonal codes 9 Table 3: The difference table of B = {(0, 0), (1, 1), (1,), (1,2}. (0, 0) (1, 1) (1,) (1,2) (0, 0) • (1,1) (1,) (1,2) (1, 1) (1, 1) • (0, 1 ) (0, 1 2) (1,) (1,) (0, 1) • (0, 2) (1,2) (1,2) (0,2 1) (0,2 ) • We have checked that in the range [1, 105], Theorem 4.1 works in 1533 out of 2399 of the cases. Remark 4.2. Using formula (1.2), the optimal difference packing D constructed in The- orem 4.1 can be more explicitly written in terms of Fibonacci numbers. Indeed we have D = {Bi | 1  i  n} with Bi = {(0, 0), (1, F3i1+ F3i2), (1, F3i+ F3i1), (1, F3i+1+ F3i)}. By way of illustration we explicitly construct a golden (142, 4, 1)-OOC using the above remark. We have 142 = 2p with p = 71 ⌘ 11 (mod 30) prime, and we can write p = 6n + 5 with n = 11. A golden element of Zp clearly is = 9; indeed we have 92 ⌘ 10 (mod 71). The maximal proper divisors of (p1)/2 are 5 and 7 and neither 105 nor 107 is 1 (mod p). This guarantees that 10 has order (p 1)/2 in Z⇤p, namely + 1 is a primitive square of Zp. Thus Theorem 4.1 can be applied. The reduction modulo p of the Fibonacci sequence up to its 34-th term is (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 18, 2, 20, 22, 42, 64, 35, 28, 63, 20, 12, 32, 44, 5, 49, 54, 32, 15, 47, 62, 38, 29, 67, 25). Thus, applying Remark 4.2, the blocks of an optimal (Z2⇥Z71, 4, 1) difference packing are the following: {(0, 0), (1,+ 1), (1, 2+ 1), (1, 3+ 2)} {(0, 0), (1, 5+ 3), (1, 8+ 5), (1, 13+ 8)} {(0, 0), (1, 21+ 13), (1, 34+ 21), (1, 55+ 34)} {(0, 0), (1, 18+ 55), (1, 2+ 18), (1, 20+ 2)} {(0, 0), (1, 22+ 20), (1, 42+ 22), (1, 64+ 42)} {(0, 0), (1, 35,+ 64), (1, 28+ 35), (1, 63+ 28)} {(0, 0), (1, 20+ 63), (1, 12+ 20), (1, 32+ 12)} {(0, 0), (1, 44+ 32), (1, 5+ 44), (1, 49+ 5)} {(0, 0), (1, 54+ 49), (1, 32+ 54), (1, 15+ 32)} {(0, 0), (1, 47+ 15), (1, 62+ 47), (1, 38+ 62)} {(0, 0), (1, 29+ 38), (1, 67+ 29), (1, 25+ 67)} 10 Art Discrete Appl. Math. 1 (2018) #P2.02 The isomorphism f : (x, y) 2 Z2 ⇥ Z71 ! 71x + 72y 2 Z142 turns the above blocks into the following eleven codeword-sets forming the desired golden (142, 4, 1)-OOC with difference leave {0, 1, 8, 9, 70, 71, 72, 133, 134, 141}: {0, 81, 19, 29}, {0, 119, 77, 125}, {0, 131, 43, 103}, {0, 75, 107, 111}, {0, 5, 45, 121}, {0, 95, 3, 27}, {0, 101, 57, 87}, {0, 73, 89, 91}, {0, 109, 129, 25}, {0, 83, 37, 49}, {0, 15, 135, 79}. Although the above optimal OOC is probably new, the same cannot be said for its parameters for the same reason explained in the end of Section 3. References [1] R. J. R. Abel and M. Buratti, Some progress on (v, 4, 1) difference families and optical orthog- onal codes, J. Comb. Theory Ser. A 106 (2004), 59–75, doi:10.1016/j.jcta.2004.01.003. [2] R. J. R. Abel and M. Buratti, Difference families, in: C. J. Colbourn and J. H. 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Chang, Combinatorial constructions for maximum optical orthogonal signature pattern codes, Discrete Math. 313 (2013), 2918–2931, doi:10.1016/j.disc.2013.09.005. [23] S. Szabó and A. D. Sands, Factoring Groups into Subsets, volume 257 of Lecture Notes in Pure and Applied Mathematics, CRC Press, Boca Raton, FL, 2009, doi:10.1201/9781420090475. [24] J. Yin, Some combinatorial constructions for optical orthogonal codes, Discrete Math. 185 (1998), 201–219, doi:10.1016/s0012-365x(97)00172-6. [25] J. Yin, X. Yang and Y. Li, Some 20-regular CDP(5, 1; 20u) and their applications, Finite Fields Appl. 17 (2011), 317–328, doi:10.1016/j.ffa.2011.01.002. ISSN 2590-9770 The Art of Discrete and Applied Mathematics 1 (2018) #P2.03 https://doi.org/10.26493/2590-9770.1220.3a1 (Also available at http://adam-journal.eu) Subspace restrictions and affine composition for covering perfect hash families⇤ Charles J. Colbourn †, Erin Lanus Computing, Informatics, and Decision Systems Engineering, Arizona State University, PO Box 878809, Tempe, AZ, 85287-8809, U.S.A. To Mario Gionfriddo on his seventieth birthday. Received 19 November 2017, accepted 1 August 2018, published online 3 August 2018 Abstract Covering perfect hash families provide a very compact representation of a useful family of covering arrays, leading to the best asymptotic upper bounds and fast, effective algo- rithms. Their compactness implies that an additional row in the hash family leads to many new rows in the covering array. In order to address this, subspace restrictions constrain cov- ering perfect hash family so that a predictable set of many rows in the covering array can be removed without loss of coverage. Computing failure probabilities for random selections that must, or that need not, satisfy the restrictions, we identify a set of restrictions on which to focus. We use existing algorithms together with one novel method, affine composition, to accelerate the search. We report on a set of computational constructions for covering arrays to demonstrate that imposing restrictions often improves on previously known upper bounds. Keywords: Covering array, covering perfect hash family, affine composition, subspace restriction. Math. Subj. Class.: 05B40, 05B15, 51E26, 68R05 1 Introduction We develop effective construction techniques for combinatorial arrays called covering per- fect hash families, which form a compact representation of covering arrays. Covering arrays arise in numerous applications in which interactions among options or factors are ⇤Thanks to Ryan Dougherty, Kaushik Sarkar, and Violet Syrotiuk for useful discussions, and to two anony- mous referees for helpful comments. †Research of CJC was supported in part by the National Science Foundation under Grant No. 1421058. E-mail addresses: colbourn@asu.edu (Charles J. Colbourn), elanus@asu.edu (Erin Lanus) cb This work is licensed under http://creativecommons.org/licenses/by/3.0/ 2 Art Discrete Appl. Math. 1 (2018) #P2.03 to be measured; they are used in, for example, software testing [12, 13], hardware testing [10, 20], design of composite materials [2], computational learning [1, 9], and biological networks [14]. Computational methods to construct covering arrays often encounter diffi- culties when the array has many rows, many columns, or both. To alleviate this concern, covering perfect hash families were introduced in [21] and shown to provide a succinct rep- resentation of a class of covering arrays. In [6] they were used to establish the best known asymptotic upper bound on the fewest rows in a covering array. Also in [6], effective and simple algorithms were examined for their construction. Covering perfect hash families have proved instrumental in obtaining many sizes of covering arrays that are the best currently known. Despite the compactness of the represen- tation that they provide, their use lessens but does not remove the computational burden. We propose and analyze a method, affine composition, to combine small covering perfect hash families to make larger ones; this extends the range of array sizes for which computa- tional methods are feasible. Moreover, the very compactness of the representation severely limits the possible numbers of rows in the covering arrays produced. We develop a method using subspace restrictions to produce covering arrays that are guaranteed to have at least a specified number of duplicated rows, which can be removed without altering the coverage. This provides finer control on the number of rows in the covering array, and hence often improves upon the coarser use of covering perfect hash families without restrictions. Both of these contribute to the construction of covering arrays with fewer rows than the best previously known, and hence to a reduction in testing and measurement cost when the covering arrays are applied. In order to develop these notions, we first provide formal definitions and background. Let q be a prime power. Let Fq be the finite field of order q. Let Rt,q = {r0, . . . , rqt1} be the set of all (row) vectors of length t with entries from Fq , and let Tt,q be the set of all column vectors of length t with entries from Fq , not all 0. A vector x 2 Tt,q is a permutation vector [21]. Lemma 1.1 (see [21]). Let X = {x1, . . . ,xt} be a set of vectors from Tt,q . The array A = (aij) formed by setting aij to be the product of ri and xj is a qt ⇥ t matrix in which every row is distinct if and only if the t⇥ t matrix X = [x1 · · ·xt] is nonsingular. When µ 2 Fq \ {0}, substituting µxi for xi does not change the multiset of rows produced, just their order. Define hxi = {µx : µ 2 Fq, µ 6= 0}. When x is not all 0, we can select as the representative of hxi the unique vector whose first nonzero coordinate is the multiplicative identity element. Let Vt,q be the set of representatives of the column vectors in Tt,q . Let Ut,q be the set of vectors in Vt,q whose first coordinate is not zero. Then |Vt,q| = q t1 q1 = Pt1 i=0 q i, and |Ut,q| = qt1. A covering perfect hash family CPHF(n; k, q, t) is an n ⇥ k array C = (cij) with entries from Vt,q so that, for every set {1, . . . , t} of distinct column indices, there is at least one row index ⇢ of C for which [c⇢1 · · · c⇢t ] is nonsingular; call this a covering t-set and say that the t-set of columns is covered. It is a Sherwood covering perfect hash family, SCPHF(n; k, q, t), if in addition each entry is in Ut,q . Let N, t, k, and v be positive integers with k t 2 and v 2. A covering array CA(N ; t, k, v) is an N ⇥ k array A in which each entry is from a v-ary alphabet ⌃, and for every N ⇥ t sub-array B of A and every x 2 ⌃t, there is a row of B that equals x. When k is a positive integer, [k] denotes the set {1, . . . , k}. A t-way interaction is {(ci, ai) : 1  i  t} where ci 2 [k], ci 6= cj for i 6= j, and ai 2 ⌃. Such an interaction C. J. Colbourn and E. Lanus: Subspace restrictions and affine composition for covering . . . 3 is an assignment of values from ⌃ to t of the k columns. An N ⇥ k array A covers the interaction ◆ = {(ci, ai) : 1  i  t, ci 2 [k], ci 6= cj for i 6= j, and ai 2 ⌃} if there is a row r in A such that A(r, ci) = ai for 1  i  t. When there is no such row in A, ◆ is not covered in A. Hence a CA(N ; t, k, v) covers all the t-way interactions on k columns on an alphabet of v symbols. Covering arrays are used extensively for interaction testing in complex engineered sys- tems. The k columns represent factors and the v values are the levels of the factors. The N rows form a test suite (each row is a test); and the coverage of interactions among the factors is limited to the strength t. Denote by CAN(t, k, v) the smallest value of N for which a CA(N ; t, k, v) exists. This is a covering array number, and for essentially all applications the goal is to minimize it. Applications also require that actual covering arrays be generated, and hence the focus is on explicit, practical construction methods. Online tables at [5] give the least upper bound on CAN(t, k, v) by an explicit construction for 2  t  6, 2  v  25, and k  10 000. The connection between CPHFs and CAs is central in this paper, so we include a standard proof for the following correspondence. Lemma 1.2 (see [21]). 1. There exists a CA(n(qt 1) + 1; t, k, q) if C is a CPHF(n; k, q, t); and 2. there exists a CA(n(qt q) + q; t, k, q) if C is an SCPHF(n; k, q, t). Proof. Let C be the covering perfect hash family. Replace each entry cij of C by the column vector obtained by multiplying cij by each r` 2 Rt,q in the specified order. By Lemma 1.1, this produces a CA(nqt; t, k, q). The product of each cij with (0, . . . , 0) 2 Rt,q is 0, so the resulting array contains n rows that contain only 0 entries. Remove n 1 of these rows to form the CA(n(qt 1) + 1; t, k, q). Now when cij 2 Ut,q , multiplication by (, 0, . . . , 0) 2 Rt,q always yields . For each 2 Fq , remove n 1 of the rows in which each entry is to form the CA(n(qt q) + q; t, k, q). In generating the covering array from the CPHF, it may happen that rows are generated that only cover t-way interactions that are also covered by other rows. Such a row is re- dundant, and could be removed. Indeed by tracking coverage as rows of the covering array are generated, one could avoid generating some of these redundant rows. More generally, a post-optimization method [17] may reveal or produce further redundant rows. Because the covering array is typically much larger than the covering perfect hash family, however, effort can be saved by determining in advance certain rows of the covering array that are guaranteed to be redundant, thereby avoiding their generation and subsequent elimination. The simplest way to ensure that a row is redundant in the covering array generated is that it be identical to another row; then it is replicated or repeated. Lemma 1.2 already accounts for the redundancy of n 1 rows for CPHFs, and of (n 1)q rows for SCPHFs, by noting that they are replicated. Our goal here is to restrict the CPHF in such a way that many more rows are guaranteed to be replicated, and so reduce the size of the covering array generated without having to analyze its coverage during and after its generation. Whether restricted or not, CPHFs are needed to apply Lemma 1.2. Few general direct constructions are known [18, 21, 24]; most arise from computation. Computational meth- ods for SCPHFs include backtracking [21] and tabu search [25]. In [6], CPHFs are shown to lead to the best known asymptotic results on the existence of covering arrays. Indeed 4 Art Discrete Appl. Math. 1 (2018) #P2.03 the probabilistic methods lead to two classes of efficient algorithms for constructing cov- ering arrays for much larger parameters than had been earlier handled, and the best known bounds were improved on for a wide range of parameters as a result. In [6], a means to restrict the CPHFs to ensure that certain rows are replicated is out- lined, and applied for strength t = 3. In Section 2, we define restrictions and consider the effect of imposing various restrictions on the expectation that a t-set of columns is covered, in order to determine the types of restrictions that appear to be promising. In Section 3 we develop a recursive composition strategy to accelerate the computational search for re- stricted CPHFs. In Section 4 we report on new bounds obtained by subspace restrictions, at the same time updating some of the computational results from [6]. 2 Subspace restrictions We limit how entries are placed in a CPHF so that redundant rows are generated in the application of Lemma 1.2; these can be removed. We denote by Ft,p the set of all p-tuples of distinct entries from {0, . . . , t 1}. A subspace restriction for n rows of dimension p and replication r is an r-tuple (x1, . . . , xr) of distinct entries from {1, . . . , n} and an r-tuple (U1, . . . , Ur) for which each Ui 2 Ft,p. Let A = (aij) be a CPHF(n; k, q, t) in which each entry aij is a permutation vector of length t. Write aij` for the `th entry of this vector. Let S be the subspace restriction (for n rows), given by (x1, . . . , xr) and (U1, . . . , Ur). Denote by uab the element of Ua in position b. Then A satisfies or meets the restriction if, when 1  c, d  r, axc,j,uc` = axd,j,ud` for all 1  j  k and 1  `  p; for short, A is S-restricted. Table 1: A CPHF(4; 18, 3, 4). Each permutation vector (h0, h1, h2, h3)T is written as h0h1h2h3. 1020 1002 1211 1112 1122 1001 1002 1202 1111 1222 1100 1010 1101 1010 1220 1022 1122 1110 1020 1001 1212 1112 1120 1001 1000 1200 1111 1222 1100 1011 1102 1010 1220 1020 1121 1111 1022 1221 1212 1001 1100 1110 1012 1020 1111 1122 1112 1202 1010 1102 1200 1220 1222 1201 1021 1220 1202 1011 1121 1120 1002 1000 1110 1100 1122 1212 1020 1121 1222 1201 1211 1221 Table 1 shows an example. Verifying that this is a CPHF entails checking, for each 4-set of columns, that in at least one row the four permutation vectors are covering. For instance, checking that the second, fifth, sixth, and seventh columns have a covering 4-set in some row is the same as checking that at least one of the matrices 2 664 1 1 1 1 0 1 0 0 0 2 0 0 2 2 1 2 3 775 , 2 664 1 1 1 1 0 1 0 0 0 2 0 0 1 0 1 0 3 775 , 2 664 1 1 1 1 2 1 1 0 2 0 1 1 1 0 0 2 3 775 , 2 664 1 1 1 1 2 1 1 0 2 2 2 0 0 1 0 2 3 775 is nonsingular over F3. The first two are not because they repeat columns; the third is not because the sum of the first and fourth rows equals the second. But the fourth matrix is nonsingular, so we have verified that one of the 3 060 possible 4-sets of columns is covering (the diligent reader can verify the rest). Every permutation vector in the CPHF of Table 1 has a 1 in coordinate 0; hence the CPHF satisfies the restriction (1, 2, 3, 4) with U1 = U2 = U3 = U4 = {0}. More is true. In the third and fourth rows, in each column the C. J. Colbourn and E. Lanus: Subspace restrictions and affine composition for covering . . . 5 two permutation vectors agree in the first two coordinates, and hence the CPHF satisfies the restriction (3, 4) with U3 = U4 = {0, 1}. Moreover, in the first and second rows, in each column the two permutation vectors agree in the first three coordinates, and hence the CPHF satisfies the restriction (1, 2) with U1 = U2 = {0, 1, 2}. When S is a set of restrictions for n rows, and a CPHF(n; k, q, t) meets each S 2 S , it is S-restricted. Now suppose that A is an S-restricted CPHF(n; k, q, t). Suppose that S 2 S consists of (x1, . . . , xr) and (U1, . . . , Ur), and that each Ui is a p-tuple. For each 1  i  r, in the qt rows obtained from the expansion of row xi, let Ei be the qp rows that arise using evaluations on t-sets that are 0 on all elements not in Ui. Then E1 = · · · = Er; that is, each row in E1 is replicated r 1 further times in the expansion of A by Lemma 1.2, and hence (r 1)qp rows are redundant in the covering array generated (although Lemma 1.2 removes some of them already). In the example of Table 1, we noted the presence of three restrictions. The restriction of dimension 1 and replication 4 results in 3 · 3 = 9 redundant rows. The restriction of dimension 2 results in 32 = 9 redundant rows, of which three were already found to be redundant using the restriction of dimension 1. Finally the restriction of dimension 3 results in 33 = 27 redundant rows; three of these were already found to be redundant using the restriction of dimension 1, while none among the remaining 24 are made redudant by the restriction of dimension 2. Hence for our example, we can ensure that at least 9+6+24 = 39 rows are redundant; rather than getting 4 · 34 3 = 321 rows, we get 285. A general example of a subspace restriction is straightforward. When the restriction S has r = n, (x1, . . . , xn) = (1, 2, . . . , n), and Ui = (0) for 1  i  n, an S-restricted CPHF(n; k, q, t) is precisely an SCPHF(n; k, q, t). (The entry in coordinate 0 must be nonzero for the array to be a CPHF.) Enforcing restrictions of larger dimension can in- crease the redundancy [6], but enforcing too many restrictions or restrictions of too large a dimension might result in more rows or fewer columns. Next we explore the effect of imposing a restriction on the expectation that an array is a CPHF. To simplify the presentation, we fix a strength t and a prime power q, and denote the product Qb i=a qtqi qt by ⇡a,b. We consider a fixed set of t columns and ask for the prob- ability that the columns are covered within r rows of the CPHF. In the basic process, we impose no restriction, choosing each of the coordinates of each of t permutation vectors independently and uniformly at random from Fq for each of the r rows. The probability that the chosen set of columns is not covered in the basic process is (1⇡0,t1)r. In the re- stricted process, we first choose the p entries specified in the restriction independently and uniformly at random for one row, using the same choice for all. The remaining coordinates of each permutation vector are then chosen randomly. The probability that the chosen set of columns is not covered in the restricted process is (1⇡0,p1)+⇡0,p1 [(1 ⇡p,t1)r]. The restricted process has a larger failure probability; indeed it is larger by an amount equal to (1 ⇡p,t1) h 1 [1 ⇡0,t1]r1 i . We consider two cases to examine the effect of the dimension and replication of a restriction. We tabulate failure probabilities within r rows when there is a restriction of size r and dimension p. (When p = 0, there is no restriction.) First we give failure probabilities for q = 25 and t = 6 (see Table 2). 6 Art Discrete Appl. Math. 1 (2018) #P2.03 Table 2: Failure probabilities with restrictions for q = 25 and t = 6. p # r ! 2 3 4 0 .1730551453⇥ 102 .7199076267⇥ 104 .2994808332⇥ 105 1 .1730555215⇥ 102 .7199483800⇥ 104 .2998903189⇥ 105 2 .1730649273⇥ 102 .7209672112⇥ 104 .3101274621⇥ 105 3 .1733000718⇥ 102 .7464379962⇥ 104 .5660560231⇥ 105 4 .1791790606⇥ 102 .1383210872⇥ 103 .6964257727⇥ 104 5 .3263893163⇥ 102 .1730452999⇥ 102 .1669115393⇥ 102 As one might expect, restrictions increase the failure probability, but for those of ‘low’ dimension the increase is modest. Next we examine failure probabilities for q = 3 and t = 5 (see Table 3). Table 3: Failure probabilities with restrictions for q = 3 and t = 5. p # r ! 2 3 4 0 .1924749034 .0844425161 .0370465884 1 .1937767036 .0868835477 .0402353448 2 .1977309220 .0942611526 .0498358606 3 .2100498330 .1168880686 .0789332757 4 .2516261575 .1892616707 .1684735084 Naturally, because q is much smaller the failure probabilities are much larger than for q = 25, even though the strength here is lower. Now suppose that our goal is to make 2qp rows redundant in the generated covering array. We can choose a restriction of dimension p and replicate it three times. An alternative is to choose a second restriction of dimension p. If we choose the two restrictions so that they have no rows in common, we can replicate each twice to get the same number of redundant rows that we would get by selecting one with replication three. Which should we prefer? An easy calculation shows that the failure probability after four rows is lower with two restrictions of replication two, than one of replication three along with an unrestricted row. In general when two restrictions of dimension p with replications r1 and r2 restrict disjoint sets of rows, failure probabilities are minimized when r1 and r2 are as equal as possible. Because setting r2 = 1 imposes no restriction at all, our quick example says that r1 = r2 = 2 is better than r1 = 3 and r2 = 1. Hence we strive to choose many restrictions with replication two. This ignores the fact that there may be too few rows to specify the desired number of restrictions. However, the requirement that the restrictions share no rows is too severe. For the analysis, we only require that every two restrictions sharing a row select different coordinates within that row. This ensures that the rows made redundant by one are not those made redundant by the other (with the exception of the all zero row, which is redundant in every case). Moreover, in an analysis of failure probabilities the effects of two restrictions are independent, because the entries in each are chosen independently of one another. Consider restrictions of dimensions d1 and d2 that share s coordinates within a row. The impact is that the qd1 rows made redundant by the first and the qd2 rows made redundant by C. J. Colbourn and E. Lanus: Subspace restrictions and affine composition for covering . . . 7 the second have qs rows in common (a subspace). Consequently, fewer rows are redundant than if the two restrictions acted independently. When restrictions share the same rows, even when on different coordinates, the effect on the failure probability can be dramatic: When t = 5, for example, a restriction of dimension 4 on (0, 1, 2, 3) and a restriction of dimension 2 on (3, 4), if replicated on the same two rows, are in fact a restriction of dimension 5, forcing a replicated row in the CPHF itself. Nevertheless, when s is small, the double coverage is also small, and there are cases in which permitting sharing is sensible. 3 Affine composition Algorithms employed for unrestricted CPHFs from [6, 21, 25] extend in a natural way to search for restricted CPHFs, so we do not repeat them here. Instead we describe an additional approach. When one wants to make a covering array with ‘many’ columns, computational methods either require too much time, or yield an array with many more rows than anticipated. Yet the same methods can yield arrays with ‘few’ rows quickly when the number of columns is small. To take advantage of the efficacy of computational methods for smaller arrays, and still construct larger ones, recursive methods have been developed to use small arrays in a cut–and–paste method; for example, see [3, 4, 7, 8, 15, 16, 19]. Roughly speaking, a cut–and–paste (or composition) method starts with an array on k columns, forms m copies of the array written side by side to obtain mk columns, and then uses further rows to cover the as-yet-uncovered interactions. Writing two copies of a CPHF(n; k, q, t) side by side is equivalent to duplicating each column. But then in the (n ⇥ 2k) array produced, every choice of t columns contain- ing a duplicate leads to a singular matrix for every row of the CPHF. Hence exactlyPbt/2c s=1 k s ks t2s 2 t2s sets of t columns are noncovering. Although the uncovered t-sets are easily counted and characterized, there are many of them. Let A be a CPHF(n; k, q, t). Consider the effect of multiplying coordinate c of every permutation vector in row r by a nonzero element µr,c of the finite field. This does not affect the (non)singularity of any of the t⇥ t matrices that determine coverage. Indeed we can apply different nonzero multipliers µr,c for every 1  r  n and 1  c  t 1 to form a new array B that is again a CPHF(n; k, q, t). Then we no longer simply duplicate columns, we can change them in a controlled way. Similarly for any row we can choose a field element m and any two different coordi- nates c and d; then for each column, add m times the entry in coordinate c to the entry in coordinate d. Again, this does not affect the (non)singularity of any of the t ⇥ t matrices that determine coverage. This is most easily accomplished using an SCPHF, whose per- mutation vectors always have first coordinate equal to 1. Then setting c = 0, this amounts to m being an adder, which can be added to the entry of coordinate d of every permutation vector in the row. We can choose adders ↵r,c for every 1  r  n and 1  c  t 1, provided that the array is an SCPHF. In general, given an SCPHF(n; k, q, t) A, multipliers µr,c and adders ↵r,c for 1  r  n and 1  c  t 1, we can create a new array by, for each row r, for every column, replacing the entry x in coordinate c by µr,cx + ↵r.c, with arithmetic in the field. Hence each coordinate of each row undergoes an affine transformation. No matter how this is done, the resulting array is an SCPHF(n; k, q, t). There are (q(q 1))n(t1) ways to choose these multipliers and adders. Affine composition applied to an SCPHF(n; k, q, t) A0 selects m 1 arrays A1, . . . , Am1, each obtained by affine transformations of A. 8 Art Discrete Appl. Math. 1 (2018) #P2.03 When affine composition is applied to A0 to form A1, . . . , Am1, not only is each an SCPHF when A0 is, but each Ai meets all of the restrictions that A0 does. In fact, although [A0 A1 · · · Am1] need not be an SCPHF because certain t-sets of columns are not covered, it does meet all restrictions that A0 does. The question remains: Which affine composition should we apply in order to leave the fewest, or to leave a particular set, of uncovered t-sets of columns? We consider various SCPHF(n; k, q, t)s, taking m = 2. Because considering all (q(q 1))n(t1) affine transformations is too time-consuming, we adopt a greedy strat- egy. We consider each row in turn, and determine for some or all of the (q(q 1))t1 affine transformations how many uncovered t-sets of columns having at least one column from the original and from the transformed copy remain, if this transformation is applied in the current row. We choose transformations that lead to a smallest number of t-sets yet to cover. After the last row is processed, this smallest number is the number that must be dealt with in additional rows not produced in the composition. Tie-breaking is carried out by choosing the lexicographically first, so the method as implemented is deterministic. Random tie-breaking, or selection methods more clever than greedy, might result in further reductions. Table 4: Numbers of noncovering t-sets after affine compositions. n; k, v, t ⇥1 + 0 ⇥1 + ↵ ⇥µ+ 0 ⇥µ+ ↵ ⇥1 + ↵c ⇥µc + 0 ⇥µc + ↵c 2;12,4,4 2706 548 678 535 486 619 463 3;21,4,4 16170 1876 2015 1739 1572 1964 1452 4;31,4,4 54405 3239 2941 2783 2621 2594 2099 5;45,4,4 171270 4584 4176 3698 4228 3765 3312 6;59,4,4 391819 5161 3911 4835 3546 2;15,5,4 5565 1016 871 848 915 871 803 3;24,5,4 24564 1795 1611 1409 1634 1442 1236 4;40,5,4 119340 4001 3299 3037 3441 2869 2432 5;59,5,4 391819 5253 3635 4938 3337 6;88,5,4 1320660 8451 4689 7964 4367 2;18,7,4 9945 1109 985 940 1081 972 855 3;34,7,4 72369 2883 2421 2619 2202 4;57,7,4 352716 4627 3140 4386 2983 2;20,8,4 13870 1296 1242 1117 1224 1172 1069 3;38,8,4 101935 2979 2499 2737 2249 4;67,8,4 577071 5461 3441 5191 3278 2;22,9,4 18711 1606 1448 1330 1539 1364 1251 2;11,4,5 11550 2280 1763 1500 2070 1618 1389 3;15,4,5 46410 4856 2706 2350 4550 2706 2;10,3,6 25320 7984 5081 4573 7524 4766 4274 In Table 4, we report results for the number of uncovered t-sets of columns after affine composition when the allowed affine transformations are limited in various ways. The column (⇥1 + 0) gives numbers when every multiplier is 1 and every adder 0. This, of C. J. Colbourn and E. Lanus: Subspace restrictions and affine composition for covering . . . 9 course, is another way to say that the second array is an exact copy of the first. The column (⇥1 + ↵) always uses multiplier 1, and selects the same adder for every coordinate in the row. The column (⇥µ + 0) always uses adder 0, and selects the same multiplier for every coordinate in the row. The column (⇥µ + ↵) selects the same adder, and the same multiplier, for every coordinate in the row. The column (⇥1 + ↵c) always uses multiplier 1, and considers all qt1 ways to select adders for the coordinates in the row. The column (⇥µc + 0) always uses adder 0, and considers all (q 1)t1 ways to select multipliers for the coordinates in the row. The column (⇥µc + ↵c) considers all (q(q 1))t1 ways to select adders and multipliers for the coordinates in the row. Consider an arbitrary permutation vector in Ut,q and its images under the (q(q1))t1 affine transformations. Every permutation vector in Ut,q appears precisely (q1)t1 times among these images. Hence when q(t1) is large compared to the number of columns, the probability that duplicate columns arise is reduced. Indeed simply making a copy (without any nontrivial transformation) leaves far more uncovered sets of columns than even very restricted sets of affine transformations do. This is part of the explanation for the effective- ness of applying affine transformations. Choosing the best affine transformation to apply seems impractical; indeed we did not fill in the blank entries in Table 4 because even the greedy strategy has either too many op- tions to consider or too large an array to check repeatedly. Of course, it would be better to determine the most effective transformations without having to conduct a large search, but because this depends heavily on the structure of the SCPHF, we know of no way to do this. Hence we choose them randomly. The expected number of t-sets of columns that remain uncovered after affine composition depends not only on the parameters of the SCPHF and the number of copies made, but also on the structure of the SCPHF. Indeed it depends on the number of rows in which ` columns of the SCPHF are linearly independent, for every way to choose ` columns with 2  ` < t. Consider, for example, the situation with strength t = 3, and consider two columns. For any row with identical permutation vectors in these two columns, there is no possibility for affine composition to cover a 3-set of columns con- taining this pair. Then the probability that randomly chosen affine transformations succeed on a particular triple of columns depends upon in how many rows of the CPHF the two have identical entries. In general, we wish to maximize the number of sets of ` columns that are linearly independent for all `  t. Because the definition of a CPHF does not include such a strong condition, there can be SCPHFs on which affine composition yields a poor result. After any affine composition is carried out, we anticipate that not all t-sets of columns will be covered (although within each of the m copies formed. all t-sets of columns are covered). Hence affine composition is not a means to avoid doing any computation, but rather a means to reduce to a substantially smaller problem. 4 Computations with subspace restrictions Subspace restrictions give a natural way for redundant rows to be formed in the expansion of a CPHF into a covering array. For this to be worthwhile, the restricted CPHF must have more columns than does the unrestricted CPHF with one fewer row. (Otherwise the resulting covering array would in general have more rows without increasing the number of columns.) In the results to follow, we find that restrictions not only reduce the number of rows needed, but in many cases do not reduce the achievable number of columns. 10 Art Discrete Appl. Math. 1 (2018) #P2.03 We construct SCPHFs satisfying specific sets of restrictions. We always enforce the restriction (x1, . . . , xn) = (1, 2, . . . , n), and Ui = (0) for 1  i  n; this means we are talking about SCPHFs and not general CPHFs, and hence can employ affine composition as described. We also enforce restrictions of higher dimension. In particular, a restriction on two distinct rows i and j with Ui = Uj = (0, 1, . . . , p 1) is denoted by hpii,j or, more simply, hpi. When multiple restrictions of this type are enforced, we require that they refer to disjoint sets of rows of the SCPHF; provided that they do, we can simply list the dimensions of the restrictions imposed. We use exponential notation, so that hpia requires that the restriction hpi be imposed on a disjoint pairs of rows. In order to accelerate the computation, we use affine composition in some cases to make a ‘large’ fraction of the CPHF. We employ column resampling, random extension, and conditional expectation algorithms from [6]. The adaptation of each to incorporate any number of restrictions is straightforward. There are more intensive search techniques that yield more accurate results, but the simpler methods can be effectively applied for some- what larger numbers of columns and symbols. So when we report that a certain number of columns can be realized, we fully expect that a more intensive search can find a solution with more columns (and, in some cases, has). Despite this, certain trends are evident, as one expects based on the failure probabilities. Given a prime power q, number n of rows, strength t, and set S of restrictions, we tabulate the largest number k of columns found in an S-restricted SCPHF(n; k, q, t). When q = 23 and t = 4, rows in Table 5 indicate the value of n; columns indicate the restrictions enforced. Here C reports results for CPHFs, while reports results for SCPHFs (that is, no restrictions beyond the basic one). Table 5: Improvements (shown in bold) on known covering array numbers when t = 4 and q = 23. n C h2i h2i2 h3i h3ih2i h3i2 h3i3 2 39 39 39 30 3 98 98 98 85 4 250 245 240 227 196 194 170 5 603 600 585 569 497 484 389 6 1461 1365 1333 1192 1184 1174 1003 874 Entries shown in bold are those that improve upon the previously best known size of a covering array for these parameters (all from [6]). It may be disappointing that by imposing three h3i restrictions, our methods construct only 874 columns rather than 1365. However, one must bear in mind that these restrictions force (at least) 36 432 redundant rows in the covering array. This accomplishes what we set out to do. Although we may have fewer columns, we generate fewer rows. Table 6 shows similar results for t = 5 and q = 5, using a more extensive set of restrictions. Improvements are again frequent. Table 7 gives a complete set of results for strength t = 4 with 4  q  25. (Henceforth we do not display every improvement for a covering array number in bold; see [5] to determine when an array is the best known.) Table 8 displays the results for t = 5 with 4  q  25 having two, three, and four rows, while Table 9 displays results with five and six rows, and Table 10 gives results having seven or more rows. C. J. Colbourn and E. Lanus: Subspace restrictions and affine composition for covering . . . 11 Table 6: Improvements (shown in bold) on known covering array numbers when t = 5 and q = 5. n C h2i h2i2 h3i h3ih2i h3i2 h4i h4ih2i h4ih3i h4i2 h4i3 2 12 12 12 12 12 3 17 17 17 16 16 4 24 24 23 23 23 22 22 21 21 21 21 5 32 32 32 32 32 31 30 30 30 28 28 6 44 44 44 41 42 40 40 42 39 39 39 35 7 59 59 59 58 58 57 56 54 52 52 51 45 8 81 81 81 79 78 76 75 73 70 68 67 62 9 107 107 107 106 106 106 103 95 93 93 91 84 10 143 142 142 141 141 139 138 131 128 126 119 110 11 196 196 196 196 191 191 187 175 174 171 162 152 12 266 266 266 266 262 261 255 242 242 239 220 200 13 346 346 346 340 333 333 327 333 333 327 286 265 12 Art Discrete Appl. Math. 1 (2018) #P2.03 Table 7: Number of columns found for various restrictions and strength four. q n C h2i h3i h3i2 h3i3 4 2 13 12 11 9 4 3 21 21 20 17 4 4 32 31 31 27 23 4 5 45 45 43 38 34 5 2 15 15 14 11 5 3 26 24 24 21 5 4 41 40 40 34 30 5 5 60 59 58 54 48 5 6 90 89 88 83 74 65 5 7 141 138 132 125 110 103 5 8 206 205 198 172 157 157 5 9 306 301 286 265 232 214 5 10 465 457 434 392 355 314 5 11 700 680 657 559 502 480 5 12 1102 1013 1012 820 751 712 5 13 1607 1431 1425 1264 1127 1090 7 2 18 18 17 14 7 3 34 34 33 30 7 4 57 57 56 51 45 7 5 101 99 98 89 79 7 6 169 169 166 154 134 119 7 7 282 277 275 242 209 186 7 8 475 475 457 394 364 311 7 9 814 764 742 631 551 527 7 10 1338 1336 1334 1064 956 897 8 2 20 20 19 15 8 3 38 38 37 33 8 4 67 67 67 60 53 8 5 122 121 118 109 94 8 6 220 218 214 177 154 148 8 7 379 370 359 316 277 239 8 8 657 657 611 565 445 415 8 9 1161 1159 1155 918 808 732 9 2 22 22 21 17 9 3 43 43 41 37 9 4 79 78 76 69 60 9 5 148 147 143 133 114 9 6 270 276 275 223 188 183 9 7 487 484 462 405 356 309 9 8 896 847 845 681 596 556 9 9 1621 1503 1475 1243 1106 1025 q n C h2i h3i h3i2 h3i3 11 2 24 24 24 20 11 3 49 49 47 43 11 4 100 99 96 89 80 11 5 207 202 199 166 141 11 6 388 388 374 321 286 241 11 7 745 735 734 572 494 456 11 8 1508 1507 1474 1112 1009 934 13 2 28 27 27 21 13 3 57 57 55 51 13 4 124 124 121 106 94 13 5 260 265 262 214 177 13 6 524 524 494 431 371 324 13 7 1102 1027 1002 839 721 654 16 2 31 31 31 24 16 3 71 71 69 61 16 4 159 158 157 140 120 16 5 357 357 344 293 246 16 6 778 763 745 579 506 468 16 7 1666 1578 1527 1260 1102 1040 17 2 34 34 32 25 17 3 75 74 71 63 17 4 175 174 171 152 122 17 5 382 380 366 320 258 17 6 873 822 818 654 550 517 17 7 1778 1743 1323 1193 19 2 35 35 35 28 19 3 83 83 79 72 19 4 203 199 197 167 136 19 5 455 455 443 381 301 19 6 1056 1001 934 802 692 642 23 2 39 39 39 30 23 3 98 98 98 85 23 4 250 245 240 196 170 23 5 603 600 585 497 389 23 6 1461 1365 1333 1174 1003 874 25 2 42 42 41 33 25 3 107 107 104 90 25 4 277 274 265 215 187 25 5 694 688 668 529 443 25 6 1706 1584 1575 1286 1115 1037 C. J. Colbourn and E. Lanus: Subspace restrictions and affine composition for covering . . . 13 Table 8: Number of columns found for various restrictions and strength five, n 2 {2, 3, 4}. n = 2 n = 3 n = 4 q C h2i h3i h4i C h2i h3i h4i C h2i h3i h4i h4i2 3 12 11 10 10 8 13 13 12 12 12 16 16 15 15 15 14 4 11 11 11 11 10 15 15 15 15 14 20 20 20 19 18 17 5 12 12 12 12 12 17 17 17 16 16 24 24 23 23 21 21 7 14 14 14 14 12 22 21 21 21 19 31 31 31 31 29 27 8 15 15 15 15 13 23 23 23 23 22 36 36 36 35 33 30 9 16 16 15 15 14 25 25 24 24 22 39 39 39 39 36 33 11 17 17 17 16 15 29 29 28 28 26 47 47 47 47 42 38 13 19 19 18 18 16 31 31 31 31 29 55 55 55 53 48 44 16 20 20 20 20 18 38 38 38 36 33 66 66 64 64 56 53 17 21 20 20 20 18 39 39 39 37 34 70 70 70 68 61 56 19 22 22 22 22 19 41 41 40 40 37 78 78 77 75 68 60 23 24 24 24 24 21 48 48 47 46 43 90 90 88 87 81 72 25 25 25 24 24 21 49 49 49 49 42 98 97 96 94 88 76 Table 9: Number of columns found for various restrictions and strength five, n 2 {5, 6}. n = 5 n = 6 q C h2i h3i h4i h4i2 C h2i h3i h4i h4i2 h4i3 3 19 19 19 19 18 17 23 23 23 22 22 21 19 4 26 26 26 25 24 23 34 34 32 32 31 28 27 5 32 32 32 32 30 28 44 44 44 42 42 39 35 7 47 47 47 47 43 39 69 69 68 67 61 59 50 8 55 55 55 55 50 45 83 83 82 82 74 67 60 9 62 62 61 60 55 51 95 95 92 91 88 77 73 11 79 79 79 77 69 61 127 125 125 120 109 100 90 13 93 93 92 88 84 75 157 157 156 153 140 120 106 16 119 119 119 111 103 92 210 207 206 197 185 165 146 17 128 128 128 120 112 97 228 223 219 217 199 181 157 19 143 143 143 136 126 107 269 262 262 258 233 207 180 23 181 178 175 171 155 136 345 344 342 332 299 259 228 25 197 197 194 184 166 147 406 406 391 386 332 288 249 14 Art Discrete Appl. Math. 1 (2018) #P2.03 Table 10: Number of columns found for various restrictions and strength five, n 7. q n C h2i h3i h4i h4i2 h4i3 4 7 42 42 41 41 41 38 33 5 7 59 59 59 58 54 51 45 5 8 81 81 81 78 73 67 62 5 9 107 107 107 106 95 91 84 5 10 143 142 142 141 131 119 110 5 11 196 196 196 191 175 162 152 5 12 266 266 266 262 242 220 200 5 13 346 346 346 333 333 288 265 5 14 458 458 455 447 435 394 363 5 15 609 609 606 603 569 527 484 5 16 779 721 672 647 7 7 101 101 99 96 93 82 77 7 8 149 149 148 140 132 121 112 7 9 211 209 205 201 195 179 159 7 10 318 312 312 315 284 253 230 7 11 455 454 451 447 418 361 341 7 12 661 661 660 628 592 546 495 7 13 702 679 8 7 123 123 122 116 109 99 90 8 8 186 185 184 183 163 150 134 8 9 283 283 277 278 252 229 210 8 10 430 429 427 417 371 343 320 8 11 622 622 604 595 569 510 473 8 12 812 749 675 q n C h2i h3i h4i h4i2 h4i3 9 7 151 149 149 147 133 118 110 9 8 231 229 229 224 214 184 167 9 9 374 374 355 353 324 283 254 9 10 568 568 546 542 483 445 405 9 11 818 755 690 629 11 7 209 209 209 196 182 161 147 11 8 342 340 332 328 289 248 228 11 9 533 529 524 505 472 425 389 11 10 827 730 680 614 13 7 272 272 271 261 237 209 186 13 8 454 452 450 442 379 349 324 13 9 767 767 764 750 653 567 523 16 7 379 379 376 360 336 284 248 16 8 656 656 650 641 582 515 454 16 9 766 17 7 408 407 407 400 358 309 278 17 8 721 721 718 663 620 555 480 19 7 473 470 466 461 446 373 343 19 8 851 838 723 671 597 23 7 656 656 654 648 564 493 430 23 8 785 25 7 712 712 709 701 636 568 483 C. J. Colbourn and E. Lanus: Subspace restrictions and affine composition for covering . . . 15 We illustrate a further useful application of restrictions using strength t = 6. Every time a restriction ht 1i is enforced, qt1 q additional rows become redundant in the covering array. By enforcing restrictions ht1iq , qt q2 rows are redundant. On the other hand, each row of the SCPHF would employ only a slightly larger number of rows, qt q. To take advantage of this, compare the four situations with q = 3 and t = 6 in Table 11. The SCPHF produced is permitted to have n rows, but must satisfy the specified number of ht 1i restrictions. The covering array generated has N rows; notice how close the values of N are. Which should we prefer? By computing failure probabilities after n rows subject to the restrictions, one finds that enforcing more restrictions gives lower failure probability, primarily because more restrictions allow more rows. Table 11: Four restricted SCPHF(n; k, 3, 6)s that yield covering arrays with similar num- bers of rows. n N #h5i Failure k 15 10893 0 .0000044079 57 16 10899 3 .0000043357 57 17 10905 6 .0000042646 58 18 10911 9 .0000041948 61 We applied the simple computational methods to each; the computed failure probabil- ities suggest that we should choose more restrictions, and the largest number of columns produced agrees. This is not an isolated example. In Table 12 we report on similar compu- tations for q = 3 and t = 6 with different numbers of rows and restrictions. In order to read this effectively, an entry ought to be compared with the one that is three columns to the left and one row above, because the resulting covering arrays have comparable number of rows. We reiterate that we have not found the maximum number of columns in general; indeed we may be very far from it. Nevertheless, it is important that when enough restrictions (and the right ones) are enforced, there is a possibility of improving on an unrestricted SCPHF with fewer rows. Improvements arise frequently when t = 6 for larger values of q as well; we summarize the results in Table 13. Naturally, other search techniques can be applied to make improvements, and other restrictions may prove useful in the construction of covering arrays. What we have shown is that worthwhile restrictions can often be enforced with little penalty in failure probability or in number of columns generated. In order to avoid substantial computaion, it would be of substantial interest to develop further geometric constructions of CPHFs using finite projective or affine spaces, particularly with an eye to which nontrivial restrictions can be enforced. 5 Concluding remarks Two extensions of research on covering perfect hash families have been developed here. The first provides a flexible recursive technique for making large CPHFs from smaller ones; the remarkable feature of this approach is that rather than simply juxtaposing copies of smaller arrays, each copy can be transformed by affine mappings in order to enhance the coverage obtained. We have demonstrated that different affine transformations can have a 16 Art Discrete Appl. Math. 1 (2018) #P2.03 Table 12: Number of columns found in an SCPHF(n; k, 3, 6) satisfying h5i`. Number ` of h5i restrictions n C 0 1 2 3 4 5 6 7 8 9 10 2 10 10 8 3 11 11 11 4 13 13 12 5 15 16 15 14 6 18 18 17 16 16 7 20 21 19 19 18 8 23 23 21 21 20 19 9 27 27 24 24 24 23 10 31 30 28 27 27 25 24 11 34 33 31 30 30 29 27 12 39 38 35 35 33 33 31 31 13 44 43 40 40 37 39 36 35 14 50 50 50 47 42 44 42 39 39 15 57 57 54 51 50 48 47 46 44 16 66 66 62 59 57 55 54 52 51 49 17 73 71 70 68 66 64 62 58 58 53 18 85 82 82 77 75 74 72 68 64 63 61 19 99 98 93 87 86 82 77 73 73 70 69 20 108 108 101 96 92 85 82 78 77 72 21 123 122 102 97 93 88 85 22 144 142 102 97 C. J. Colbourn and E. Lanus: Subspace restrictions and affine composition for covering . . . 17 Table 13: Number of columns found for various restrictions and strength six. q n C h2i h3i h4i h5i h5i2 h5i3 4 2 11 11 11 11 10 9 4 3 13 13 13 13 13 12 4 4 16 16 16 16 15 15 14 4 5 20 19 19 19 19 18 17 4 6 23 23 23 23 23 22 21 19 4 7 28 28 27 27 26 26 24 22 4 8 33 33 33 33 32 31 31 28 4 9 40 40 40 39 38 38 36 35 4 10 49 49 48 48 46 46 43 43 4 11 58 58 57 57 54 54 53 50 4 12 73 73 73 72 69 66 63 60 4 13 90 86 86 84 82 78 76 74 4 14 107 107 107 103 100 96 93 89 4 15 128 128 128 127 127 119 112 107 4 16 157 156 153 150 146 141 134 131 5 2 11 11 11 11 11 10 5 3 15 15 14 14 14 13 5 4 19 19 18 18 18 17 16 5 5 23 23 23 23 23 22 20 5 6 30 30 28 27 27 26 25 24 5 7 36 36 35 35 34 33 32 29 5 8 46 45 45 44 44 43 38 37 5 9 59 59 56 56 54 53 48 47 5 10 75 74 74 72 69 68 64 59 5 11 93 93 93 92 90 85 82 76 5 12 121 121 121 120 118 111 103 95 5 13 156 156 149 149 146 142 131 121 7 2 13 13 13 12 12 11 7 3 17 17 17 17 17 15 7 4 23 23 22 22 22 21 20 7 5 31 31 31 31 29 27 27 7 6 41 41 40 39 38 38 36 33 7 7 55 55 55 52 51 49 46 44 7 8 76 76 76 75 70 68 63 58 7 9 103 103 103 101 95 93 85 80 7 10 140 140 138 135 132 127 116 106 8 2 13 13 13 13 13 12 8 3 18 18 18 18 17 17 8 4 26 26 26 25 25 23 22 8 5 33 33 33 32 32 31 29 8 6 47 47 47 46 45 42 39 36 8 7 65 65 65 65 61 58 54 48 8 8 94 94 94 91 88 83 77 69 8 9 130 130 130 128 127 118 109 102 q n C h2i h3i h4i h5i h5i2 h5i3 9 2 14 14 13 13 13 12 9 3 20 20 19 19 19 18 9 4 27 26 26 26 25 24 23 9 5 38 38 37 37 36 35 33 9 6 55 55 53 53 51 49 44 39 9 7 78 76 75 75 72 69 64 58 9 8 112 112 112 108 107 99 92 83 11 2 15 15 14 14 14 13 11 3 22 22 22 22 21 19 11 4 31 31 31 30 30 28 26 11 5 46 46 45 44 43 41 35 11 6 68 67 66 66 61 60 54 47 11 7 100 100 100 97 95 88 83 74 11 8 150 150 147 145 141 132 121 107 13 2 16 16 16 16 15 14 13 3 23 23 22 22 22 20 13 4 35 34 34 33 33 30 29 13 5 54 54 52 52 51 49 41 13 6 82 81 81 81 77 71 65 59 13 7 124 124 124 124 121 111 102 91 16 2 16 16 16 16 16 15 16 3 26 26 25 25 25 22 16 4 41 40 39 39 39 35 33 16 5 66 66 63 63 58 56 51 16 6 104 104 104 103 98 89 82 73 16 7 164 163 159 156 150 141 127 113 17 2 17 17 16 16 16 15 17 3 27 27 25 25 25 23 17 4 42 42 41 41 41 38 35 17 5 66 66 65 65 62 57 53 17 6 108 108 108 106 105 96 86 77 17 7 178 177 176 171 167 156 139 125 19 3 29 29 27 27 27 25 19 4 46 45 45 45 45 41 37 19 5 76 74 74 74 70 65 59 19 6 124 124 124 122 118 110 100 87 23 3 31 31 30 29 29 26 23 4 52 52 51 51 51 45 41 23 5 87 87 87 87 85 78 70 23 6 148 148 146 145 145 133 119 106 25 3 31 31 31 31 30 28 25 4 56 56 54 54 54 49 43 25 5 96 95 95 93 91 82 76 25 6 169 169 166 163 161 147 132 116 18 Art Discrete Appl. Math. 1 (2018) #P2.03 dramatic effect on the composition, but theoretical guarantees for the observed improve- ments are needed. The second extension provides finer control on the number of rows obtained in the resulting covering array, using the notion of subspace restrictions. Extensive computa- tions demonstrate the effectiveness of such restrictions on reducing the number of rows in the covering arrays produced, over a wide range of parameters. Although the algorithmic methods used are relatively fast, we have repeatedly remarked that there is no reason to believe that they yield sizes that are besr possible in general. More computationally expen- sive methods such as backtracking [21] and tabu search [25], when feasible, may improve upon the results obtained. Indeed, after this paper was completed, simulated annealing and post-optimization have been used to obtain more accurate sizes for a smaller range of parameters [11, 23]. Certainly further computation will yield further improvements, and the use of subspace restrictions and affine composition accelerate these computations. However, we anticipate that direct constructions (extending those in [18, 22, 24]) would be of most interest, partic- ularly if they accommodate a variety of subspace restrictions. References [1] N. H. Bshouty and A. Costa, Exact learning of juntas from membership queries, in: R. Ortner, H. U. Simon and S. Zilles (eds.), Algorithmic Learning Theory, Springer, Cham, volume 9925 of Lecture Notes in Artificial Intelligence, 2016 pp. 115–129, doi:10.1007/978-3-319-46379-7 8, proceedings of the 27th International Conference (ALT 2016) held in Bari, October 19 – 21, 2016. [2] J. N. Cawse, Experimental design for combinatorial and high throughput materials develop- ment, Technical Report 2002GRC253, GE Global Research, November 2002. [3] M. Chateauneuf and D. L. Kreher, On the state of strength-three covering arrays, J. Combin. Des. 10 (2002), 217–238, doi:10.1002/jcd.10002. [4] M. B. Cohen, C. J. Colbourn and A. C. H. Ling, Constructing strength three covering arrays with augmented annealing, Discrete Math. 308 (2008), 2709–2722, doi:10.1016/j.disc.2006. 06.036. [5] C. J. Colbourn, Covering array tables: 2  v  25, 2  t  6, t  k  10000, 2005–2017, http://www.public.asu.edu/˜ccolbou/src/tabby/catable.html. [6] C. J. Colbourn, E. Lanus and K. Sarkar, Asymptotic and constructive methods for covering perfect hash families and covering arrays, Des. Codes Cryptogr. 86 (2018), 907–937, doi:10. 1007/s10623-017-0369-x. [7] C. J. Colbourn, S. S. Martirosyan, G. L. Mullen, D. Shasha, G. B. Sherwood and J. L. Yucas, Products of mixed covering arrays of strength two, J. Combin. Des. 14 (2006), 124–138, doi: 10.1002/jcd.20065. [8] C. J. Colbourn, S. S. Martirosyan, Tran Van Trung and R. A. Walker, II, Roux-type construc- tions for covering arrays of strengths three and four, Des. Codes Cryptogr. 41 (2006), 33–57, doi:10.1007/s10623-006-0020-8. [9] P. Damaschke, Adaptive versus nonadaptive attribute-efficient learning, Mach. Learn. 41 (2000), 197–215, doi:10.1023/a:1007616604496. [10] N. Graham, F. Harary, M. Livingston and Q. F. Stout, Subcube fault-tolerance in hypercubes, Inf. Comput. 102 (1993), 280–314, doi:10.1006/inco.1993.1010. C. J. Colbourn and E. Lanus: Subspace restrictions and affine composition for covering . . . 19 [11] I. Izquierdo-Marquez, J. Torres-Jimenez, B. Acevedo-Juárez and H. Avila-George, A greedy- metaheuristic 3-stage approach to construct covering arrays, Inform. Sci. 460–461 (2018), 172– 189, doi:10.1016/j.ins.2018.05.047. [12] D. R. Kuhn, R. N. Kacker and Y. Lei, Introduction to Combinatorial Testing, Chapman & Hall/CRC Innovations in Software Engineering and Software Development Series, CRC Press, 2013. [13] D. R. Kuhn, D. R. Wallace and A. M. Gallo, Software fault interactions and implications for software testing, IEEE Trans. Software Eng. 30 (2004), 418–421, doi:10.1109/tse.2004.24. [14] L. V. Lejay, D. E. Shasha, P. M. Palenchar, A. Y. Kouranov, A. A. Cruikshank, M. F. Chou and G. M. Coruzzi, Adaptive combinatorial design to explore large experimental spaces: approach and validation, IEE Proceedings - Syst. Biol. 1 (2004), 206–212, doi:10.1049/sb:20045020. [15] S. Martirosyan and Tran Van Trung, On t-covering arrays, Des. Codes Cryptogr. 32 (2004), 323–339, doi:10.1023/b:desi.0000029232.40302.6d. [16] S. S. Martirosyan and C. J. Colbourn, Recursive constructions of covering arrays, Bayreuth. Math. Schr. 74 (2005), 266–275. [17] P. Nayeri, C. J. Colbourn and G. Konjevod, Randomized post-optimization of covering arrays, European J. Combin. 34 (2013), 91–103, doi:10.1016/j.ejc.2012.07.017. [18] S. Raaphorst, L. Moura and B. Stevens, A construction for strength-3 covering arrays from linear feedback shift register sequences, Des. Codes Cryptogr. 73 (2014), 949–968, doi:10. 1007/s10623-013-9835-2. [19] G. Roux, k-propriétés dans des tableaux de n colonnes: cas particulier de la k-surjectivité et de la k-permutivité, Ph.D. thesis, Université de Paris, 1987, http://www.theses.fr/ 1987PA066096. [20] G. Seroussi and N. H. Bshouty, Vector sets for exhaustive testing of logic circuits, IEEE Trans. Inform. Theory 34 (1988), 513–522, doi:10.1109/18.6031. [21] G. B. Sherwood, S. S. Martirosyan and C. J. Colbourn, Covering arrays of higher strength from permutation vectors, J. Combin. Des. 14 (2006), 202–213, doi:10.1002/jcd.20067. [22] J. Torres-Jimenez and I. Izquierdo-Marquez, Covering arrays of strength three from extended permutation vectors, Des. Codes Cryptogr. (2018), doi:https://doi.org/10.1007/ s10623-018-0465-6. [23] J. Torres-Jimenez and I. Izquierdo-Marquez, A simulated annealing algorithm to construct covering perfect hash families, Math. Probl. Eng. 2018 (2018), Article ID 1860673, doi: 10.1155/2018/1860673. [24] G. Tzanakis, L. Moura, D. Panario and B. Stevens, Constructing new covering arrays from LFSR sequences over finite fields, Discrete Math. 339 (2016), 1158–1171, doi:10.1016/j.disc. 2015.10.040. [25] R. A. Walker, II and C. J. Colbourn, Tabu search for covering arrays using permutation vectors, J. Statist. Plann. Inference 139 (2009), 69–80, doi:10.1016/j.jspi.2008.05.020. ISSN 2590-9770 The Art of Discrete and Applied Mathematics 1 (2018) #P2.04 https://doi.org/10.26493/2590-9770.1239.e1f (Also available at http://adam-journal.eu) Reaction graphs of double Fano planes Mariusz Meszka AGH University of Science and Technology, Kraków, Poland Alexander Rosa McMaster University, Hamilton, ON, Canada Dedicated to Mario Gionfriddo on the occasion of his 70th birthday. Received 3 January 2018, accepted 16 February 2018, published online 6 August 2018 Abstract We consider various reaction graphs on the set of distinct double Fano planes. Keywords: Double Fano plane, reaction graph, strongly regular graph. Math. Subj. Class.: 05B07, 05C62 The concept of a reaction graph, which has its origin in chemistry, has been explored in several papers, for example, [5, 7, 8, 9, 10, 11]. The reaction graph(s) of the Fano plane (i.e. projective plane of order 2, Steiner triple system of order 7, or BIBD(7, 3, 1)) are considered in detail in [8, 9], with some additional comments provided in [7]. The vertices of such reaction graph are the 30 distinct Fano planes (on a fixed 7-element set). The reaction graph is of degree 14, 8, and 7, respectively, according to how adjacency is defined: namely, whenever two vertices (Fano planes) have one, zero, or three triples in common, respectively. The graph of degree 14 is actually iso- morphic to 2K15, that is, two disjoint complete graphs K15 as components. Each of these corresponds to a maximal set of MAD STS(7)s (mutually almost disjoint STSs, cf. [6]). It is well known that the simple BIBD(7, 3, 2) (i.e. with no repeated blocks) is unique up to an isomorphism and consists of two disjoint Fano planes. It contains 14 blocks (triples) and its automorphism group is of order 42. The blocks (triples) of one such design can be represented as {0, 1, 3}, {0, 2, 3} mod 7. We shall call any simple BIBD(7, 3, 2) a double Fano plane. Thus there are 7!42 = 120 distinct double Fano planes on any 7-element set. A double Fano plane will be denoted (a, b) provided a and b are the two disjoint Fano planes that constitute it. Let (a, b), (c, d) be two distinct double Fano planes. Due to the structure of the reaction graphs of the single Fano plane, whenever |{a, b, c, d}| = 4, two of the 4 intersections E-mail addresses: meszka@agh.edu.pl (Mariusz Meszka), rosa@mcmaster.ca (Alexander Rosa) cb This work is licensed under http://creativecommons.org/licenses/by/3.0/ 2 Art Discrete Appl. Math. 1 (2018) #P2.04 between a and c, a and d, b and c, and b and d must contain exactly one triple. Without loss of generality we may assume that the intersection between a and c, and also between b and d both contain one triple. The remaining two intersections, namely between a and d, and between b and c, may (i) both contain zero triples, or (ii) both contain three triples, or (iii) one contains zero and the other contains three triples. The edges of our reaction graph on K120 can now be one of four kinds: either |{a, b, c, d}| = 3, or it is one of the three types above (see Figure 1). We shall use the following “colourful” terminology. A green edge joins two double Fano planes (a, b), (c, d) when |{a, b, c, d}| = 3, that is, one of a, b equals one of c, d. A yellow, blue or red edge, respectively, joins two double Fano planes (a, b), (c, d) when |{a, b, c, d}| = 4 and case (i), (ii) or (iii), respectively, occurs. • • {a, b} • • {c, d} ................................................................. ................................................................. .... .... .... .... .... .... .... .... .... .... .... .... ................................................. ..... .... ... . .... .... .... .... .... .... .... ... ... .... .... ................................................................ ..... .... .... .... .... .... .... .... .... .... .... .... .... .... .... ........................................................... 1 1 3 3 blue edge • • {a, b} • • {c, d} ................................................................. ................................................................. .... .... .... .... .... .... .... .... .... .... .... .... ................................................. ..... .... ... . .... .... .... .... .... .... .... ... ... .... .... ................................................................ ..... .... .... .... .... .... .... .... .... .... .... .... .... .... .... ........................................................... 1 1 0 3 red edge • {a, b} • {c, d} • ................................................................. ........... .... .... ... ... ... ... ... .... .... .... .... .... .... .... ................................................................. .... .... .... .... .... .... .... ... ... ... ... ... .... .... ................................................................1 green edge • • {a, b} • • {c, d} ................................................................. ................................................................. .... .... .... .... .... .... .... .... .... .... .... .... ................................................. ..... .... ... . .... .... .... .... .... .... .... ... ... .... .... ................................................................ ..... .... .... .... .... .... .... .... .... .... .... .... .... .... .... ........................................................... 1 1 0 0 yellow edge Figure 1: Pairs of double Fano planes. According to the aforementioned intersections, one may define the following four re- action graphs: I. The green graph (the subgraph of K120 induced by the green edges). This graph is regular of degree 14, and is quasi-strongly regular of grade 3 (cf. [4]) with parameters (120, 14, 6, (0, 1, 2)). II. The yellow graph (the subgraph of K120 induced by the yellow edges). This graph is regular of degree 21, and is quasi-strongly regular of grade 2 (cf. [4]) with parameters (120, 21, 0, (3, 6)). III. The blue graph (the subgraph of K120 induced by the blue edges). This graph is regular of degree 28 and is quasi-strongly regular of grade 3 with pa- rameters (120, 28, 6, (4, 6, 12)). M. Meszka and A. Rosa: Reaction graphs of double Fano planes 3 IV. The red graph (the subgraph of K120 induced by the red edges). This graph is regular of degree 56. and is strongly regular with parameters (120, 56, 28, 24). A strongly regular graph with these parameters and automorphism group of order 348 364 800 is known to exist (cf. [3]; see also [12, 13]). The four coloured graphs together form a 4-class association scheme. The intersection numbers for this scheme can be found at http://home.agh.edu.pl/˜meszka/ reaction_graphs.html. Next we want to investigate the structure of so-called neighbourhood graphs. For a vertex {a, b} of the reaction graph, a vertex {c, d} joined to it by a green edge is called a green neighbour, and similarly for yellow, blue or red edges we have yellow, blue, or red neighbours. Given a vertex of the reaction graph, the green neighbourhood graph is the complete graph K14 on its green neighbours. Its edges are coloured green, yellow or red – there are no blue edges. The green edges induce graph consisting of two disjoint K7’s, the yellow edges induce the Heawood graph (cf. [1]) , and the red edges induce the bipartite complement of the Heawood graph. It is well-known that the automorphism group of the Heawood graph is PGL(2, 7) of order 336. The coloured edges form a 3-class association scheme. The yellow neighbourhood graph of a vertex is the complete graph K21 on its yellow neighbours. Its edges are 3-coloured: green, blue and red; there are no yellow edges. The graph induced by the green edges is regular of degree 4 and is distance-transitive with intersection array [4, 2, 2; 1, 1, 2], with automorphism group PGL(2, 7) of order 336. The graphs induced by the blue and red edges, respectively, both have degree 8, and the same automorphism group as the graph induced by green edges. In this case too the coloured edges form a 3-class association scheme. The blue neighbourhood graph of a vertex is the complete graph K28 on its blue neigh- bours. Its edges are 4-coloured. The graph induced by green and blue edges, respectively, is regular of degree 6, while the graph induced by the yellow edges is cubic, and is actu- ally isomorphic to the Coxeter graph (cf. [2]). The automorphism group of each of these three graphs is again PGL(2, 7). The graph induced by the red edges is the so-called 8- triangular graph; it is regular of degree 12, and is distance-transitive with intersection array [12, 5; 1, 4]. Its automorphism group is S8 of order 40 320. The coloured edges form a 4-class association scheme. Finally, the red neighbourhood graph of a vertex is the complete graph K56 on its red neighbours. Its edges are 4-coloured. The graph induced by green edges is of degree 6, and its automorphism group has order 225 792. Those induced by the yellow, blue and red edges, respectively, are of degree 9, 12, and 28, respectively, where the first two of these have automorphism group PGL(2, 7) of order 336, while the last one has large automorphism group of order 2 903 040. In this case, the coloured edges do not form an association scheme. Let us remark that the graph induced by the union of the green and blue edges is of degree 42, and turns out to be a quasi-strongly regular graph of grade 2 (cf. [4]) with parameters (120, 42, 18, (6, 15)). Of course, the graph induced by the union of green, yellow and blue edges is complementary to the red graph, and so is strongly regular with parameters (120, 63, 30, 36). 4 Art Discrete Appl. Math. 1 (2018) #P2.04 References [1] A. E. Brouwer, A. M. Cohen and A. Neumaier, Distance-Regular Graphs, volume 18 of Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag, Berlin, 1989, doi: 10.1007/978-3-642-74341-2. [2] H. S. M. Coxeter, My graph, Proc. London Math. Soc. 46 (1983), 117–136, doi:10.1112/plms/ s3-46.1.117. [3] D. Crnković, S. Rukavina and A. Švob, New strongly regular graphs from orthogonal groups O +(6, 2) and O(6, 2), Discrete Math. 341 (2018), 2723–2728, doi:10.1016/j.disc.2018.06. 029. [4] F. Goldberg, On quasi-strongly regular graphs, Linear Multilinear Algebra 54 (2006), 437–451, doi:10.1080/03081080600867210. [5] M. H. Klin, S. S. Tratch and N. S. Zefirov, Group-theoretical approach to the investigation of reaction graphs for highly degenerate rearrangements of chemical compounds: I. Criterion of the connectivity of a graph, J. Math. Chem. 7 (1991), 135–151, doi:10.1007/bf01200820. [6] C. C. Lindner and A. Rosa, Construction of large sets of almost disjoint Steiner triple systems, Canad. J. Math. 27 (1975), 256–260, doi:10.4153/cjm-1975-031-3. [7] C. C. Lindner and A. Rosa, Reaction graphs of the K4e design of order 6, Bull. Inst. Combin. Appl. 47 (2006), 43–47. [8] E. K. Lloyd, The reaction graph of the Fano plane, in: T.-H. Ku (ed.), Combinatorics and Graph Theory ’95, Volume 1, World Scientific Publishing Co., River Edge, NJ, pp. 260–274, 1995, proceedings of the Summer School and International Conference on Combinatorics (SSICC ’95) held in Hefei, May 25 – June 5, 1995. [9] E. K. Lloyd, Some graphs associated with the seven point plane, Amer. J. Math. Management Sci. 20 (2000), 85–101, doi:10.1080/01966324.2000.10737502. [10] E. K. Lloyd and G. A. Jones, Reaction graphs, Acta Appl. Math. 52 (1998), 121–147, doi: 10.1023/a:1005954907636. [11] Z. Masárová, Reaction graphs for some complete graph decompositions, to appear in J. Com- bin. Math. Combin. Comput. [12] R. Mathon and A. P. Street, Overlarge sets and partial geometries, J. Geom. 60 (1997), 85–104, doi:10.1007/bf01252220. [13] R. Mathon and A. P. Street, Partitions of sets of designs on seven, eight and nine points, J. Statist. Plann. Inference 58 (1997), 135–150, doi:10.1016/s0378-3758(96)00066-3. ISSN 2590-9770 The Art of Discrete and Applied Mathematics 1 (2018) #P2.05 https://doi.org/10.26493/2590-9770.1234.37b (Also available at http://adam-journal.eu) Mixed hypergraphs and beyond⇤ Zsolt Tuza Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, H-1053 Budapest, Reáltanoda u. 13–15, Hungary and Department of Computer Science and Systems Technology, University of Pannonia, H-8200 Veszprém, Egyetem u. 10, Hungary Dedicated to Mario Gionfriddo on the occasion of his 70th birthday. Received 31 December 2017, accepted 12 May 2018, published online 9 July 2019 Abstract Some open problems are collected on hypergraphs, graphs, and designs, presented at the HyGraDe conference celebrating Mario Gionfriddo’s 70th birthday. Keywords: Mixed hypergraph coloring, stably bounded interval hypergraph, chromatic spectrum, chromatic polynomial, WORM coloring, coloring of Steiner systems. Math. Subj. Class.: 05C15, 05B05 The conference HyGraDe took its name from HYpergraphs, GRAphs and DEsigns, three important areas of the research activities of Mario Gionfriddo, to whom we happily dedicated all our talks. Those are also the subjects of my collaborations with colleagues in Catania. For the celebration conference I collected some open problems which are related to the coloring theory of mixed hypergraphs; here they are organized in this three-sided structure. The sources of the problems are mentioned in the text, rather than specified inside the statement of each one. 1 Hypergraph coloring A hypergraph H is a pair (X, E), where X is the underlying set called vertex set and E is a set system over X , called edge set. A hypergraph is uniform if all its edges have the same cardinality; more specifically, if |E| = r for all E 2 E , then H is said to be r- uniform. (Hence, the 2-uniform hypergraphs are precisely the graphs.) In order to avoid some anomalies, we shall restrict our attention to hypergraphs in which each edge contains at least two vertices. ⇤Research supported in part by the National Research, Development and Innovation Office – NKFIH, grant SNN 116095. E-mail address: tuza@dcs.uni-pannon.hu (Zsolt Tuza) cb This work is licensed under http://creativecommons.org/licenses/by/3.0/ 2 Art Discrete Appl. Math. 1 (2018) #P2.05 As a general term, by coloring we mean any assignment ' : X ! N, and call '(x) the color of vertex x 2 X . The classical notion of proper coloring means a coloring such that every edge E 2 E contains two vertices of distinct colors; in other words, no edge is monochromatic. The chromatic number of H, denoted by (H), is the smallest possible number of colors in a proper coloring of H. The opposite side is where each edge E 2 E contains two vertices of the same color; i.e., no edge is multicolored.1 Motivated by Voloshin’s works, we use the term C-coloring for a coloring of this type, and if a hypergraph has to be colored in this way, it will be called a C-hypergraph. The upper chromatic number of a C-hypergraph H, denoted by (H), is the largest possible number of colors in a C-coloring of H. Proper hypergraph coloring is a direct generalization of the fundamental notion of proper graph coloring; research in this direction started in the mid-1960’s. On the other hand, C-coloring in graphs is not really interesting as it simply means that each connected component is monochromatic. For hypergraphs, however, such problems become highly nontrivial; the first such questions arose in the first half of the 1970’s. But it took two decades until Voloshin independently introduced the notion and also created a model far beyond that, as we shall discuss below. A comparison of some basic properties of proper colorings and C-colorings is given in Table 1. It is important to emphasize that every number of colors is possible between min- imum and maximum; indeed, in a proper coloring it is feasible to split any non-singleton color class into two, while in a C-coloring any two color classes may be united. This simple observation will have a relevance later. Table 1: Some coloring properties. proper coloring C-coloring excluded edge type monochromatic multicolored always colorable with |X| colors (= max) 1 color (= min) interesting parameter = min # of colors = max # of colors A general overview on hypergraph colorings — not only these two types — can be found in [13]; and a comprehensive survey on C-coloring is given in [9]. Mixed hypergraphs. A new dimension in the theory of hypergraph coloring was opened in the works of Voloshin [28, 29] where he invented the following complex model. A mixed hypergraph has two types of edges, namely C-edges and D-edges; formally we may write H = (X, C,D). The requirement for a coloring ' : X ! N is that every C-edge E 2 C has to contain two vertices with common color and every D-edge E 2 D has to contain two vertices with distinct colors. In other words, ' should be a proper coloring of (X,D) and a C-coloring of (X, C) at the same time. There is no a priori assumption on the relation between C and D, they may or may not be disjoint. Edges in C \ D are termed bi-edges, and if C = D then H is called a bi- 1By ‘multicolored’ we mean that the colors of the elements are mutually distinct. Such a set is often called a rainbow set in the literature. Zs. Tuza: Mixed hypergraphs and beyond 3 hypergraph. A coloring of a bi-hypergraph — termed bi-coloring — is a proper coloring and a C-coloring at the same time. Contrary to proper colorings and C-colorings, which always exist for every hypergraph, a mixed hypergraph may not admit any coloring; in this case it is called uncolorable. For instance, the bi-hypergraph whose bi-edges are the ten 3-element subsets of a 5-element vertex set, is uncolorable because either at least three colors occur (violating the condition of C-coloring) or some color occurs on at least three vertices (violating proper coloring). If a mixed hypergraph H = (X, C,D) is colorable, its lower chromatic number denoted by (H) is the smallest possible number of colors, and its upper chromatic number (H) is the largest possible number of colors. The feasible set (H) of H is the set of those integers k for which H admits a coloring with precisely k colors. A comprehensive account on the first decade of results and methods concerning mixed hypergraphs is the monograph [30]. Stably bounded hypergraphs. A structure more general than mixed hypergraphs was introduced in two steps, in the papers [5, 7] and [6], and studied further in a series of papers. A stably bounded hypergraph is a hypergraph H = (X, E) for which also four functions s, t, a, b : E ! N are given. The first two of them prescribe lower and upper bounds on the number of colors occurring inside the edges, and the other two prescribe bounds for each edge on the multiplicity of the color occurring most frequently in it. We assume 1  s(E)  t(E)  |E| and 1  a(E)  b(E)  |E| for every E 2 E . A coloring ' is feasible if, for each E 2 E , we have: • ' uses at least s(E) colors inside E, • ' uses at most t(E) colors inside E, • there exists a color which is assigned to least a(E) vertices of E, • no color is assigned to more than b(E) vertices of E. Hence, if E is a C-edge of a mixed hypergraph then its requirements are t(E) = |E| 1 and a(E) = 2; and if it is a D-edge, then s(E) = 2 and b(E) = |E| 1. In fact one of a and t suffices to describe a C-edge, and one of s and b suffices to describe a D-edge. Stating the conditions in other words, the functions s and t restrict the sizes of largest multicolored subsets inside the edges, while a and b restrict the sizes of their largest monochromatic subsets. The lower chromatic number (H), the upper chromatic number (H), and the feasible set (H) are naturally defined in the same way as for mixed hypergraphs. The conditions s(E) = 1, t(E) = |E|, a(E) = 1, b(E) = |E| put no restriction on the coloring of edge E. We obtain functional subclasses of stably bounded hyper- graphs if we prescribe the set of functions which are allowed to be restrictive. For instance, (S,T,A)-hypergraph means that the functions s, t, and a can put restrictions on (some of) the edges, but b must be non-restrictive for all edges. An interesting subclass is that of (S,T)-hypergraphs, termed color-bounded hypergraphs. Earlier examples in the literature may be interpreted as B-hypergaphs [1, 24] and S-hypergaphs [15]. 4 Art Discrete Appl. Math. 1 (2018) #P2.05 Table 2: Coloring restrictions determined by the functions s, t, a, b. function meaning s at least s(E) colors inside E t at most t(E) colors inside E a some color at least a(E) times inside E b each color at most b(E) times inside E Interval hypergraphs. A hypergraph H = (X, E) is called an interval hypergraph if its vertex set X admits an ordering x1, x2, . . . , xn such that every edge E 2 E is a set of consecutive vertices in this order. Interval hypergraphs have many nice properties and admit efficient algorithms for various problems which are intractable on general structures. Problem 1.1. Determine the time complexity of the following problems over the given functional subclasses of stably bounded interval hypergraphs: 1. Colorability of (S,T)-hypergraphs. 2. Lower chromatic number of (S,A)-hypergraphs. 3. Lower chromatic number of (S,T,A)-hypergraphs. 4. Upper chromatic number of (S,T)-hypergraphs. 5. Upper chromatic number of A-hypergraphs. 6. Upper chromatic number of (T,A)-hypergraphs. 7. Upper chromatic number of (T,B)-hypergraphs. 8. Upper chromatic number of (S,T,B)-hypergraphs. Table 3: Solved and unsolved cases — time complexity of basic coloring problems on seven functional subclasses of stably bounded interval hypergraphs; ??? = open, o = obvious, lin = solvable in linear time, NP-c = NP-complete, NP-h = NP-hard. S,T A / T,A S,A / S,T,A T,B / S,T,B exists? ??? o NP-c NP-c min lin o ??? NP-h max ??? ??? NP-h ??? On interval hypergraphs, complexity is known for all the other combinations of the four functions s, t, a, b. Subsets of those results are proved in different papers, the last pieces appearing in [10], where also a detailed summary for several further classes of hypergraphs is given. Note that each of T, A, and (T,A) admits a monochromatic X , whereas each of S, B, and (S,B) admits a multicolored X . Zs. Tuza: Mixed hypergraphs and beyond 5 Table 4: The other functional subclasses, time complexity solved completely on interval hypergraphs. T S / B / S,B A,B / any larger exists? o o NP-c min o lin NP-h max lin o NP-h Gaps. The feasible set (H) of a colorable hypergraph H is called gap-free if it is an interval of integers. If this property does not hold, then we say that H has a gap at k (also called ‘gap in the chromatic spectrum’) if k is an integer such that (H) < k < (H) and k /2 (H). If 1 2 (H), then the feasible set is gap-free. On the other hand, for every finite set W of positive integers with 1 /2 W , in [20] a mixed hypergraph H is constructed such that (H) = W . Since |X| 2 (H) also guarantees that (H) is gap-free, a hypergraph with gaps in (H) necessarily has both C-edges and D-edges. It is interesting to investigate which classes of hypergraphs have members with gaps in the chromatic spectrum, and which are completely gap-free. For instance, a gap-free class is that of interval hypergraphs [21], and the property remains valid also for interval (S,T)-hypergraphs. Also, mixed ‘hypertrees’ — hypergraphs H = (X, C,D) which can be represented over a tree graph such that each hyperedge E 2 C [ D induces a subtree — have a gap-free (H) [23], but this property does not extend for (S,T)-hypertrees [8]. Another famous example is the class of planar mixed hypergraphs, which admit con- structions with (H) = {2, 4}, hence a gap at 3 [22]. Problem 1.2. 1. Can interval (S,A)-hypergraphs have gaps? 2. Can interval (T,B)-hypergraphs have gaps? 3. Can interval (A,B)-hypergraphs have gaps? 4. What gaps can occur in stably bounded planar hypergraphs and in their functional subclasses? The planar case is open also for color-bounded hypergraphs. Chromatic polynomials. Let H = (X, E) be a hypergraph in any of the models above (mixed, stably bounded, etc.), and assume that H is colorable. For running over the natural numbers, it is known that the number of allowed colorings ' : X ! {1, . . . ,} is a polynomial in , more precisely a polynomial of degree (H). It is called the chromatic polynomial of H, denoted by P (H,). For any class H of hypergraphs, one can consider the class {P (H,) | H 2 H} 6 Art Discrete Appl. Math. 1 (2018) #P2.05 of chromatic polynomials. From this point of view, the partial order for functional sub- classes of mixed and stably bounded hypergraphs is determined in [6], as illustrated in Figure 1. Also, the chromatic polynomials of non-1-colorable hypergraphs (i.e., of those containing at least one D-edge) is characterized [7], in terms of Stirling numbers of the second kind. any Y ✓ {S,T,A,B} such that {S,A} ✓ Y or {A,B} ✓ Y . & A / A,T S,T,B / S,T / T,B / M & . & T / C S / S,B # B / D Figure 1: Hierarchy of classes of chromatic polynomials; M = mixed hypergraphs, C = only C-edges, D = only D-edges. Problem 1.3. 1. Characterize those polynomials which are chromatic polynomials of a given type of 1-colorable hypergraphs. 2. Determine the hierarchy analogous to the one exhibited in Figure 1 when the hyper- graphs have a structural property (e.g., interval hypergraphs). How does the hierarchy depend on the structure? The requirement of 1-colorability in Problem 1.3.1 means the restriction to C-hyper- graphs for mixed, T-hypergraphs for color-bounded, and A-hypergraphs or (T,A)-hyper- graphs for those subclasses of stably bounded hypergraphs which are not color-bounded. 2 Graphs There are many problems in graph theory which can be interpreted in terms of colorings of mixed hypergraphs. Here we discuss only one of them. F-WORM colorings. Let F be a fixed graph with at least three vertices. For a graph G = (V,E), a vertex coloring ' is an F -WORM coloring if the vertex set of every subgraph isomorphic to F in G is neither monochromatic nor multicolored. (‘WORM’ abbreviates ‘without rainbow or monochromatic’.) The notion was introduced not much time ago, in [19], which actually appeared later than the second paper [18]. Further early works on the subject are [11] and [12]. Three basic coloring problems, also for F -WORM colorings, are whether a given G is colorable, and if it is, then what is the minimum and maximum number of colors in an F -WORM coloring of G. This similarity to the previous section is no surprise because one can observe that F -WORM coloring of G precisely means a feasible coloring of the bi-hypergraph whose bi-edges are the subsets B ⇢ V such that |B| = |V (F )| and the induced subgraph G[B] contains a subgraph isomorphic to F . Zs. Tuza: Mixed hypergraphs and beyond 7 Many aspects of mixed hypergraphs can be raised for F -WORM colorings as well, and also further questions arise. Here we mention only some of the interesting problems. Problem 2.1. Let F be a connected graph with at least three vertices. 1. Is it NP-complete to decide whether a generic graph G admits an F -WORM color- ing? 2. What is the necessary and sufficient condition for F to ensure that the minimum number of colors in an F -WORM coloring is bounded above by a universal constant for all F -WORM colorable graphs? 3. What is the time complexity of computing the minimum number of colors? 4. What is the complexity of deciding whether the feasible set (set of those numbers k of colors for which a generic input graph G admits an F -free coloring with precisely k colors) is gap-free? 5. Can the F -WORM feasible set contain any large gaps? 6. Study similar problems assuming that G belongs to a particular class of graphs. Partial results are known to these questions, but the case of general F seems to be open. 3 Designs A Steiner system S(t, k, v) is a k-uniform hypergraph with v vertices, such that each t-tuple of vertices is contained in precisely one edge (also called block). Viewing such systems from the direction of mixed hypergraphs, several interesting approaches arise. For in- stance, if each block is considered as a C-edge, we obtain a C-S(t, k, v) system. Another possibility2 is to assume that each block is a bi-edge; then we have a B-S(t, k, v) system. Particular types are the systems B-STS(v), C-STS(v), B-SQS(v), C-SQS(v), derived from Steiner triple and quadrulpe systems (where (t, k) = (2, 3) or (t, k) = (3, 4), respectively), cf. also [27]. Besides, we consider here finite geometries, too. Finite projective planes. It is proved in [3] that if the points of a projective plane of order q are colored in such a way that no line is multicolored, then the number of colors cannot exceed q2 q ⇥(q1/2) as q ! 1; i.e., this function is an upper bound on the upper chromatic number. The bound is tight for an infinite sequence of planes, and it is even proved in [2] that an optimal C-coloring is obtained by making a ‘double blocking set’ (a set that meets every line in at least two points) monochromatic and assigning a distinct color to every point outside this set, provided that the plane is a Desarguesian plane PG(2, q) of sufficiently large order. Problem 3.1. 1. Find a tight general lower bound on the upper chromatic number for every finite projective plane of order q. 2. Find estimates on the upper chromatic number of other types of finite geometries. 3. Study further types of colorings of finite geometries. 2In fact many more possibilities arise when larger block sizes are considered. 8 Art Discrete Appl. Math. 1 (2018) #P2.05 Steiner quadruple systems. It is known that for every fixed t 2 the upper chromatic number of a C-S(t, t+1, v) system is at most ct log v for some constant ct [26]. However, a tight estimate is available only for triple systems, as we shall mention below. For quadruple systems of order v = 2m a repeated application of the ‘doubling construction’ shows that the upper chromatic number can be at least m+1 in general. The method is: start with two vertex-disjoint systems H1 = (X1, E1) and H2 = (X2, E2) of order v, take 1-factorizations (F i1, . . . , F i v1) of the complete graphs whose vertex set is Xi for i = 1, 2; and then the blocks in the system of order 2v are those in H1 [ H2 moreover the 4-tuples of the form e1j [ e 2 k where e 1 j 2 F 1 j and e2k 2 F 2 k , for all combinations (j, k) with 1  j, k  v 1. Problem 3.2. 1. Do there exist uncolorable B-SQS(v) systems? 2. Does every H = C-SQS(2m) have (H)  m+ 1? 3. Does there exist an infinite sequence of B-SQS(v) systems with unbounded upper chromatic number? A complete answer to parts 2 and 3 seems to be unknown even for quadruple systems obtained by the repeated application of the doubling construction, starting from a single 4- element block on four vertices. (Such systems always admit a bi-coloring — their feasible set is {2, 3} when viewed as bi-hypergraphs — hence they are not relevant concerning part 1.) Steiner triple systems. The ‘doubling plus one’ construction builds an STS(2v + 1) from an STS(v). The method is: start with a triple system H = (X, E) of order v, where X = {x1, . . . , xv}; let X 0 be a set of v + 1 further vertices, disjoint from X; take a 1-factorization3 (F1, . . . , Fv) of the complete graphs whose vertex set is X 0; and create the triples of the form xi [ e where e 2 Fi. Together with the edges of H, this yields a Steiner triple system over X [X 0. The coloring requirement on a B-STS system means that each block (triple) has to contain precisely two colors. In a C-STS system, monochromatic blocks may also occur (and the lower chromatic number is 1). In [25], the first paper dealing with C-STS and B-STS (and also with SQS) systems, it is proved that if v < 2m, then the upper chromatic number of every STS(v) is at most m; moreover, = m is attained for exactly those systems which are obtained by a sequence of doubling plus one constructions starting from the trivial system of order 3 with one triple. For such B-STS systems, Mario Gionfriddo raised the following attractive problem in [16]. Conjecture 3.3. If a B-STS(2m 1) system H is obtained from B-STS(3) by a sequence of doubling plus one constructions, then it has (H) = (H) = m. In other words, no bi-coloring of B-STS(2m11) can be extended to the B-STS(2m 1) without increasing the number of colors, i.e., the latter system does not admit any ‘ex- tended bi-coloring’. 3Since v is odd — more precisely v ⌘ 1 _ 3 (mod 6) — we have v + 1 even, therefore the edge set of the complete graph Kv+1 can be decomposed into 1-factors. Zs. Tuza: Mixed hypergraphs and beyond 9 One approach to the conjecture is to assume that a B-STS(2m 1), obtained from a B-STS(2m1 1) by doubling plus one, admits a bi-coloring with m 1 colors, and to investigate what types of size distributions of the color classes might occur. The first necessary conditions of this kind are given in [14]. The recent paper [4] makes further steps in this direction, and also describes a doubling-plus-one sequence constructed explicitly over GF(2), which is proved to not admit any extended bi-coloring. It is important that the construction be started from B-STS(3), because other systems may admit extended bi-colorings [17]. The smallest known example is v = 13! 2v+1 = 27, which has an extended bi-coloring with 3 colors. 4 Conclusion Mixed hypergraph is a great invention. By the combination of two antipodal concepts a new dimension has been opened for coloring theory. The above collection of problems is just an appetizer, lots of interesting further ones remain unsolved, for instance to characterize nice classes of colorable hypergraphs. Moreover, mixed hypergraphs and their generalizations can describe several issues in graph theory as well. WORM coloring considered above is just one example; one can mention areas in Ramsey theory, and more. References [1] N. Ahuja and A. Srivastav, On constrained hypergraph coloring and scheduling, in: K. Jansen, S. Leonardi and V. 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ISSN 2590-9770 The Art of Discrete and Applied Mathematics 1 (2018) #P2.06 https://doi.org/10.26493/2590-9770.1230.f19 (Also available at http://adam-journal.eu) Mario Gionfriddo and mixed hypergraph coloring Vitaly Voloshin Troy University, Troy, AL, USA Received 20 December 2017, accepted 28 January 2018, published online 14 July 2019 Abstract We give a brief description of the explicit and implicit contribution of Mario Gionfriddo to mixed hypergraph coloring. Keywords: Graph and hypergraph coloring, mixed hypergraphs, block designs, Steiner systems. Math. Subj. Class.: 05C15, 05C65 1 A little bit of history It was the summer of 1992 in new independent state, Republic of Moldova, the very fresh ex-USSR country. Living in Kishinev, the capital city, and desperately looking for any con- tacts with western mathematicians, one day I went to the library of the Institute of Math- ematics and Informatics. That library was good because one could find some additional western mathematical journals compared to the university library. It was an absolutely ran- dom event (or was it?): my attention was attracted by the International Mathematical Union Canberra Circular (from Australia!), with the list of mathematical conferences all over the world. At that time the internet was not widely accessible, there was no Google, not even email available. I noticed a very brief, just one paragraph, advertisement that there will be Catania Combinatorial Conference in October 1992, in Italy. And the address of Mario Gionfriddo was simply provided as the contact information. Since I had nothing to lose, I decided to write a postcard (see both sides in Figure 1). I wrote: Dear Professor Mario Gionfriddo: I would be much obliged to you if you could send me invitation/program/information/ Proceedings of the 3rd Catania Combinatorial Conference. I am a specialist in Graph and Hypergraph Theory, Assistant Professor of the Moldova State University, Kishinev. E-mail address: vvoloshin@troy.edu (Vitaly Voloshin) cb This work is licensed under http://creativecommons.org/licenses/by/3.0/ 2 Art Discrete Appl. Math. 1 (2018) #P2.06 I would like to be friend with you. My report may be entitled :“Conditional Colourings on Hypergraphs”. Thank you very much. Sincerely yours V. Voloshin 24 July 1992. Since I was studying English from zero at that time (my foreign language was French), writing in English was a good exercise. If you look at this postcard, you may also recognize an old Soviet postcard with the stamps of Republic of Moldova over it. But it was beyond any imagination, what significant events were implied by this simple postcard. Figure 1: Postcard that changed the world. I was waiting the reply for about two months and, having none, thought that it was one of many cases left without any answer. All of the sudden, one week before the conference, the invitation letter by Mario Gionfriddo arrived. There was no possibility for me to arrange everything (visa etc.) on such short notice (later I learned it required 1–4 months!). So I had to answer that I can’t come. Then Mario asked me for my CV (correspondence by V. Voloshin: Mario Gionfriddo and mixed hypergraph coloring 3 email just started in Moldova); it was mailed to Catania. Since then I had no news for a long time and decided that I have to forget it again. But I will never forget the day of June 26, 1993 when I have received the official invitation by Mario Gionfriddo saying that CNR of Italy has awarded a research grant to me for visiting University of Catania for two months. It was a huge event because it gave me some hope that I can probably survive the hardships of that period. I must confess that there was time when I believed that as mathematician I will not survive. 1992 was the year of war in Moldova, and it felt like we just escaped the Titanic. After arranging multiple problems (like visa, ticket etc.) I arrived to Catania in early October, 1993. I could only recognize Mario at Catania railway station (we never met!) because he was holding the famous book of Claude Berge “Hypergraphs” [3]. It was the best “password” in that moment; and it was the very first application of hypergraph theory in real life for me. When we started discussions about possible research collaboration, I realized that we have very different backgrounds and not that much in common. Mario mostly worked in block designs but I was not familiar with them except occasional mentioning in the book of Berge. However, at that time I already was developing basic concepts of mixed hypergraph coloring. The basic idea of it was to allow edges that can be monochromatic but must not be polychromatic (all vertices = different colors). At the very beginning they were called “anti-edges” because this term exactly reflected the meaning. Mixing classic edges (non monochromatic subsets) with anti-edges (non polychromatic subsets) lead to the concept of mixed hypergraph coloring. It was completely new at that time (even this circumstance became clear much later!). The very first paper [33] was just published but nobody heard about it in Italy. The main paper [34] was in progress and not even published yet. But because of the generality of graph coloring, the ideas for collaboration came naturally. It was due to Mario himself and his colleagues Salvatore Milici, Gaetano Quattrocchi, Angelo Lizzio and others. It was due to the Mario’s seminar, multiple discussions with international visitors like Zsolt Tuza, Alex Rosa, Curt Lindner, Chris Rodger, Carsten Thomassen, Robin Wilson, and graduate students like Lorenzo Milazzo and many oth- ers. Very soon I realized that I got into the very best environment that a mathematician can dream: the international center of active research in graphs, hypergraphs and designs under the leadership of Mario Gionfriddo. The new direction of research aiming at application of mixed hypergraph coloring in coloring of block designs has taken off. It was a matter of not one, not even two years of research collaboration when the very first significant results have been obtained and published. As I recall, the very first fundamental questions which were worth to work on, were these: 1. What is the upper chromatic number of Steiner triple system (abbreviated by STS for short) considered as C-hypergraph? That is, what is the maximum total number of colors when coloring the vertices of each block with at most two colors? At that time, the STS were colored in old classic way: no block was monochromatic. Under this constraint, the problem on maximum number of colors did not exist since the coloring with n (number of points, or vertices in STS) colors was always feasible. 2. If we color each block with precisely two colors, are there uncolorable Steiner triple systems? That is, are there STS which cannot be colored in this way with any number of colors? 4 Art Discrete Appl. Math. 1 (2018) #P2.06 Notice that in classic coloring, all systems were trivially colorable with n colors and therefore this problem never arose. However, if the system is colorable under these new constraints, then such concepts as the minimum and maximum number of colors naturally arise; they are called the lower and upper chromatic numbers and denoted by and ̄ respectively. 3. Is the chromatic spectrum of any STS continuous? That is, whether there exist col- orings using any intermediate number of colors between and ̄. Otherwise there is a gap in chromatic spectrum meaning there is no coloring with a number of colors k such that < k < ̄. When I arrived to Catania for the first time, there were a few preliminary results in the first and second questions regarding some other hypergraph classes like interval mixed hy- pergraphs. But there was no idea, no approach, not even one fact of any mixed hypergraph with the gap in chromatic spectrum. Simultaneously, in 1993, in order to find out if the concept of mixed hypergraph col- oring was new, I submitted the current version of [34] to Paul Erdős with only one this question. The assumption was that Erdős knew everything; and, to my great satisfaction, the answer was yes. In 1993–1996, I received six letters from Paul Erdős, and in the fifth letter he wrote, see Figure 2: 1994 IX 13 Dear Professor Voloshin, Many thanks for your letter, I hope you will have a pleasant and fruitful time in Catania, please give my regards to Professor Gionfriddo. Keep me informed of your further plans. Kind regards Paul Erdős Figure 2: Paul Erdős sends regards to Mario Gionfriddo, 1994. This letter was an additional evidence of how high was the international recognition of Mario Gionfriddo long before I came to Catania. V. Voloshin: Mario Gionfriddo and mixed hypergraph coloring 5 2 Mathematical results obtained in Catania We use the terminology from [4]. Let V = {v1, v2, . . . , vn} be a finite set of elements called vertices, and let E = {E1, E2, . . . , Em} be a family of subsets of V called edges or hyperedges. The pair H = (V, E) is called a hypergraph with vertex set V = V (H) and edge-set E = E(H). The hypergraph H = (V, E) is sometimes called a set system. If each edge of a hypergraph contains precisely two vertices, then it is a graph. If every edge of H is of size r, then H is called an r-uniform hypergraph; evidently, a simple graph is a 2-uniform hypergraph. Let {1, 2, . . . ,} be a set of colors. A proper -coloring of a hypergraph H = (V, E) is a mapping c : V ! {1, 2, . . . ,} for which every edge E 2 E has at least two vertices of different colors. The number of proper -colorings of H is a polynomial in ; it is denoted by P (H,) and is called the chromatic polynomial. The minimum value of for which there exists a proper -coloring of a hypergraph H is called the chromatic number of H, denoted by (H). A hypergraph H is k-colorable if (H)  k. The concept of a mixed hypergraph coloring was introduced in [33]. Instead of H = (V, E), the basic idea was to consider a structure H = (V, C,D), termed a mixed hyper- graph, with two families of subsets called C-edges and D-edges. By definition, a proper -coloring of a mixed hypergraph H = (V, C,D) is a mapping c : V ! {1, 2, . . . ,} for which two conditions hold: • every C 2 C has at least two vertices of a Common color; • every D 2 D has at least two vertices of Different colors. A mixed hypergraph H is called colorable if it admits at least one proper coloring; and it is uncolorable if no such colorings exist. The chromatic spectrum is the vector (r1, r2, . . . , rn), where each rk is the number of partitions of the vertex set induced by proper colorings using precisely k colors. A gap in the chromatic spectrum is an integer k for which (H) < k < ̄(H) and rk = 0. A mixed hypergraph H = (V, C,D) is called a bihypergraph if the families of C-edges and D-edges coincide, i.e., C = D. A Steiner system S(t, k, v) is a k-uniform hypergraph of order v, for which each t- tuple of vertices is contained in precisely one edge. To mention some examples, a system S(2, 3, v) is a Steiner triple system STS(v), an S(3, 4, v) is a Steiner quadruple system SQS(v), and an S(2, q + 1, q2 + q + 1) is a finite projective plane of order q. The edges are called blocks. We may view each block as a C-edge (when an STS(v) is denoted by CSTS(v)) or as a bi-edge – that is, a C-edge and a D-edge at the same time (when an STS(v) is denoted by BSTS(v)). The notations CS(t, k, v), BS(t, k, v), CSQS(v) and BSQS(v) are derived for the respective systems in a similar way. The study of the upper chromatic number in Steiner triple systems started in Catania in 1993 and resulted in a series of publications, see [26, 27, 28, 29, 31, 30, 32]. For example, in [30] the authors proved that ̄(BSTS(v))  ̄(CSTS(v))  k, for all v  2k 1. This upper bound on ̄ is tight for all k 2, and the systems attaining equality were also characterized. In particular, in them the cardinalities of color classes are powers of 2. The first PhD Thesis in this field (and in mixed hypergraph coloring in 6 Art Discrete Appl. Math. 1 (2018) #P2.06 general) was defended under supervision of Mario Gionfriddo by Lorenzo Milazzo in 1997, see [28]. A coloring of a Steiner triple system STS(n) in a way that every block receives pre- cisely two colors is also called a bicoloring [7]. All bicolorable STS(2h 1)s have upper chromatic number ̄  h. If ̄ = h < 10, then lower and upper chromatic numbers co- incide, i.e., = ̄ = h. In 2003, Mario raised a subtle and challenging conjecture [13] that this equality holds whenever ̄ = h 2. Until today it remains open, intriguing and motivating for further research. Some of the most recent results in this direction discuss extensions of bicolorings of STS(v) to bicolorings of STS(2v + 1) obtained by using the so called doubling plus one construction, see [5]. The problem of colorability of BSTS was also first formulated in Catania in 1993 though no example of uncolorable BSTS was found. The first such example was con- structed by Ganter at TU Dresden, and all uncolorable BSTS(15)s have been characterized by Rosa [35]: BSTS(15) is colorable if and only if it contains BSTS(7) as a subsystem. Out of the 80 non-isomorphic BSTS(15)s, only 23 meet this criterion and are therefore colorable. The other 57 are uncolorable. It follows that uncolorable BSTS(n)s exist for each admissible n 15. As to BSQS, the situation is much more difficult. Even though the problem to find at least one uncolorable BSQS was formulated first in Catania in 1993 as well, no one such system has been found. The conjecture is that they exist. The best result related to this problem is by Lo Faro, Milazzo and Tripodi [22]: all BSQS(n) are colorable for all admissible n  16. Therefore, the smallest admissible n for which uncolorable BSTS(n) may exist is n = 20. There is a significant series of important results and publications by the whole Catania group of mathematicians like these [2, 6, 13, 14, 15, 16, 17, 20, 22, 23, 24, 25, 26, 27, 28, 29, 31, 30, 32] just to name a few. Here is the right point to mention that Catania group is very closely related to Messina group of mathematicians who work in the same direction: Giovanni Lo Faro, Enzo Li Marzi, Corinna Marino and Antoinette Tripodi. Because of this connection, as one can see, they also have published many papers generally speaking in mixed hypergraph coloring. It was October of 1998. After half year stay in USA, I arrived to Catania where I met Zsolt Tuza as usual, again. We were working on some problems when I received an email from Dhruv Mubayi, a graduate student of Doug West, University of Illinois at Urbana-Champaign. In that email, Dhruv communicated that while looking at a completely different problem, he found an example of mixed hypergraph on 16 vertices, 36 D-edges and 144 (!) C-edges with the gap in the chromatic spectrum. It was a shocking discovery, literally a breakthrough! As I recall, when we realized it, we immediately started searching for the smallest example, and it took just one night for Zsolt to come up next morning with the example depicted in Figure 3. It had 6 vertices {1, 2, 3, 4, 5, 6}, 2 D-edges {1, 6} and {2, 3, 4, 5}) (solid line and curve), 3 C-edges {2, 3, 4}, {3, 4, 5} and {2, 4, 5} (dashed curves), and 4 bi-edges {1, 2, 3}, {1, 4, 5}, {6, 2, 3}, and {6, 4, 5} (bold curves). One can easily see that if vertices 2 and 3 are colored with the same color, say A, then this coloring extends in a unique way to vertices 4 and 5 with color B, vertex 1 with color C and vertex 6 with color D. If, on the contrary, vertices 2 and 3 are colored differently, say colors A and B, then this coloring extends to vertex 1 with color A, vertex 4 with color A, and vertices 5, 6 with color B. Actually, there are four distinct extensions of this coloring, all with colors A and B. So, there is no coloring using 3 colors, and the chromatic spectrum of this example V. Voloshin: Mario Gionfriddo and mixed hypergraph coloring 7 is R(H) = (0, 4, 0, 1, 0, 0). The chromatic polynomial P (H,) = (1)(25+10). Figure 3: The very first smallest example with gap, Catania, 1998. The first results have been published in [21]. It was the very beginning of the chase for the gaps in the chromatic spectrum in mixed hypergraphs which with variable success continues until today. Around the year of 2000, Catania group was reinforced by new researcher, Lucia Gion- friddo. After defending PhD Thesis, she got interested in problems related to mixed hy- pergraph coloring, namely the gaps in the chromatic spectrum. For an impressively short period of time, Lucia has discovered the first designs, namely P3-designs with the gaps in the chromatic spectrum. The idea was to consider decompositions of complete graphs into P3 (a path on 3 vertices) and declare every block as a bi-edge, i.e., colorable with precisely two colors. It is a special case of bi-hypergraph. She proved that there are many such struc- tures with many gaps, in particular, big gaps, even gaps, odd gaps, etc., see [8, 9, 10, 11]. Later other designs, namely P4-designs with the gaps have been found when considering equicolorings in [1]. Surprisingly, until today, these are the only examples of block designs with the gaps. We do not know anything about continuity of the chromatic spectrum of BSTS or BSQS, for example. Based on Lucia’s results, while my stay in Catania, we were able to carry out some computational experiments and construct the smallest 3-uniform bi-hypergraph with the gap in the chromatic spectrum, see Figure 4: it contains 7 vertices and 9 bi-edges and its chromatic spectrum is R(H) = (0, 12, 0, 3, 0, 0, 0), see [12]. The chromatic polynomial of this bi-hypergraph P (H,) = 3(1)(25+10). It is interesting that the chromatic spectra and chromatic polynomials of hypergraphs in Figure 3 and in Figure 4 are related (compare). They were found independently. There is one more result that is worth to mention. It is about the upper chromatic index of a multi-graph, which represents a type of anti-Vizing theorem. It was first formulated in [34] as Problem 13 and was implied by the duality of mixed hypergraphs. Consider the colorings of the edges of a multi-graph in such a way that every non-pendant vertex is incident to at least two edges of the same color. The maximum number of colors that can be used in such colorings is the upper chromatic index of a multi-graph G, denoted 8 Art Discrete Appl. Math. 1 (2018) #P2.06 Figure 4: The smallest 3-uniform example with gap, Catania, 2002. Figure 5: Mixed Hypergraph Coloring encoded in BSTS(7). V. Voloshin: Mario Gionfriddo and mixed hypergraph coloring 9 by ̄0(G). The exact value of it was found in [18]. It was proved that if a multi-graph G has n vertices, m edges, p pendant vertices and maximum number c of disjoint cycles, then ̄0(G) = c+m n+ p. This result was reported by Lorenzo Milazzo at the Second Lethbridge Workshop on Designs, Codes, Cryptography and Graph Theory, July 9 – 14, 2001. One of the basic results in applications of mixed hypergraph coloring to block designs, namely, about the cardinality of color classes being powers of 2 in the optimal coloring of BSTS(7), was encoded in the picture of it on the book cover of [19]: one vertex in yellow color, two vertices in red color and four vertices in blue, see Figure 5. In contrast to classic drawing, it is depicted as a hypergraph. Every block is colored with two colors and any other proper coloring has the same distribution of color classes. 3 Conclusion In conclusion, what I witnessed through decades, was a multilateral activity by Mario Gion- friddo which can be summarized in this way (I do not pretend to be complete): Mario Gionfriddo: 1. Created an outstanding scientific school of researchers in Graphs, Hypergraphs and Designs with many publications in top journals all over the world. 2. Turned University of Catania and University of Messina into major centers of in- ternational collaboration. Proved that he is a great teacher, educator, researcher, organizer, and in general, a great leader in contemporary mathematics. 3. Played and still plays an outstanding role in Italian and especially Sicilian Discrete Mathematics, namely, the Theory of Graphs, Hypergraphs and Designs. 4. Played an outstanding explicit and implicit role in developing the theory of Mixed Hypergraph Coloring. Explicit: by personal participation in research and actually proving many theorems. Implicit: by inviting researchers and organizing seminars, workshops and conferences where actual collaboration occurred. Dear Mario, I congratulate you on the occasion of 70th anniversary, thank you for your great role in my life and wish you a good health and further achievements in developing Graphs, Hypergraphs and Designs! Remark. Dear reader! If it were not that postcard, randomly mailed in 1992 (Figure 1), at this very same moment you would read a very different paper. 10 Art Discrete Appl. Math. 1 (2018) #P2.06 Figure 6: With Mario Gionfriddo, Messina, 2003. References [1] A. Amato, M. Gionfriddo and L. Milazzo, 2-regular equicolourings for P4-designs, Discrete Math. 312 (2012), 2252–2261, doi:10.1016/j.disc.2012.03.030. [2] A. Amato, M. Gionfriddo and L. Milazzo, A survey of Lucia Gionfriddo’s results about colour- ings in BP3-designs, in: M. Buratti, C. Lindner, F. Mazzocca and N. Melone (eds.), Recent Results in Designs and Graphs: A Tribute to Lucia Gionfriddo, Aracne, Rome, volume 28 of Quaderni di Matematica, pp. 39–60, 2012. [3] C. Berge, Hypergraphs: Combinatorics of Finite Sets, volume 45 of North-Holland Mathemat- ical Library, North-Holland, Amsterdam, 1989. [4] Cs. Bujtás, Zs. Tuza and V. Voloshin, Hypergraph colouring, in: L. W. Beineke and R. J. Wilson (eds.), Topics in Chromatic Graph Theory, Cambridge Univiversity Press, Cambridge, volume 156 of Encyclopedia of Mathematics and its Applications, pp. 230–254, 2015, doi: 10.1017/cbo9781139519793.014. [5] Cs. Bujtás, M. Gionfriddo, E. Guardo, L. Milazzo, Zs. Tuza and V. Voloshin, Extended bicol- orings of Steiner triple systems of order 2h 1, Taiwanese J. Math. 21 (2017), 1265–1276, doi:10.11650/tjm/8042. [6] M. Buratti, M. Gionfriddo, L. Milazzo and V. Voloshin, Lower and upper chromatic numbers for BSTSs(2h 1), Comput. Sci. J. Moldova 9 (2001), 259–272, http://www.math.md/ publications/csjm/issues/v9-n2/8294/. [7] C. J. Colbourn, J. H. Dinitz and A. Rosa, Bicoloring Steiner triple systems, Electron. J. Combin. 6 (1999), #R25 (16 pages), https://www.combinatorics.org/ojs/index.php/ eljc/article/view/v6i1r25. [8] L. Gionfriddo, Extremal gaps in BP3-designs, Comput. Sci. J. Moldova 9 (2001), 305–320, http://www.math.md/publications/csjm/issues/v9-n3/8302/. V. Voloshin: Mario Gionfriddo and mixed hypergraph coloring 11 [9] L. Gionfriddo, P3-designs with gaps in the chromatic spectrum, Rend. Sem. Mat. Messina Ser. II 8(24) (2001/02), suppl., 49–58. [10] L. 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ISSN 2590-9770 The Art of Discrete and Applied Mathematics 1 (2018) #P2.07 https://doi.org/10.26493/2590-9770.1219.034 (Also available at http://adam-journal.eu) An alternate description of a (q + 1, 8)-cage Marièn Abreu ⇤ Dipartimento di Matematica Informatica ed Economia, Università degli Studi della Basilicata, Viale dell’Ateneo Lucano 10, I-85100 Potenza, Italy Gabriela Araujo-Pardo † Instituto de Matemáticas, Universidad Nacional Autónoma de México, México D. F., México Camino Balbuena Departament de Matemática Aplicada III, Universitat Politècnica de Catalunya, Campus Nord, Edifici C2, C/ Jordi Girona 1 i 3 E-08034, Barcelona, Spain Domenico Labbate ‡ Dipartimento di Matematica Informatica ed Economia, Università degli Studi della Basilicata, Viale dell’Ateneo Lucano 10, I-85100 Potenza, Italy Received 15 November 2017, accepted 4 June 2018, published online 23 July 2019 Abstract Let q 2 be a prime power. In this note we present an alternate description of the known (q + 1, 8)-cages which has allowed us to construct small (k, g)-graphs for k = q 1, q and g = 7, 8 in other papers on this same topic. Keywords: Cages, girth, Moore graphs, perfect dominating sets. Math. Subj. Class.: 05C35, 05C69, 05B25 ⇤This research was carried out within the activity of INdAM-GNSAGA and supported by the Italian Ministry MIUR. Research supported by PAPIIT-México under Project IN107218. †Research supported by CONACyT-México under Project 282280 and PAPIIT-México under Projects IN106318 and IN107218. ‡Corresponding author. This research was carried out within the activity of INdAM-GNSAGA and supported by the Italian Ministry MIUR. Research supported by PAPIIT-México under Project IN107218. E-mail addresses: marien.abreu@unibas.it (Marièn Abreu), garaujo@matem.unam.mx (Gabriela Araujo-Pardo), m.camino.balbuena@upc.edu (Camino Balbuena), domenico.labbate@unibas.it (Domenico Labbate) cb This work is licensed under http://creativecommons.org/licenses/by/3.0/ 2 Art Discrete Appl. Math. 1 (2018) #P2.07 1 Introduction Throughout this paper, only undirected simple graphs without loops or multiple edges are considered. Unless otherwise stated, we follow the book by Bondy and Murty [14] for terminology and notation. Let G be a graph with vertex set V = V (G) and edge set E = E(G). The girth of G is the number g = g(G) of edges in a shortest cycle. For every v 2 V , NG(v) denotes the neighbourhood of v, i.e. the set of all vertices adjacent to v, and NG[v] = NG(v) [ {v} is the closed neighbourhood of v. The degree of a vertex v 2 V is the cardinality of NG(v). Let S ⇢ V (G), then we denote by NG(S) = [s2SNG(s)S and by NG[S] = S[NG(S). A graph is called regular if all its vertices have the same degree. A (k, g)-graph is a k-regular graph with girth g. Erdős and Sachs [15] proved the existence of (k, g)-graphs for all values of k and g provided that k 2. Since then most work carried out has focused on constructing a smallest (k, g)-graph (cf. e.g. [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 16, 18, 20, 21, 22]). A (k, g)-cage is a k-regular graph with girth g having the smallest possible number of vertices. Cages have been intensely studied since they were introduced by Tutte [25] in 1947. More details about constructions of cages can be found in the recent survey by Exoo and Jajcay [17]. In this note we are interested in (k, 8)-cages. Counting the number of vertices in the distance partition with respect to an edge yields the following lower bound on the order of a (k, 8)-cage: n0(k, 8) = 2(1 + (k 1) + (k 1)2 + (k 1)3). (1.1) A (k, 8)-cage with n0(k, 8) vertices is called a Moore (k, 8)-graph (cf. [14]). These graphs have been constructed as the incidence graphs of generalized quadrangles Q(4, q) and W (q) [12, 17, 24], which are known to exist for q a prime power and k = q + 1 and no example is known when k 1 is not a prime power (cf. [11, 13, 19, 27]). Since they are incidence graphs, these cages are bipartite and have diameter 4. Recall also that if q is even, Q(4, q) is isomorphic to the dual of W (q) and viceversa. Hence, the corresponding (q + 1, 8)-cages are isomorphic. In this note we present an alternate description of the known (q+1, 8)-cages with q 2 a prime power as follows: Definition 1.1. Let Fq be a finite field with q 2 a prime power and % be a symbol not belonging to Fq . Let q = q[W0,W1] denote a bipartite graph with vertex sets Wi = F3q [ {(%, b, c)i, (%, %, c)i : b, c 2 Fq} [ {(%, %, %)i}, i = 0, 1, and edge set defined as follows: For all a, b, c 2 Fq Nq ((a, b, c)1) = {(w, aw + b, a2w + 2ab+ c)0 : w 2 Fq} [ {(%, a, c)0}; Nq ((%, b, c)1) = {(c, b, w)0 : w 2 Fq} [ {(%, %, c)0}; Nq ((%, %, c)1) = {(%, c, w)0 : w 2 Fq} [ {(%, %, %)0}; Nq ((%, %, %)1) = {(%, %, w)0 : w 2 Fq} [ {(%, %, %)0}. M. Abreu et al.: An alternate description of a (q + 1, 8)-cage 3 Or equivalently, For all i, j, k 2 Fq Nq ((i, j, k)0) = {(w, j wi, w2i 2wj + k)1 : w 2 Fq} [ {(%, j, i)1} Nq ((%, j, k)0) = {(j, w, k)1 : w 2 Fq} [ {(%, %, j)1} Nq ((%, %, k)0) = {(%, w, k)1 : w 2 Fq} [ {(%, %, %)1}; Nq ((%, %, %)0) = {(%, %, w)1 : w 2 Fq} [ {(%, %, %)1}. Note that % is just a symbol not belonging to Fq and no arithmetical operation will be performed with it. Theorem 1.2. The graph q given in Definition 1.1 is a Moore (q + 1, 8)-graph for each prime power q 2. The proof of the above theorem shows that the graph q described in Definition 1.1 is in fact a labelling for a (q + 1, 8)-cage, for each prime power q 2. We need to settle this alternate description because it is used in [2, 3, 4] to construct small (k, g)-graphs for k = q 1, q and g = 7, 8. 2 Proof of Theorem 1.2 2.1 Preliminaries: the graphs Hq and Bq In order to prove Theorem 1.2 we will first define two q-regular bipartite graphs Hq and Bq (cf. Definitions 2.1 and 2.4). The graph Hq was also introduced by Lazebnik, Ustimenko and Woldar [20] with a different formulation. Definition 2.1. Let Fq be a finite field with q 2. Let Hq = Hq[U0, U1] be a bipartite graph with vertex set Ur = F3q , r = 0, 1, and edge set E(Hq) defined as follows: For all a, b, c 2 Fq NHq ((a, b, c)1) = {(w, aw + b, a2w + c)0 : w 2 Fq}. Note that throughout the proofs equalities and operations are intended in Fq . Lemma 2.2. Let Hq be the graph from Definition 2.1. For any given a 2 Fq , the vertices in the set {(a, b, c)1 : b, c 2 Fq} are mutually at distance at least four. And, for any given i 2 Fq , the vertices in the set {(i, j, k)0 : j, k 2 Fq} are mutually at distance at least four. Proof. Suppose that there exists a path of length two between distinct vertices of the form (a, b, c)1 (w, j, k)0 (a, b0, c0)1 in Hq . By Definition 2.1, j = aw + b = aw + b0 and k = a 2 w+ c = a2w+ c0. Combining the equations we get b = b0 and c = c0 which implies that (a, b, c)1 = (a, b0, c0)1 contradicting the assumption that the path has length two. Similarly suppose that there exists a path of length two (i, j, k)0 (a, b, c)1 (i, j0, k0)0. Reasoning as before, we obtain j = ai + b = j0, and k = a2i + c = k0 yielding (i, j, k)0 = (i, j0, k0)0 which is a contradiction. Proposition 2.3. The graph Hq from Definition 2.1 is q-regular, bipartite, of girth 8 and order 2q3. 4 Art Discrete Appl. Math. 1 (2018) #P2.07 Proof. For q = 2 it can be checked that H2 consists of two disjoint cycles of length 8. Thus we assume that q 3. Clearly Hq has order 2q3 and every vertex of U1 has degree q. Let (x, y, z)0 2 U0. By definition of Hq , NHq ((x, y, z)0) = (a, y ax, z a2x)1 : a 2 Fq . (2.1) Hence every vertex of U0 has also degree q and Hq is q-regular. Next, let us prove that Hq has no cycles of length smaller than 8. Otherwise suppose that there exists in Hq a cycle C2t+2 = (a0, b0, c0)1 (x0, y0, z0)0 (a1, b1, c1)1 · · · (xt, yt, zt)0 (a0, b0, c0)1 of length 2t + 2 with t 2 {1, 2}. By Lemma 2.2, ak 6= ak+1 and xk 6= xk+1 (subscripts being taken modulo t+ 1). Then yk = akxk + bk = ak+1xk + bk+1, k = 0, . . . , t, zk = a 2 k xk + ck = a 2 k+1xk + ck+1, k = 0, . . . , t, subscripts k being taken modulo t+ 1. Summing all these equalities we get t1X k=0 (ak ak+1)xk = (a0 at)xt, t = 1, 2; t1X k=0 (a2 k a2 k+1)xk = (a 2 0 a2t )xt, t = 1, 2. (2.2) If t = 1, then (2.2) leads to (a0 a1)(x1 x0) = 0. System (2.2) gives x0 = x1 = x2 which is a contradiction to Lemma 2.2. This means that Hq has no squares so that we may assume that t = 2. The coefficient matrix of (2.2) has a Vandermonde determinant, i.e. a1 a0 a0 a2 a 2 1 a20 a20 a22 = 1 1 1 a1 a0 a2 a 2 1 a 2 0 a 2 2 = Y 0k 1. It is known that: Theorem 1.1. A P3-design of order v exists if and only if v ⌘ 0 or 1 (mod 3), v 4. Observe that if a path P3 has vertices x, y, z and edges {x, y}, {y, z}, we will denote it by [x, y, z]. 2 Transversals and blocking sets in G-designs The following results were proved in [4, 8]: Theorem 2.1. If ⌃ = (X,B) is a G-design of order v and T is a blocking set of cardinality p of ⌃ such that p  v1r , then: ✓ p 2 ◆ + p ·  v 1 r (p 1) |B|. Theorem 2.2. If ⌃ = (X,B) is a P3-design of order v and T is a transversal of cardinality p of ⌃, then: ✓ p 2 ◆ + p · (v p) v(v 1)/4. where the inequality is the best possible. Proof. To see that the inequality is the best possible, consider the system ⌃ = (X,B), defined in X = {1, 2, . . . , 8}, and having for blocks: B : [1, 2, 3], [1, 3, 4], [1, 4, 2], [5, 6, 7], [5, 7, 8], [5, 8, 6], [1, 5, 3], [2, 5, 4], [1, 6, 3], [2, 6, 4], [1, 7, 3], [2, 7, 4], [1, 8, 3], [2, 8, 4]. We can see that ⌃ is a P3-design of order v = 8 and that T = {1, 2, 5} is a blocking set of ⌃. Further, we can verify that, from Theorem 2.2, the minimum possible value of p for v = 8 is p = 3. Observe that the minimum cardinality of a blocking set depends on the system and not only on its order. The following two systems ⌃1 = (X,B1) and ⌃2 = (X,B2), are defined both in X = {1, 2, . . . , 9}. Therefore their order is v = 9. However, the minimum cardinality of a blocking set in them is different: B1 : [1, 2, 4], [1, 3, 4], [2, 3, 5], [1, 4, 7], [1, 5, 4], [1, 6, 4], [1, 7, 8], [1, 8, 4], [1, 9, 4], [2, 5, 8], [2, 6, 5], [2, 7, 5], [2, 8, 9], [2, 9, 5], [3, 6, 9], [3, 7, 6], [3, 8, 6], [3, 9, 7]. P. Bonacini et al.: Perfect blocking sets in P3-designs 3 B2 : [1, 2, 3], [1, 3, 4], [1, 4, 2], [5, 6, 7], [5, 7, 8], [5, 8, 6], [1, 5, 3], [2, 5, 4], [1, 6, 3], [2, 6, 4], [1, 7, 3], [2, 7, 4], [1, 8, 3], [2, 8, 4], [1, 9, 7], [2, 9, 6], [3, 5, 9], [4, 9, 8]. We can see that: • the minimum possible cardinality in ⌃1 is exactly p = 3 and that T1 = {1, 2, 3} is a blocking set of ⌃2; • the minimum possible cardinality in ⌃2 is p = 4 and that T2 = {1, 2, 5, 9} is a blocking set of ⌃1. Definition 2.3. Let ⌃ = (X,B) be a G-design. We say that a blocking set T of ⌃ is perfect if there exists a constant C 2 N such that in every block B 2 B there are exactly C edges having an extreme in T and the other extreme in CXT . Observe that the blocking set T1 of the P3-design ⌃1, defined above, is perfect; while the blocking set T2 of ⌃2 is not perfect. 3 Perfect blocking sets in P3-designs We see the exact cardinality of any perfect blocking set in in P3-designs. Theorem 3.1. If T is a perfect blocking set of any P3-design of order v, then |T | = v ± p v 2 , and v must be a square. Proof. Let ⌃ = (X,B) be a P3-design of order v and let T be a perfect blocking set of ⌃. Observe that, from the definition of perfect blocking set, every block of ⌃ contains exactly one edge having an extreme in T and the other extreme in CXT . Therefore, since |T | = p and |CXT | = v p, it follows: p(v p) = v(v 1) 4 , hence: p = v ± p v 2 . Since p is a positive integer, it follows that v must be a square. From Theorem 3.2, if we consider a P3-design of order v having perfect blocking set, since v ⌘ 0 or 1 (mod 4), then there is a positive integer k such that v = (2k)2 or v = (2k + 1)2. 4 Art Discrete Appl. Math. 1 (2018) #P2.08 Theorem 3.2. Let T be a perfect blocking set of any P3-design of order v. If |T |  |CXT |, then: (i) if v ⌘ 0 (mod 4), then v = (2k)2 and |T | = k(2k 1), for any k 2 N ; (ii) if v ⌘ 1 (mod 4), then v = (2k + 1)2 and |T | = k(2k + 1), for any k 2 N . Proof. Let ⌃ = (X,B) be a P3-design of order v and let T be a perfect blocking set of ⌃, such that |T |  |CXT |. (i): If v ⌘ 0 (mod 4), then there exists k 2 N such that v = (2k)2. Further: |T | = v p v 2 = 4k2 2k 2 = k(2k 1). (ii): If v ⌘ 1 (mod 4), then there exists k 2 N such that v = (2k + 1)2. Further: |T | = v p v 2 = (4k2 + 4k + 1) (2k + 1) 2 = k(2k + 1). 4 Main results In this section we determine the spectrum of P3-designs having perfect blocking sets. Theorem 4.1. There exist P3-designs of order v having perfect blocking sets if and only if v is a square. Proof. Let ⌃ = (X,B) be a P3-design of order v and let T be a perfect blocking set of ⌃. From Theorems 3.1 and 3.2, it follows that v must be a square. Therefore, let v be an odd [resp. even] square number. This implies that v = (2k+ 1)2 and p = |T | = k(2k + 1) [resp. v = (2k)2 and p = |T | = k(2k 1)]. If X is a set of cardinality v, partition X into 3 classes X1, X2, X3, defined as follows: X1 = {a1, a2, . . . , ak(2k+1)} [resp.: X1 = {a1, a2, . . . , ak(2k1)}]; X2 = {b1, b2, . . . , bk(2k+1)} [resp.: X2 = {b1, b2, . . . , bk(2k1)}]; X3 = {c1, c2, . . . , c2k+1} [resp.: X3 = {c1, c2, . . . , c2k}]. To simplify, indicate by q the cardinality of X3, i.e. q = 2k + 1 [resp. q = 2k], and of course p = k(2k + 1) [resp. p = k(2k 1)]. Define in X the following families of paths P3: F : [a1, a2, b1], [a1, a3, b1], . . . , [a1, ap, b1], [a2, a3, b2], [a2, a4, b2], . . . , [a2, ap, b2], ... [ap2, ap1, bp2], [ap2, ap, bp2], [ap1, ap, bp1]; P. Bonacini et al.: Perfect blocking sets in P3-designs 5 G1,2 : [a1, b1, c1], [a2, b2, c1], . . . , [aq1, bq1, c1], [aq, bq, c2], [aq+1, bq+1, c2], . . . , [a2q3, b2q3, c2], (the last index 2q 3 is because of 2q 3 = (q 1) + (q 2)), [a2q2, b2q2, c3], [a2q1, b2q1, c3], . . . , [a3q6, b3q6, c3], (the last index 3q 6 is because of 2q 3 = (q 1) + (q 2) + (q 3)), ... [ap2, bp2, cq2], [ap1, bp1, cq2], [ap, bp, cq1]; G2,2 : [a1, c1, c2], [a2, c1, c3], [a3, c1, c4], . . . , [aq1, c1, cq], [aq, c2, c3], [aq+1, c2, c4], [aq+2, c2, c5], . . . , [a2q3, c2, c2q3], ... [ap2, cq2, cq1], [ap1, cq2, cq], [ap, cq1, cq]; H1 : [a1, b2, b1], [a1, b3, b1], . . . , [a1, bp, b1], [a2, b3, b2], [a2, b4, b2], . . . , [a2, bp, b2], [a3, b4, b3], [a3, b5, b3], . . . , [a3, bp, b3], ... [aq1, bq, bq1], [aq1, bq+1, bq1], . . . , [aq1, bp, bq1], [aq, bq+1, bq], [aq, bq+2, bq], . . . , [aq, bp, bq], [aq+1, bq+2, bq+1], [aq+1, bq+3, bq+1], . . . , [aq+1, bp, bq+1], ... [a2q3, b2q2, b2q3], [a2q3, b2q1, b2q3], . . . , [a2q3, bp, b2q3], ... [ap2, bp1, bp2], [ap2, bp, bp2], [ap1, bp, bp1]; H2 : [a1, c2, b1], [a1, c3, b1], . . . , [a1, cq, b1], [a2, c2, b2], [a2, c3, b2], . . . , [a2, cq, b2], [a3, c2, b3], [a3, c3, b3], . . . , [a3, cq, b3], ... [aq1, c2, bq1], [aq1, c3, bq1], . . . , [aq1, cq, bq1], [aq, c1, bq], [aq, c3, bq], . . . , [aq, cq, bq], 6 Art Discrete Appl. Math. 1 (2018) #P2.08 [aq+1, c1, bq+1], [aq+1, c3, bq+1], . . . , [aq+1, cq, bq+1], ... [a2q3, c1, b2q3], [a2q3, c3, b2q3], . . . , [a2q3, cq, b2q3], ... [ap2, c1, bp2], [ap2, c2, bp2], [ap2, c3, bp2], . . . , [ap2, cq3, bp2], [ap2, cq1, bp2], [ap2, cq, bp2], [ap1, c1, bp1], [ap1, c2, bp1], [ap1, c3, bp1], . . . , [ap1, cq3, bp1], [ap1, cq1, bp1], [ap1, cq, bp1], [ap, c1, bp], [ap, c2, bp], [ap, c3, bp], . . . , [ap, cq3, bp], [ap, cq2, bp], [ap, cq, bp]. If X = X1[X2[X3 and B = F [G1[G2[H1[H2, then it is possible to verify that ⌃ = (X,B) is a P3-design of order v = (2k + 1)2 [resp. v = (2k)2], for any k 2 N , and that X1 having is a perfect blocking set of ⌃ of cardinality k(2k+1) [resp. v = k(2k1)]. Indeed, observe that: 1. the family F has cardinality |F| = p 2 and its blocks contain exactly an edge having both extremes in X1; no block of B F contains two elements of X1; further they contain all the edges {aj , bi}, for every i = 1, 2, . . . , p1 and j = i+1, i+2, . . . , p; 2. the family G1 contains all the blocks of type [ai, bi, cj ], where: • j = 1, for i = 1, 2, . . . , q 1; • j = 2, for i = q, q + 1, . . . , 2q 3; ... • j = q 1, for i = p = q 2 = k(2k + 1) [resp. = k(2k 1)]; 3. the family G2 contains blocks of type {a, c0, c00}, where {a, c0} 2 X1 ⇥ X3 and {c0, c00} 2 X3 ⇥X3; 4. the family H1 contains all the blocks of type [ai, bj , bi], for every i = 1, 2, . . . , p 1 and j = i+ 1, . . . , p; 5. the family H2 contains all the blocks of type [ai, cj , bi], for every i = 1, 2, . . . , p and j = 1, . . . , q, with exception for: • j = 1, for i = 1, 2, . . . , q 1; • j = 2, for i = q, q + 1, . . . , 2q 3; ... • j = q 1, for i = p = q. References [1] Y. Chang, G. Lo Faro and A. Tripodi, Tight blocking sets in some maximum packings of Kn, Discrete Math. 308 (2008), 427–438, doi:10.1016/j.disc.2006.11.060. P. Bonacini et al.: Perfect blocking sets in P3-designs 7 [2] Y. Chang, J. Zhou, G. Lo Faro and A. Tripodi, On tight blocking set in minimum coverings, Electron. Notes Discrete Math. 40 (2013), 365–370, doi:10.1016/j.endm.2013.05.064. [3] F. Franek, T. S. Griggs, C. C. Lindner and A. Rosa, Completing the spectrum of 2-chromatic S(2, 4, v), Discrete Math. 247 (2002), 225–228, doi:10.1016/s0012-365x(01)00308-9. [4] M. Gionfriddo, Blocking sets in G-designs and K3,3-designs, J. Discrete Math. Sci. Cryptogr. 16 (2013), 201–206, doi:10.1080/09720529.2013.838409. [5] M. Gionfriddo, E. Guardo and L. Milazzo, Extending bicolorings for Steiner triple systems, Appl. Anal. Discrete Math. 7 (2013), 225–234, doi:10.2298/aadm130827019g. [6] M. Gionfriddo, C. C. Lindner and C. A. Rodger, 2-colouring K4 e designs, Aus- tralas. J. Combin. 3 (1991), 211–229, https://ajc.maths.uq.edu.au/pdf/3/ ocr-ajc-v3-p211.pdf. [7] M. Gionfriddo and G. Lo Faro, 2-colourings in S(t, t + 1, v), Discrete Math. 111 (1993), 263–268, doi:10.1016/0012-365x(93)90161-l. [8] M. Gionfriddo, L. Milazzo and V. Voloshin, Hypergraphs and Designs, Mathematics Research Developments, Nova Science Publishers, New York, 2015. [9] D. G. Hoffman, C. C. Lindner and K. T. Phelps, Blocking sets in designs with block size 4, European J. Combin. 11 (1990), 451–457, doi:10.1016/s0195-6698(13)80027-3. [10] C. C. Lindner, M. Meszka and A. Rosa, On a problem of Lo Faro concerning blocking sets in Steiner systems S(2, 4, v), in: R. G. Stanton and J. L. 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ISSN 2590-9770 The Art of Discrete and Applied Mathematics 1 (2018) #P2.09 https://doi.org/10.26493/2590-9770.1310.6e2 (Also available at http://adam-journal.eu) Parallelism in Steiner systems ⇤ Maria Di Giovanni , Mario Gionfriddo Department of Mathematics and Computer Science, University of Catania, Catania, Italy Antoinette Tripodi Department of Mathematical and Computer Science, Physical Sciences and Earth Sciences, University of Messina, Italy Received 28 December 2017, accepted 27 April 2018, published online 23 July 2019 Abstract The authors give a survey about the problem of parallelism in Steiner systems, pointing out some open problems. Keywords: Steiner system, (partial) parallel class. Math. Subj. Class.: 05B05, 51E10 1 Introduction A Steiner system S(h, k, v) is a k-uniform hypergraph ⌃ = (X,B) of order v, such that every subset Y ✓ X of cardinality h has degree d(Y ) = 1 [4]. In the language of classical design theory, an S(h, k, v) is a pair ⌃ = (X,B) where X is a finite set of cardinality v, whose elements are called points (or vertices), and B is a family of k-subsets B ✓ X , called blocks, such that for every subset Y ✓ X of cardinality h there exists exactly one block B 2 B containing Y . Using more modern terminology, if Kun denotes the complete u-uniform hypergraph of order n, then a Steiner system S(h, k, v) is a Khk -decomposition of K h v , i.e. a pair ⌃ = (X,B), where X is the vertex set of Khv and B is a collection of hypergraphs all isomorphic to Khk (blocks) such that every edge of K h v belongs to exactly one hypergraph of B. An S(2, 3, v) is usually called Steiner Triple System and denoted by STS(v); it is well-kown that an STS(v) exists if and only if v ⌘ 1, 3 (mod 6), and contains v(v 1)/6 triples. An S(3, 4, v) is usually called Steiner Quadruple System and denoted by SQS(v); ⇤Supported by I.N.D.A.M. (G.N.S.A.G.A.). E-mail address: mariadigiovanni1@hotmail.com (Maria Di Giovanni), gionfriddo@dmi.unict.it (Mario Gionfriddo), atripodi@unime.it (Antoinette Tripodi) cb This work is licensed under http://creativecommons.org/licenses/by/3.0/ 2 Art Discrete Appl. Math. 1 (2018) #P2.09 it is well-kown that an SQS(v) exists if and only if v ⌘ 2, 4 (mod 6), and contains v(v 1)(v 2)/24 quadruples. Given a Steiner system ⌃ = (X,B), two distinct blocks B0, B00 2 B are said to be parallel if B0 \B00 = ;. A partial parallel class of ⌃ is a family ⇧ ✓ B of parallel blocks. If ⇧ is a partition of X , then it is said to be a parallel class of ⌃. Of course not every Steiner system S(h, k, v) has a parallel class (for example, when v is not a multiple of k) and so it is of considerable interest to determine in general just how large a partial parallel class a Steiner system S(h, k, v) must have. Open problem 1.1 (Parallelism Problem). Let 2  h < k < v, determine the maximum integer ⇡(h, k, v) such that any S(h, k, v) has at least ⇡(h, k, v) distinct parallel blocks. In this paper we will survey known results on the parallelism problem and give some open problems, including Brouwer’s conjecture. 2 A result of Lindner and Phelps In [6] C. C. Lindner and K. T. Phelps proved the following result. Theorem 2.1. Any Steiner system S(k, k+1, v), with v k4 +3k3 + k2 +1, has at least d vk+1k+2 e parallel blocks. Proof. Let ⌃ = (X,B) be a Steiner system S(k, k + 1, v), with v k4 + 3k3 + k2 + 1. Let ⇧ be a partial parallel class of maximum size, say t, and denote by P the set of vertices belonging to the blocks of ⇧. Since ⇧ is a partial parallel class of maximum size, every Y ✓ X P , |Y | = k, is contained in one block B 2 B which intersects P in exactly one vertex. Denote by ⌦ the set of all blocks having k elements in XP (and so the remaining vertex in P ) and by A the set of all vertices belonging to P and to some block of ⌦: ⌦ = {B 2 B : |B \ (X P )| = k}, A = {x 2 P : x 2 B,B 2 ⌦}. For every x 2 A, set T (x) = {B {x} : B 2 ⌦}. We can see that ⌃0 = (X P, T (x)) is a partial Steiner system of type S(k 1, k, v (k + 1)t), with |T (x)|  v(k+1)t k1 k , and {T (x)}x2A is a partition of Pk(XP ), i.e., the set of all k-subsets of XP . Observe that, if B is a block of ⇧ containing at least two vertices of A, then for each x 2 A\B we must have |T (x)|  k v(k+1)t1 k2 k 1 . Indeed, otherwise, let y be any other vertex belonging to A \ B and B1 be a block of T (y). Since at most k v(k+1)t1 k2 /(k 1) of the blocks in T (x) can intersect the block B1, then T (x) must contain a block B2 such that B1 \ B2 = ;. Hence, the family ⇧0 = M. Di Giovanni et al.: Parallelism in Steiner systems 3 (⇧ {B}) [ {B1, B2} is a partial parallel class of blocks having size |⇧0| > |⇧|, a contradiction. It follows that, for every block B 2 ⇧ containing at least two vertices of A, X x2A\B |T (x)|  (k + 1)k v(k+1)t1 k2 k 1 . Therefore, if we denote by r the number of blocks of ⇧ containing at most one vertex of A and by s the number of blocks of ⇧ containing at least two vertices of A, then ✓ v (k + 1)t k ◆ = X x2A |T (x)|  " (k + 1)k v(k+1)t1 k2 k 1 # r + "v(k+1)t k1 k # s. Now, consider the following two cases: Case 1. h (k + 1)k v(k+1)t1 k2 /(k 1) i  v(k+1)t k1 /k. It follows ✓ v (k + 1)t k ◆ = X x2A |T (x)|  (r + s) v(k+1)t k1 k  t "v(k+1)t k1 k # , from which t vk+1k+2 . Case 2. h (k + 1)k v(k+1)t1 k2 /(k 1) i > v(k+1)t k1 /k. In this case, it follows t v k3 k2 /(k + 1) and so t v k 3 k2 k + 1 v k + 1 k + 2 , for v k4 + 3k3 + k2 + 1. Combining Cases 1 and 2 completes the proof of the theorem. For Steiner triple and quadruple systems Theorem 2.1 gives the following result. Corollary 2.2. (i) Any STS(v), with v 45, has at least ⌃ v1 4 ⌥ parallel blocks. (ii) Any SQS(v), with v 172, has at least ⌃ v2 5 ⌥ parallel blocks. Regarding STS(v)s, the cases of v < 45 has been studied by C. C. Lindner and K. T. Phelps in [6] and by G. Lo Faro in [7, 8], while for SQS(v)s, the cases of v < 172 has been examined by G. Lo Faro in [9]. Collecting together their results gives the following theorem. Theorem 2.3. (i) Any STS(v), with v 9, has at least ⌃ v1 4 ⌥ parallel blocks. (ii) Any SQS(v) has at least ⌃ v2 5 ⌥ parallel blocks, with the possible exceptions for v = 20, 28, 34, 38. 4 Art Discrete Appl. Math. 1 (2018) #P2.09 The following result due to D. E. Woolbright [12] improves the inequality of Lindner- Phelps for Steiner triple systems of order v 139. Theorem 2.4. Any STS(v) has at least 3v7010 parallel blocks. For large values of v (greater then v0 ⇡ 10000), the above result in turn is improved by the following theorem which is due to A. E. Brouwer [1] and is valid for every admissible v 127. Theorem 2.5. Any Steiner triple system of sufficiently large order v has at least l v5v2/3 3 m parallel blocks. In 1981 A. E. Brouwer stated the following open problem. Open problem 2.6 (Brouwer’s Conjecture). Any STS(v) has at least ⌃ vc 3 ⌥ parallel blocks, for a constant c 2 N . By similar arguments as in Theorem 2.1, C. C. Lindner and R. C. Mullin [11] proved a further result for an arbitrary Steiner system S(h, k, v). Theorem 2.7. Any Steiner system S(h, k, v), with v 2k[2k(k 1) 2(k h) (h 1)(k h 1)] + h 1 k2 kh h+ 1 , has at least 2(vh+1)(k+1)(kh+1) parallel blocks. 3 A result on parallelism in S(k, k + 1, v), for k 3 For k 3, in [3] (for k = 3) and in [2] (for k > 3) the author proved the following result. Theorem 3.1. Any Steiner system S(k, k + 1, v), with k 3, has at least ⌅ v+2 2k ⇧ parallel blocks. Proof. Let ⌃ = (X,B) be a Steiner system S(k, k + 1, v), with k 3, and ⇧ be a family of parallel blocks of ⌃ such that P = [ B2⇧ B and |X P | (k 1)|⇧|+ 2(k 1), which implies v 2k(|⇧|+1)2. We will prove that ⌃ has a family ⇧0 of parallel blocks such that |⇧0| > |⇧|. This is trivial if there exists a block B 2 ⌃ such that B ✓ X P . Therefore, we suppose that for every block B 2 B, B * X P . Note that, for any Y ✓ X P, |Y | = k 1, if R = (X P ) Y , then there exists an injection ' : R ! P defined as follows: for every x 2 R, '(x) is the element of P such that Y [ {x,'(x)} 2 B. Now let {ai,1, ai,2, . . . , ai,k+1} 2 ⇧, for i = 1, 2, . . . , r, such that {ai,1, ai,2, . . . , ai,k} ✓ '(R); M. Di Giovanni et al.: Parallelism in Steiner systems 5 let {bji,1, b j i,2, . . . , b j i,k+1} 2 ⇧, for j = 1, 2, . . . , k 1 and i = 1, 2, . . . , pj , such that {bji,1, b j i,2, . . . , b j i,j} ✓ '(R) and {b j i,j+1, . . . , b j i,k+1} \ '(R) = ;; and let {ci,1, ci,2, . . . , ci,k+1} 2 ⇧, for i = 1, 2, . . . , h, such that {ci,1, ci,2, . . . , ci,k+1} \ '(R) = ;. Necessarily, (k + 1)r + k1X i=1 ipi |'(R)| = |X P | (k 1) (k 1)t+ k 1. Since t = r + Pk1 i=1 pi + h, it follows that (k + 1)r + k1X i=1 ipi (k 1)r + (k 1) k1X i=1 pi + (k 1)h+ k 1, and so r 1 2 " k2X i=1 pi(k 1 i) + h(k 1) + (k 1) # . Let xi,j 2 R such that '(xi,j) = ai,j and let yji,u 2 R such that '(yui,j) = b j i,u. Case 1. Suppose ai,k+1 /2 '(R), for each i = 1, 2, . . . , r. It follows that |X P | (k 1) = k1X i=1 ipi + kr. Since |X P | (k 1) (k 1)t+ (k 1) and t = h+ r + Pk1 i=1 pi, it follows k1X i=1 ipi + kr (k 1)t+ k 1 = (k 1)h+ (k 1)r + (k 1) k1X i=1 pi + k 1, hence r k2X i=1 pi(k 1 i) + h(k 1) + (k 1). Now, consider the injection : R0 ! P , where R0 = {xi,j 2 R : i 6= 1}, such that for all xi,j 2 R0, (xi,j) is the element of '(R) satisfying the condition {x1,1, x1,2, . . . , x1,k1, xi,j , (xi,j)} 2 B. If is the family of the blocks {x1,1, x1,2, . . . , x1,k1, xi,j , (xi,j)} and L = {ci,j : i = 1, 2, . . . , h, j = 1, 2, . . . , k + 1} [ {b1i,1 : i = 1, 2, . . . , p1} [ {a1,k}, 6 Art Discrete Appl. Math. 1 (2018) #P2.09 then || = k(r 1) and |L| = (k + 1)h+ p1 + 1, with || = k(r 1) = kr k k2X i=1 pi(k i 1) + hk(k 1) + k2 2k > (k + 1)h+ p1 + 1 = |L|, where we used the following inequalities, which hold for k 3, r k2X i=1 pi(k i 1) + h(k 1) + k 1, hk(k 1) > h(k + 1), k2 k > 1. Then, it is possible to find an element x 2 P L such that {x1,1, x1,2, . . . , x1,k1, 1(x), x} 2 B. Further, there exists at least an element y 2 '(R), y 6= x, with x and y belonging to the same Bx,y 2 ⇧. If ⇧0 = ⇧ {Bx,y} [ {{x1,1, x1,2, . . . , x1,k1, 1(x), x}, Y [ {'1(y), y}}, then ⇧0 is a family of parallel blocks of B with |⇧0| > |⇧|. Case 2. Suppose there is at least one element ai,k+1 such that {ai,1, ai,2, . . . , ai,k+1} ✓ '(R). Assume that {ai,1, ai,2, . . . , ai,k+1} ✓ '(R), for each i = 1, 2, . . . , r0 and {ai,1, ai,2, . . . , ai,k} ✓ '(R), ai,k+1 /2 '(R), for each i = r0 + 1, . . . , r. If r 2, consider the injection µ : R00 ! P , where R00 = xi,j 2 R : (i, j) 6= (1, 1), (1, 2), . . . , 1, ⌃ k1 2 ⌥ , (2, 1), (2, 2), . . . , 2, ⌃ k1 2 ⌥ , such that for every xi,j 2 R00, µ(xi,j) is the element of '(R) satisfying the condition n x1,1, x1,2, . . . , x1,d k12 e, x2,1, x2,2, . . . , x2,b k12 c, xi,j , µ(xi,j) o 2 B. If 0 is the family of blocks n x1,1, x1,2, . . . , x1,d k12 e, x2,1, x2,2, . . . , x2,b k12 c, xi,j , µ(xi,j) o and L0 = {ci,j : i = 1, 2, . . . , h, j = 1, 2, . . . , k + 1} [ {b1i,1 : i = 1, 2, . . . , p1}, M. Di Giovanni et al.: Parallelism in Steiner systems 7 it follows that |0| (k 1)t k1X i=1 ipi = (k 1)(r + k1X i=1 pi + h) k1X i=1 ipi = (k 1)r + (k 1)h+ k2X i=1 (k 1 i)pi k + 1 2 k2X i=1 (k 1 i)pi + h(k2 1) 2 + (k 1)2 2 > (k + 1)h+ p1 + 1 = |L0|+ 1, where we used t = r + h+ k1X i=1 pi, r 1 2 " k2X i=1 pi(k i 1) + h(k 1) + (k 1) # , k 3. Therefore, it is possible to find at least two distinct elements x0, x00 belonging to two distinct blocks B0, B00 of 0: B0 = n x1,1, x1,2, . . . , x1,d k12 e, x2,1, x2,2, . . . , x2,b k12 c, µ 1(x0), x0 o , B00 = n x1,1, x1,2, . . . , x1,d k12 e, x2,1, x2,2, . . . , x2,b k12 c, µ 1(x00), x00 o , such that x0, x00 2 P L0. Since x0 6= x00, we can suppose that x0 6= a2,d k12 e. Therefore, it is possible to find an element y 2 '(R), y 6= x0, with x0 and y belonging to the same block Bx,y of ⇧. It follows that there exists a family ⇧0 of parallel blocks with |⇧0| = |⇧|+ 1. If r = 1, then r0 = r = 1. Since r 1 2 " k2X i=1 pi(k i 1) + h(k 1) + (k 1) # , then k2X i=1 pi(k i 1) + h(k 1) + (k 1)  2. It follows necessarily: k = 3, h = 0, p1 = 0. Hence t = p2 + 1, |X P | = 2p2 + 6, and v = 6p2 + 10. If p2 = 0, then v = 10 and t = 1, and it is well-known that the unique STS(10) has two parallel blocks. 8 Art Discrete Appl. Math. 1 (2018) #P2.09 If p2 1, consider the blocks B0 = {x1,1,'1(b21,1), X1,2, x0}, B00 = {x1,1,'1(b21,1), X1,3, x00}, where x0, x00 2 '(R). Since x0 6= x00, we can assume x0 6= b21,2 and by applying the same technique as the previous cases we can find a family ⇧0 of parallel blocks with |⇧0| = |⇧|+ 1. Therefore, it is proved that if ⌃ = (X,B) is any S(k, k + 1, v), with k 3, and ⇧ is a family of parallel blocks of ⌃ such that |⇧| = t and |X P | (k 1)t + 2(k 1), where P = S B2⇧ B, then ⌃ has a partial parallel class ⇧ 0 of cardinality |⇧0| > |⇧|. It follows that, if t = b v2(k1)2k c, then ⌃ has a partial parallel class of cardinality t0 = t+ 1 = ⌅ v+2 2k ⇧ . By applying the same technique used in the previous proof, M. C. Marino and R. S. Rees [10] improved the lower bound stated by Theorem 3.1 to j 2(v+2) 3(k+1) k . 4 Open problems (a) Remove the exceptions of Theorem 2.3. It is known that ⇡(3, 4, v) = ⌅ v 4 ⇧ for v = 4, 8, 10, 14. In [5] by means of an exhaus- tive computer search the authors classified the Steiner quadruple systems of order 16 up to isomorphism; following a private conversation, it turned out that the computer search showed that every SQS(16) has a parallel class and so ⇡(3, 4, 16) = 4. (b) Determine the smallest v such that ⇡(3, 4, v) 6= ⌅ v 4 ⇧ . Concerning the parallelism in Steiner systems, an interesting question arises when we consider resolvable systems. A Steiner system ⌃ = (X,B) is said to be resolv- able provided B admits a partition R (resolution) into parallel classes. A resolvable Steiner triple system is called Kirkman Triple System (KTS, in short). It is well- known that a KTS(v) exists if and only if v ⌘ 3 (mod 6) (any resolution contains (v 1)/2 parallel classes of size v/3). (c) Problem of A. Rosa (1978): Let ⌃ = (X,B) be any KTS(v) and R be a resolution of ⌃. Determine a lower bound for the size of partial parallel classes of ⌃ in which no two triples come from the same parallel class of R. The problem of A. Rosa can be posed for any resolvable Steiner systems S(h, k, v): (c’) Problem of A. Rosa: Let ⌃ = (X,B) be any Steiner system S(h, k, v) and R be a resolution of ⌃. Determine a lower bound for the size of partial parallel classes of ⌃ in which no two blocks come from the same parallel class of R. References [1] A. E. Brouwer, On the size of a maximum transversal in a Steiner triple system, Canadian J. Math. 33 (1981), 1202–1204, doi:10.4153/cjm-1981-090-7. M. Di Giovanni et al.: Parallelism in Steiner systems 9 [2] M. Gionfriddo, On the number of pairwise disjoint blocks in a Steiner system, in: C. J. Colbourn and R. Mathon (eds.), Combinatorial Design Theory, North-Holland, Amster- dam, volume 149 of North-Holland Mathematics Studies, pp. 189–195, 1987, doi:10.1016/ s0304-0208(08)72886-x. [3] M. Gionfriddo, Partial parallel classes in Steiner systems, Colloq. Math. 57 (1989), 221–227, doi:10.4064/cm-57-2-221-227. [4] M. Gionfriddo, L. Milazzo and V. Voloshin, Hypergraphs and Designs, Mathematics Research Developments, Nova Science Publishers, New York, 2015. [5] P. Kaski, P. R. J. Östergård and O. Pottonen, The Steiner quadruple systems of order 16, J. Comb. Theory Ser. A 113 (2006), 1764–1770, doi:10.1016/j.jcta.2006.03.017. [6] C. C. Lindner and K. T. Phelps, A note on partial parallel classes in Steiner systems, Discrete Math. 24 (1978), 109–112, doi:10.1016/0012-365x(78)90179-6. [7] G. Lo Faro, On the size of partial parallel classes in Steiner systems STS(19) and STS(27), Discrete Math. 45 (1983), 307–312, doi:10.1016/0012-365x(83)90047-x. [8] G. Lo Faro, Partial parallel classes in Steiner system S(2, 3, 19), J. Inform. Optim. Sci. 6 (1985), 133–136, doi:10.1080/02522667.1985.10698814. [9] G. Lo Faro, Partial parallel classes in Steiner quadruple systems, Utilitas Math. 34 (1988), 113–116. [10] M. C. Marino and R. S. Rees, On parallelism in Steiner systems, Discrete Math. 97 (1991), 295–300, doi:10.1016/0012-365x(91)90445-8. [11] R. C. Mullin and C. C. Lindner, Lower bounds for maximal partial parallel classes in Steiner systems, J. Comb. Theory Ser. A 26 (1979), 314–318, doi:10.1016/0097-3165(79)90109-2. [12] D. E. Woolbright, On the size of partial parallel classes in Steiner systems, Ann. Discrete Math. 7 (1980), 203–211, doi:10.1016/s0167-5060(08)70181-x. ISSN 2590-9770 The Art of Discrete and Applied Mathematics 1 (2018) #P2.10 https://doi.org/10.26493/2590-9770.1286.d7b (Also available at http://adam-journal.eu) Open problems in the spectral theory of signed graphs ⇤ Francesco Belardo † Department of Mathematics and Applications, University of Naples Federico II, I-80126 Naples, Italy Sebastian M. Cioabă Department of Mathematical Sciences, University of Delaware, USA Jack Koolen School of Mathematical Sciences, University of Science and Technology of China, Wen-Tsun Wu Key Laboratory of the Chinese Academy of Sciences, Anhui, 230026, China Jianfeng Wang School of Mathematics and Statistics, Shandong University of Technology, Zibo, China Received 30 December 2018, accepted 10 July 2019, published online 7 August 2019 Abstract Signed graphs are graphs whose edges get a sign +1 or 1 (the signature). Signed graphs can be studied by means of graph matrices extended to signed graphs in a natural way. Recently, the spectra of signed graphs have attracted much attention from graph spectra specialists. One motivation is that the spectral theory of signed graphs elegantly generalizes the spectral theories of unsigned graphs. On the other hand, unsigned graphs do not disappear completely, since their role can be taken by the special case of balanced signed graphs. Therefore, spectral problems defined and studied for unsigned graphs can be considered in terms of signed graphs, and sometimes such generalization shows nice properties which cannot be appreciated in terms of (unsigned) graphs. Here, we survey some general results ⇤The authors are indebted to Thomas Zaslavsky for reading and revising with many comments an earlier draft of this note. The research of the first author was supported by the INDAM-GNSAGA and by NRF (South Africa) with the grant ITAL170904261537, Ref. No. 113144. The research of the second author was supported by the grants NSF DMS-1600768 and CIF-1815922. The research of the third author is partially supported by the National Natural Science Foundation of China (Grant No. 11471009 and Grant No. 11671376) and by Anhui Initiative in Quantum Information Technologies (Grant No. AHY150200). The research of the fourth author was supported by the National Natural Science Foundation of China (No. 11461054). †Corresponding author. cb This work is licensed under http://creativecommons.org/licenses/by/3.0/ 2 Art Discrete Appl. Math. 1 (2018) #P2.10 on the adjacency spectra of signed graphs, and we consider some spectral problems which are inspired from the spectral theory of (unsigned) graphs. Keywords: Signed graph, adjacency matrix, eigenvalue, unbalanced graph. Math. Subj. Class.: 05C22, 05C50 1 Introduction A signed graph = (G,) is a graph G = (V,E), with vertex set V and edge set E, together with a function : E ! {+1,1} assigning a positive or negative sign to each edge. The (unsigned) graph G is said to be the underlying graph of , while the function is called the signature of . Edge signs are usually interpreted as ±1. In this way, the adjacency matrix A() of is naturally defined following that of unsigned graphs, that is by putting +1 or 1 whenever the corresponding edge is either positive or negative, re- spectively. One could think about signed graphs as weighted graphs with edges of weights in {0, 1,1}, however the two theories are very different. In fact, in signed graphs the product of signs has a prominent role, while in weighted graphs it is the sum of weights that is relevant. A walk is positive or negative if the product of corresponding weights is positive or negative, respectively. Since cycles are special kinds of walks, this definition applies to them as well and we have the notions of positive and negative cycles. Many familiar notions related to unsigned graphs directly extend to signed graphs. For example, the degree dv of a vertex v in is simply its degree in G. A vertex of degree one is said to be a pendant vertex. The diameter of = (G,) is the diameter of its underlying graph G, namely, the maximum distance between any two vertices in G. Some other definitions depend on the signature, for example, the positive (resp., negative) degree of a vertex is the number of positive (negative) edges incident to the vertex, or the already mentioned sign of a walk or cycle. A signed graph is balanced if all its cycles are positive, otherwise it is unbalanced. Unsigned graphs are treated as (balanced) signed graphs where all edges get a positive sign, that is, the all-positive signature. An important feature of signed graphs is the concept of switching the signature. Given a signed graph = (G,) and a subset U ✓ V (G), let U be the signed graph obtained from by reversing the signs of the edges in the cut [U, V (G)\U ], namely U (e) = (e) for any edge e between U and V (G)\U , and U (e) = (e) otherwise. The signed graph U is said to be (switching) equivalent to and U to , and we write U ⇠ or U ⇠ . It is not difficult to see that each cycle in maintains its sign after a switching. Hence, U and have the same positive and negative cycles. Therefore, the signature is determined up to equivalence by the set of positive cycles (see [82]). Signatures equivalent to the all-positive one (the edges get just the positive sign) lead to balanced signed graphs: all cycles are positive. By ⇠ + we mean that the signature is equivalent to the all-positive signature, and the corresponding signed graph is equivalent to its underlying graph. Hence, all signed trees on the same underlying graph are switching equivalent to the all-positive signature. In fact, signs are only relevant in cycles, while the edge signs of bridges are irrelevant. Note that (unsigned) graph invariants are preserved under switching, but also by vertex E-mail addresses: fbelardo@unina.it (Francesco Belardo), cioaba@udel.edu (Sebastian M. Cioabă), koolen@ustc.edu.cn (Jack Koolen), jfwang@sdut.edu.cn (Jianfeng Wang) F. Belardo et al.: Open problems in the spectral theory of signed graphs 3 permutation, so we can consider the isomorphism class of the underlying graph. If we combine switching equivalence and vertex permutation, we have the more general concept of switching isomorphism of signed graphs. For any not given notation and basic results in the theory of signed graphs, the reader is referred to Zaslavsky [82] (see also the dynamic surveys [80, 81]). We next consider matrices associated to signed graphs. For a signed graph = (G,) and a graph matrix M = M(), the M -polynomial is M (, x) = det(xI M()). The spectrum of M is called the M -spectrum of the signed graph . Usually, M is the adjacency matrix A() or the Laplacian matrix L() = D(G)A(), but in the literature one can find their normalized variants or other matrices. In the remainder, we shall mostly restrict to M being the adjacency matrix A(). The adjacency matrix A() = (aij) is the symmetric {0,+1,1}-matrix such that aij = (ij) whenever the vertices i and j are adjacent, and aij = 0 otherwise. As with unsigned graphs, the Laplacian matrix is defined as L() = D(G) A(), where D(G) is the diagonal matrix of vertices degrees (of the underlying graph G). In the sequel we will mostly restrict to the adjacency matrix. Switching has a matrix counterpart. In fact, let and U be two switching equivalent graphs. Consider the matrix SU = diag(s1, s2, . . . , sn) such that si = ( +1, i 2 U ; 1, i 2 \ U. The matrix SU is the switching matrix. It is easy to check that A(U ) = SU A()SU and L(U ) = SU L()SU . Hence, signed graphs from the same switching class share similar graph matrices by means of signature matrices (signature similarity). If we also allow permutation of vertices, we have signed permutation matrices, and we can speak of (switching) isomorphic signed graphs. Switching isomorphic signed graphs are cospectral, and their matrices are signed- permutationally similar. From the eigenspace viewpoint, the eigenvector components are also switched in signs and permuted. Evidently, for each eigenvector, there exists a suitable switching such that all components become nonnegative. In the sequel, let 1() 2() · · · n() denote the eigenvalues of the ad- jacency matrix A() of the signed graph of order n; they are all real since A() is a real symmetric matrix. The largest eigenvalue 1() is sometimes called the index of . If contains at least one edge, then 1() > 0 > n() since the sum of the eigen- values is 0. Note that in general, the index 1() does not equal the spectral radius ⇢() = max{|i| : 1  i  n} = max{1,n} because the Perron-Frobenius The- orem is valid only for the all-positive signature (and those equivalent to it). For example, an all 1 signing (all-negative signature) of the complete graph on n 3 vertices will have eigenvalues 1 = · · · = n1 = 1 and n = (n 1). We would like to end this introduction by mentioning what may be the first paper on signed graph spectra [82]. In that paper, Zaslavsky showed that 0 appears as an L- eigenvalue in connected signed graphs if and only if the signature is equivalent to the all- positive one, that is, is a balanced signed graph. For notation not given here and basic results on graph spectra, the reader is referred to [23, 22], for some basic results on the spectra of signed graphs, to [83], and for some applications of spectra of signed graphs, to [27]. 4 Art Discrete Appl. Math. 1 (2018) #P2.10 In Section 2, we survey some important results on graph spectra which are valid in terms of the spectra of signed graphs. In Section 3 we collect some open problems and conjectures which are open at the writing of this note. 2 What do we lose with signed edges? From the matrix viewpoint, when we deal with signed graphs we have symmetric {0, 1, 1}-matrices instead of just symmetric {0, 1}-matrices. Clearly, the results coming from the theory of nonnegative matrices can not be applied directly to signed graphs. Perhaps the most important result that no longer holds for adjacency matrices of signed graphs is the Perron-Frobenius theorem. We saw one instance in the introduction and we will see some other consequences of the absence of Perron-Frobenius in the next section. Also, the loss of non-negativity has other consequences related to counting walks and the diameter of the graph (Theorem 3.10). On the other hand, all results based on the symmetry of the matrix will be still valid in the context of signed graphs with suitable modifications. In this section, we briefly describe how some well-known results are (possibly) changed when dealing with matrices of signed graphs. We start with the famous Coefficient Theorem, also known as Sachs Formula. This for- mula, perhaps better than others, describes the connection between the eigenvalues and the combinatorial structure of the signed graph. It was given for unsigned graphs in the 1960s independently by several researchers (with different notation), but possibly first stated by Sachs (cf. [23, Theorem 1.2] and the subsequent remark). The signed-graph variant can be easily given as follows. An elementary figure is the graph K2 or Cn (n 3). A basic figure (or linear subgraph, or sesquilinear subgraph) is the disjoint union of elementary figures. If B is a basic figure, then denote by C(B) the class of cycles in B, with c(B) = |C(B)|, and by p(B) the number of components of B and define (B) = Q C2C(B) (C). Let Bi be the set of basic figures on i vertices. Theorem 2.1 (Coefficient Theorem). Let be a signed graph and let (, x) =Pn i=0 aix ni be its adjacency characteristic polynomial. Then, a0 = 1 and, for i > 0, ai = X B2Bi (1)p(B)2c(B)(B). Another important connection between the eigenvalues and the combinatorial structure of a signed graph is given by the forthcoming theorem. If we consider unsigned graphs, it is well known that the k-th spectral moment gives the number of closed walks of length k (cf. [22, Theorem 3.1.1]). Zaslavsky [83] observed that a signed variant holds for signed graphs as well, and from his observation we can give the subsequent result. Theorem 2.2 (Spectral Moments). Let be a signed graph with eigenvalues 1 · · · n. If W±k denotes the difference between the number of positive and negative closed walks of length k, then W±k = nX i=1 ki . Next, we recall another famous result for the spectra of graphs, that is, the Cauchy Interlacing Theorem. Its general form holds for principal submatrices of real symmetric matrices (see [22, Theorem 1.3.11]). It is valid in signed graphs without any modification F. Belardo et al.: Open problems in the spectral theory of signed graphs 5 to the formula. For a signed graph = (G,) and a subset of vertices U , then U is the signed graph obtained from by deleting the vertices in U and the edges incident to them. For v 2 V (G), we also write v instead of {v}. Similar notation applies when deleting subsets of edges. Theorem 2.3 (Interlacing Theorem for Signed Graphs). Let = (G,) be a signed graph. For any vertex v of , 1() 1( v) 2() 2( v) · · · n1( v) n(). In the context of subgraphs, there is another famous result which is valid in the theory of signed graphs. In fact, it is possible to give the characteristic polynomial as a linear combination of vertex- or edge-deleted subgraphs. Such formulas are known as Schwenk’s Formulas (cf. [22, Theorem 2.3.4], see also [5]). As above, v ( e) stands for the signed graph obtained from in which the vertex v (resp., edge e) is deleted. Also, to make the formulas consistent, we set (;, x) = 1. Theorem 2.4 (Schwenk’s Formulas). Let be a signed graph and v (resp., e = uv) one of its vertices (resp., edges). Then (, x) = x( v, x) X u⇠v ( u v, x) 2 X C2Cv (C)( C, x), (, x) = ( e, x) ( u v, x) 2 X C2Ce (C)( C, x), where Ca denotes the set of cycles passing through a. Finally, a natural question is the following: if we fix the underlying graph, how much can the eigenvalues change when changing the signature? Given a graph with cyclomatic number ⇠, then there are at most 2⇠ nonequivalent signatures as for each independent cycle one can assign either a positive or a negative sign. However, among the 2⇠ signatures, some of them might lead to switching isomorphic graphs, as we see later. In general, the eigenvalues coming from each signature cannot exceed in modulus the spectral radius of the underlying graph, as is shown in the last theorem of this section. Theorem 2.5 (Eigenvalue Spread). For a signed graph = (G,), let ⇢() be its spectral radius. Then ⇢()  ⇢(G). Proof. Clearly, ⇢() equals 1() or n(). Let A be the adjacency matrix of (G,+), and A be the adjacency matrix of = (G,). For a vector X = (x1, . . . , xn)T , let |X| = (|x1|, . . . , |xn|)T . If X is a unit eigenvector corresponding to 1(A), by the Rayleigh quotient we get 1(G,) = X TAX  |X|TA|X|  max z:zT z=1 zTAz = 1(G,+). Similarly, if X is a unit eigenvector corresponding to the least eigenvalue n(A), by the Rayleigh quotient we get n(G,) = X TAX |X|T (A)|X| min z:zT z=1 zT (A)z = n(G,) = 1(G,+). By gluing together the two inequalities, we get the assertion. 6 Art Discrete Appl. Math. 1 (2018) #P2.10 It is evident from the preceding results that the spectral theory of signed graphs well encapsulates and extends the spectral theory of unsigned graphs. Perhaps, we can say that adding signs to the edges just gives more variety to the spectral theory of graphs. This fact was already observed with the Laplacian of signed graphs, which nicely generalizes the results coming from the Laplacian and signless Laplacian theories of unsigned graphs. It is worth mentioning that thanks to the spectral theory it was possible to give matrix-wise definitions of the signed graph products [29], line graphs [6, 83] and subdivision graphs [6]. 3 Some open problems and conjectures In this section we consider some open problems and conjectures which are inspired from the corresponding results in the spectral theory of unsigned graphs. We begin with the intriguing concept of “sign-symmetric graph” which is a natural signed generalization of the concept of bipartite graph. 3.1 Symmetric spectrum and sign-symmetric graphs One of the most celebrated results in the adjacency spectral theory of (unsigned) graphs is the following. Theorem 3.1. 1. A graph is bipartite if and only if its adjacency spectrum is symmetric with respect to the origin. 2. A connected graph is bipartite if and only if its smallest eigenvalue equals the nega- tive of its spectral radius. For the first part, one does not need Perron-Frobenius theorem. To the best of our knowledge, Perron-Frobenius is crucial for the second part (see [10, Section 3.4] or [33, Section 8.8] or [76, Chapter 31]). On the other hand, in the larger context of signed graphs the symmetry of the spectrum is not a privilege of bipartite and balanced graphs. A signed graph = (G,) is said to be sign-symmetric if is switching isomorphic to its negation, that is, = (G,). It is not difficult to observe that the signature-reversal changes the sign of odd cycles but leaves unaffected the sign of even cycles. Since bipartite (unsigned) graphs are odd-cycle free, it happens that bipartite graphs are a special case of sign-symmetric signed graphs, or better to say, if a signed graph = (G,) has a bipartite underlying graph G, then and are switching equivalent. In Figure 1 we depict an example of a sign-symmetric graph. Here and in the remaining pictures as well negative edges are represented by heavy lines and positive edges by thin lines. u u u u u u u u H H H H H H H H H H H H Figure 1: A sign-symmetric signed graph. F. Belardo et al.: Open problems in the spectral theory of signed graphs 7 If is switching isomorphic to , then A and A are similar and we immediately get: Theorem 3.2. Let be a sign-symmetric graph. Then its adjacency spectrum is symmetric with respect to the origin. The converse of the above theorem is not true, and counterexamples arise from the theory of Seidel matrices. The Seidel matrix of a (simple and unsigned) graph G is S(G) = J I 2A, so that adjacent vertices get the value 1 and non-adjacent vertices the value +1. Hence, the Seidel matrix of an unsigned graph can be interpreted as the adjacency matrix of a signed complete graph. The signature similarity becomes the famous Seidel switching. The graph in Figure 2 belongs to a triplet of simple graphs on 8 vertices sharing the same symmetric Seidel spectrum but not being pairwise (Seidel-)switching isomorphic. In [25, p. 253], they are denoted as A1, its complement Ā1 and A2 (note, A2 and its complement Ā2 are Seidel switching isomorphic). In fact, A1 and its complement Ā1 are cospectral but not Seidel switching isomorphic. In terms of signed graphs, the signed graph A01 whose adjacency matrix is S(A1) has symmetric spectrum but it is not sign-symmetric. u u u u u u u u ⌦ ⌦ ⌦ ⌦ ⌦ ⌦ ⌦ ⌦ ⌦ J J J J J J J J J J J J JJ ⌦ ⌦ ⌦ ⌦⌦ Figure 2: The graph A1. Note that the disjoint union of sign-symmetric graphs is again sign-symmetric. Since the above counterexamples involve Seidel matrices which are the same as signed complete graphs, the following is a natural question. Problem 3.3. Are there non-complete connected signed graphs whose spectrum is sym- metric with respect to the origin but they are not sign-symmetric? Observe that signed graphs with symmetric spectrum have odd-indexed coefficients of the characteristic polynomial equal to zero and all spectral moments of odd order are also zero. A simple application of Theorems 2.1 and 2.2 for i = 3 or k = 3, respectively, leads to equal numbers of positive and negative triangles in the graph. When we consider i = 5 or k = 5, we cannot say that the numbers of positive and negative pentagons are the same. The following corollary is an obvious consequence of the latter discussion (cf. also [25, Theorem 1]). Corollary 3.4. A signed graph containing an odd number of triangles cannot be sign- symmetric. Remark 3.5. As we mentioned in Section 2, a signed graph with cyclomatic number ⇠ has exactly 2⇠ not equivalent signatures (see also [55]). On the other hand, the symmetries, if any, in the structure of the underlying graph can make several of those signatures lead to isomorphic signed graphs. 8 Art Discrete Appl. Math. 1 (2018) #P2.10 3.2 Signed graphs with few eigenvalues There is a well-known relation between the diameter and the number of distinct eigenvalues of an unsigned graph (cf. [22, Theorem 3.3.5]). In fact, the number of distinct eigenvalues cannot be less than the diameter plus 1. With signed graphs, the usual proof based on the minimal polynomial does not hold anymore. Indeed, the result is not true with signed graphs. As we can see later, it is possible to build signed graphs of any diameter having exactly two distinct eigenvalues. For unsigned graphs, the identification of graphs with a small number of eigenvalues is a well-known problem. The unique connected graph having just two distinct eigen- values is the complete graph Kn. If a graph is connected and regular, then it has three distinct eigenvalues if and only if it is strongly regular (see [22, Theorem 3.6.4]). At the 1995 British Combinatorial Conference, Haemers posed the problem of finding connected graphs with three eigenvalues which are neither strongly regular nor complete bipartite. Answering Haemers’ question, van Dam [71, 72] and Muzychuk and Klin [58] described some constructions of such graphs. Other constructions were found by De Caen, van Dam and Spence [24] who also noticed that the first infinite family nonregular graphs with three eigenvalues already appeared in the work of Bridges and Mena [9]. The literature on this topic contains many interesting results and open problems. For example, the answer to the following intriguing problem posed by De Caen (see [73, Problem 9]) is still unknown. Problem 3.6. Does a graph with three distinct eigenvalues have at most three distinct degrees? Recent progress was made recently by van Dam, Koolen and Jia [70] who constructed connected graphs with four or five distinct eigenvalues and arbitrarily many distinct de- grees. These authors posed the following bipartite version of De Caen’s problem above. Problem 3.7. Are there connected bipartite graphs with four distinct eigenvalues and more than four distinct valencies? For signed graphs there are also some results. In 2007, McKee and Smyth [57] con- sidered symmetric integral matrices whose spectral radius does not exceed 2. In their nice paper, they characterized all such matrices and they further gave a combinatorial interpreta- tion in terms of signed graphs. They defined a signed graph to be cyclotomic if its spectrum is in the interval [2,2]. The maximal cyclotomic signed graphs have exactly two distinct eigenvalues. The graphs appearing in the following theorem are depicted in Figure 3. Theorem 3.8. Every maximal connected cyclotomic signed graph is switching equivalent to one of the following: • For some k = 3, 4, . . ., the 2k-vertex toroidal tessellation T2k. • The 14-vertex signed graph S14. • The 16-vertex signed hypercube S16. Further, every connected cyclotomic signed graph is contained in a maximal one. It is not difficult to check that all maximal cyclotomic graphs are sign-symmetric. Note that for k even T2k has a bipartite underlying graph, while for k odd T2k has not bipartite underlying graph but it is sign-symmetric, as well. The characteristic polynomial to T2k is (x2)k(x+2)k, so T2k is an example of a signed graph with two distinct eigenvalues and diameter bk2 c. F. Belardo et al.: Open problems in the spectral theory of signed graphs 9 S14 u1 v1 uk vk u1 v1 T2k S16 Figure 3: Maximal cyclotomic signed graphs. Problem 3.9 (Signed graphs with exactly 2 distinct eigenvalues). Characterize all con- nected signed graphs whose spectrum consists of two distinct eigenvalues. In the above category we find the complete graphs with homogeneous signatures (Kn,+) and (Kn,), the maximal cyclotomic signed graphs T2k, S14 and S16, and that list is not complete (for example, the unbalanced 4-cycle C4 and the 3-dimensional cube whose cycles are all negative must be included). There is already some literature on this problem, and we refer the readers to see [30, 62]. All such graphs have in common the property that positive and negative walks of length greater than or equal to 2 between two different and non-adjacent vertices are equal in number. In this way we can consider a signed variant of the diameter. In a connected signed graph, two vertices are at signed distance k if they are at distance k and the difference between the numbers of positive and negative walks of length k among them is nonzero, otherwise the signed distance is set to 0. The signed diameter of , denoted by diam±(), is the largest signed distance in . Recall that the (i, j)-entry of Ak equals the difference between the numbers of positive and negative walks of length k among the vertices indexed by i and j. Then we have the following result (cf. [22, Theorem 3.3.5]): Theorem 3.10. Let be a connected signed graph with m distinct eigenvalues. Then diam±()  m 1. Proof. Assume the contrary, so that has vertices, say s and t, at signed distance p m. The adjacency matrix A of has minimal polynomial of degree m, and so we may write Ap = Pm1 k=0 akA k. This yields the required contradiction because the (s, t)-entry on the right is zero, while the (s, t)-entry on the left is non-zero. Recently, Huang [47] constructed a signed adjacency matrix of the n-dimensional hy- 10 Art Discrete Appl. Math. 1 (2018) #P2.10 percube whose eigenvalues are ± p n, each with multiplicity 2n1. Using eigenvalue in- terlacing, Huang proceeds to show that the spectral radius (and therefore, the maximum degree) of any induced subgraph on 2n1 + 1 vertices of the n-dimensional hypercube, is at least p n. This led Huang to a breakthrough proof of the Sensitivity Conjecture from theoretical computer science. We will return to Huang’s construction after Theorem 3.23. 3.3 The largest eigenvalue of signed graphs In the adjacency spectral theory of unsigned graphs the spectral radius is the largest eigen- value and it has a prominent role because of its algebraic features, its connections to combi- natorial parameters such as the chromatic number, the independence number or the clique number and for its relevance in applications. There is a large literature on this subject, see [11, 20, 43, 45, 60, 68, 78] for example. As already observed, the presence of negative edges leads invalidates of the Perron- Frobenius theorem, and we lose some nice features of the largest eigenvalue: • The largest eigenvalue may not be the spectral radius although by possibly changing the signature to its negative, this can be achieved. • The largest eigenvalue may not be a simple eigenvalue. • Adding edges might reduce the largest eigenvalue. Therefore one might say that it not relevant to study signed graphs in terms of the magnitude of the spectral radius. In this respect, Theorem 2.3 and Theorem 2.5 are helpful because the spectral radius does not decrease under the addition of vertices (together with some incident edges), and the spectral radius of the underlying graph naturally limits the magnitude of the eigenvalues of the corresponding signed graph. For the same reason, the theory of limit points for the spectral radii of graph sequences studied by Hoffman in [43, 45] is still valid in the context of signed graphs. The Hoffman program is the identification of connected graphs whose spectral radii do not exceed some special limit points established by A. J. Hoffman [45]. The smallest limit point for the spectral radius is 2 (the limit point of the paths of increasing order), so the first step would be to identify all connected signed graphs whose spectral radius does not exceed 2. The careful reader notices that the latter question has already been completely solved by Theorem 3.8. Therefore, the problem jumps to the next significant limit point, which isp 2 + p 5 = ⌧ 1 2 + ⌧ 1 2 , where ⌧ is the golden mean. This limit point is approached from above (resp., below) by the sequence of positive (resp., negative) cycles with exactly one pendant vertex and increasing girth. In [11, 19], the authors identified all connected unsigned graphs whose spectral radius does not exceed p 2 + p 5. Their structure is fairly simple: they mostly consist of paths with one or two additional pendant vertices. Regarding signed graphs, we expect that the family is quite a bit larger than that of unsigned graphs. A taste of this prediction can be seen by comparing the family of Smith Graphs (the unsigned graphs whose spectral radius is 2, cf. Figure 2.4 in [23]) with the graphs depicted in Figure 3. On the other hand, the graphs identified by Cvetković et al. acts as a “skeleton” (that is, appear as subgraphs) of the signed graphs with the same bound on the spectral radius. Problem 3.11 (Hoffman Program for Signed Graphs). Characterize all connected signed graphs whose spectral radius does not exceed p 2 + p 5. F. Belardo et al.: Open problems in the spectral theory of signed graphs 11 3.4 The smallest eigenvalue of signed graphs Unsigned graphs with smallest eigenvalue at least 2 have been characterized in a veritable tour de force by several researchers. We mention here Cameron, Goethals, Seidel and Shult [15], Bussemaker and Neumaier [13] who among other things, determined a complete list of minimal forbidden subgraphs for the class of graphs with smallest eigenvalue at least 2. A monograph devoted to this topic is [21] whose Chapter 1.4 tells the history about the characterization of graphs with smallest eigenvalue at least 2. Theorem 3.12. If G is a connected graph with smallest eigenvalue at least 2, then G is a generalized line graph or has at most 36 vertices. In the case of unsigned graphs, their work was extended, under some minimum degree condition, from 2 to 1 p 2 by Hoffman [44] and Woo and Neumaier [79] and more recently, to 3 by Koolen, Yang and Yang [51]. For signed graphs, some of the above results were extended by Vijayakumar [77] who showed that any connected signed graph with smallest eigenvalue less than 2 has an induced signed subgraph with at most 10 vertices and smallest eigenvalue less than 2. Chawathe and Vijayakumar [17] determined all minimal forbidden signed graphs for the class of signed graphs whose smallest eigenvalue is at least 2. Vijayakumar’s result [77, Theorem 4.2] was further extended by Koolen, Yang and Yang [51, Theorem 4.2] to signed matrices whose diagonal entries can be 0 or 1. These authors introduced the notion of s-integrable graphs. For an unsigned graph G with smallest eigenvalue min and adjacency matrix A, the matrix A bmincI is positive semidefinite. For a natural number s, G is called s-integrable if there exists an integer matrix N such that s(A bmincI) = NNT . Note that generalized line graphs are exactly the 1-integrable graphs with smallest eigen- value at least 2. In a straightforward way, the notion of s-integrabilty can be extended to signed graphs. Now we can extend Theorem 3.12 to the class of signed graphs with essentially the same proof. Theorem 3.13. Let be a connected signed graph with smallest eigenvalue at least 2. Then is 2-integrable. Moreover, if has at least 121 vertices, then is 1-integrable. As E8 has 240 vectors of (squared) norm 2, one can take from each pair of such a vector and its negative exactly one to obtain a signed graph on 120 vertices with smallest eigenvalue 2 that is not 1-integrable. Many of these signed graphs are connected. Koolen, Yang and Yang [51] proved that if a connected unsigned graph has smallest eigenvalue at least 3 and valency large enough, then G is 2-integrable. An interesting direction would be to prove a similar result for signed graphs. Problem 3.14. Extend [51, Theorem 1.3] to signed graphs. An interesting related conjecture was posed by Koolen and Yang [52]. Conjecture 3.15. There exists a constant c such that if G is an unsigned graph with small- est eigenvalue at least 3, then G is c-integrable. Koolen, Yang and Yang [51] also introduced (3)-maximal graphs or maximal graphs with smallest eigenvalue 3. These are connected graphs with smallest eigenvalue at least 3 such any proper connected supergraph has smallest eigenvalue less than 3. Koolen 12 Art Discrete Appl. Math. 1 (2018) #P2.10 and Munemasa [50] proved that the join between a clique on three vertices and the comple- ment of the McLaughlin graph (see Goethals and Seidel [36] or Inoue [48] for a description) is (3)-maximal. Problem 3.16. Construct maximal signed graphs with smallest eigenvalue at least 3. Woo and Neumaier [79] introduced the notion of Hoffman graphs, which has proved an essential tool in many results involving the smallest eigenvalue of unsigned graphs (see [51]). Perhaps a theory of signed Hoffman graphs is possible as well. Problem 3.17. Extend the theory of Hoffman graphs to signed graphs. 3.5 Signatures minimizing the spectral radius As observed in Section 2, an unsigned graph with cyclomatic number ⇠ gives rise to at most 2⇠ switching non-isomorphic signed graphs. In view of Theorem 2.5, we know that, up to switching equivalency, the signature leading to the maximal spectral radius is the all-positive one. A natural question is to identify which signature leads to the minimum spectral radius. Problem 3.18 (Signature minimizing the spectral radius). Let be a simple and connected unsigned graph. Determine the signature(s) ̄ such that for any signature of , we have ⇢(, ̄)  ⇢(,). This problem has important connections and consequences in the theory of expander graphs. Informally, an expander is a sparse and highly connected graph. Given an integer d 3 and a real number, a -expander is a connected d-regular graph whose (unsigned) eigenvalues (except d and possibly d if the graph is bipartite) have absolute value at most . It is an important problem in mathematics and computer science to construct, for fixed d 3, infinite families of -expanders for small (see [8, 46, 56] for example). From work of Alon-Boppana (see [18, 46, 61]), we know that = 2 p d 1 is the best bound we can hope for and graphs attaining this bound are called Ramanujan graphs. Bilu and Linial [8] proposed the following combinatorial way of constructing infinite families of d-regular Ramanujan graphs. A double cover (sometimes called 2-lift or 2- cover) of a graph = (G = (V,E),) is the (unsigned) graph 0 with vertex set V ⇥ {+1,1} such that (x, s) is adjacent to (y, s(xy)) for s = ±1. It is easy to see that if is d-regular, then 0 is d-regular. A crucial fact is that the spectrum of the unsigned adjacency matrix of 0 is the union of the spectrum of the unsigned adjacency matrix A(G) and the spectrum of signed adjacency matrix A = A(), where A(x, y) = (x, y) for any edge xy of and 0 otherwise (see [8] for a short proof). Note that this result can be deduced using the method of equitable partitions (see [10, Section 2.3]), appears in the mathematical chemistry literature in the work of Fowler [26] and was extended to other matrices and directed graphs by Butler [14]. The spectral radius of a signing is the spectral radius ⇢(A) of the signed adjacency matrix A . Bilu and Linial [8] proved the important result Theorem 3.19 (Bilu-Linial [8]). Every connected d-regular graph has a signing with spec- tral radius at most c · p d log3 d, where c > 0 is some absolute constant. and made the following conjecture. F. Belardo et al.: Open problems in the spectral theory of signed graphs 13 Conjecture 3.20 (Bilu-Linial [8]). Every connected d-regular graph G has a signature with spectral radius at most 2 p d 1. If true, this conjecture would provide a way to construct or show the existence of an infinite family of d-regular Ramanujan graphs. One would start with a base graph that is d-regular Ramanujan (complete graph Kd+1 or complete bipartite graph Kd,d for example) and then repeatedly apply the result of the conjecture above. Recently, Marcus, Spielman and Srivastava [56] made significant progress towards solving the Bilu-Linial conjecture. Theorem 3.21. Let G be a connected d-regular graph. Then there exists a signature of G such that the largest eigenvalue of A is at most 2 p d 1. As mentioned before, A may have negative entries and one cannot apply the Perron- Frobenius theorem for it. Therefore, the spectral radius of A is not always the same as the largest eigenvalue of A . In more informal terms, the Bilu-Linial conjecture is about bounding all the eigenvalues of A by 2 p d 1 and 2 p d 1 while the Marcus- Spielman-Srivastava result shows the existence of a signing where all the eigenvalues of A are at most 2 p d 1. By taking the negative of the signing guaranteed by Marcus- Spielman-Srivastava, one gets a signed adjacency matrix where all eigenvalues are at least 2 p d 1, of course. There are several interesting ingredients in the Marcus-Spielman-Srivastava result. The first goes back to Godsil and Gutman [35] who proved the remarkable result that the average of the characteristic polynomials of the all the signed adjacency matrices of a graph equals the matching polynomial of . This is defined as follows. Define m0 = 1 and for k 1, let mk denote the number of matchings of consisting of exactly k edges. The matching polynomial µ(x) of is defined as µ(x) = X k0 (1)kmkxn2k, (3.1) where n is the number of vertices of . Heilmann and Lieb [42] proved the following results regarding the matching polynomial of a graph. See Godsil’s book [34] for a nice, self-contained exposition of these results. Theorem 3.22. Let be a graph. 1. Every root of the matching polynomial µ(x) is real. 2. If is d-regular, then every root of µ(x) has absolute value at most 2 p d 1. If is a d-regular graph, then the average of the characteristic polynomials of its signed adjacency matrices equals its matching polynomial µ(x) whose roots are in the desired interval [2 p d 1, 2 p d 1]. As Marcus-Spielman-Srivastava point out, just because the average of certain polynomials has roots in a certain interval, does not imply that one of the polynomials has roots in that interval. However, in this situation, the characteristic poly- nomials of the signed adjacency matrices form an interlacing family of polynomials (this is a term coined by Marcus-Spielman-Srivastava in [56]). The theory of such polynomials is developed in [56] and it leads to an existence proof that one of the signed adjacency matrices of G has the largest eigenvalue at most 2 p d 1. As mentioned in [56], The difference between our result and the original conjecture is that we do not control the smallest new eigenvalue. This is why we consider bipartite graphs. 14 Art Discrete Appl. Math. 1 (2018) #P2.10 Note that the result of Marcus, Spielman and Srivastava [56] implies the existence of an infinite family of d-regular bipartite Ramanujan graphs, but it does not provide a recipe for constructing such family. As an amusing exercise, we challenge the readers to solve Problem 3.18 by finding a signature of the Petersen graph (try it without reading [84]) or of their favorite graph that minimizes the spectral radius. A weighing matrix of weight k and order n is a square n⇥ n matrix W with 0,+1,1 entries satisfying WWT = kIn. When k = n, this is the same as a Hadamard matrix and when k = n 1, this is called a conference matrix. Weighing matrices have been well studied in design and coding theory (see [28] for example). Examining the trace of the square of the signed adjacency matrix, Gregory [39] proved the following. Theorem 3.23. If is any signature of , then ⇢(,) p k (3.2) where k is the average degree of . Equality happens if and only if is k-regular and A is a symmetric weighing matrix of weight k. This result implies that ⇢(Kn,) p n 1 for any signature with equality if and only if a conference matrix of order n exists. By a similar argument, one gets that ⇢(Kn,n,) p n with equality if and only if there is a Hadamard matrix of order n. Note also that when k = 4, the graphs attaining equality in the previous result are known from McKee and Smyth’s work [57] (see Theorem 3.8 above). Using McKee and Smyth char- acterization and the argument below, we can show that the only 3-regular graph attaining equality in Theorem 3.23 is the 3-dimensional cube. Let Qn denote the n-dimensional hypercube. Huang [47] constructed a signed adja- cency matrix An of Qn recursively as follows: A1 =  0 1 1 0 and An+1 =  An I2n I2n An , for n 1. It is not too hard to show that (An)2 = nI2n for any n 1 and thus, An attains equality in Theorem 3.23. We remark that Huang’s method can be also used to produce infinite families of regular graphs and signed adjacency matrices attaining equality in Theorem 3.23. If G is a k-regular graph of order N with signed adjacency matrix As such that ⇢(As) = p k, then define the k+1-regular graph H by taking two disjoint copies of G and adding a perfect matching between them and a signed adjacency matrix for H as B =  As IN IN As . Because A2s = kIN , we can get that B2 = (k+ 1)I2N . Thus, using any 4-regular graph G from McKee and Smyth [57] (see again Theorem 3.8) with a signed adjacency matrix As satisfying A2s = 4I , one can construct a 5-regular graph H with signed adjacency matrix B such that B2 = 5I . The following is a natural question. Problem 3.24. Are there any other 5-regular graphs attaining equality in Theorem 3.23? If the regularity assumption on G is dropped, Gregory considered a the following vari- ant of Conjecture 3.20. F. Belardo et al.: Open problems in the spectral theory of signed graphs 15 Conjecture 3.25 ([39]). If is the largest vertex degree of a nontrivial graph G, then there exists a signature such that ⇢(G,) < 2 p 1. Gregory came to the above conjecture by observing that in view of Theorem 3.22 the bound in the above conjecture holds for the matching polynomial of G and by noticing that µG(x) = 1 |C| X C2C (G,;x), where C is the set of subgraphs of G consisting of cycles and |C| is the number of cycles of C. Since the matching polynomial of G is the average of polynomials of signed graphs on G, one could expect that there is at least one signature ̄ such that ⇢(G, ̄) does not exceed the spectral radius of µG(x). As observed in [39], for odd unicyclic signed graphs the spectral radius of the matching polynomial is always less than the spectral radius of the corresponding adjacency polynomial, but the conjecture still remains valid. We ask the following question whose affirmative answer would imply Conjecture [39]. Problem 3.26. If ⇢ is the spectral radius of a connected graph G, then is there a signature such that ⇢(G,) < 2 p ⇢ 1? In view of the above facts, we expect that the signature minimizing the spectral radius is the one balancing the contributions of cycles so that the resulting polynomial is as close as possible to the matching polynomial. For example, we can have signatures whose cor- responding polynomial equals the matching polynomial, as in the following proposition. Proposition 3.27. Let be a signed graph consisting of 2k odd cycles of pairwise equal length and opposite signs. Then ⇢() < 2 p 4k 1. Is the signature in Proposition 3.27 the one minimizing the spectral radius? We leave this as an open problem (see also [85]). We conclude this section by observing that for a general graph, it is not known whether Problem 3.18 is NP-hard or not. However, progress is made in [16] where the latter men- tioned problem is shown to be NP-hard when restricted to arbitrary symmetric matrices. Furthermore, the problems described in this subsection can be considered in terms of the largest eigenvalue 1, instead of the spectral radius. 3.6 Spectral determination problems for signed graphs A graph is said to be determined by its (adjacency) spectrum if cospectral graphs are iso- morphic graphs. It is well-known that in general the spectrum does not determine the graph, and this problem has pushed a lot of research in spectral graph theory, also with respect to other graph matrices. In general, we can say that there are three kinds of research lines: (1) Identify, if any, cospectral non-isomorphic graphs for a given class of graphs. (2) Routines to build cospectral non isomorphic graphs (e.g., Godsil-McKay switching). (3) Find conditions such that the corresponding graphs are determined by their spectrum. Evidently, the same problems can be considered for signed graphs with respect to switching isomorphism. On the other hand, when considering signed graphs, there are many more possibilities for getting pairs of switching non-isomorphic cospectral signed 16 Art Discrete Appl. Math. 1 (2018) #P2.10 graphs. For example, the paths and the cycles are examples of graphs determined by their spectrum, but the same graphs as signed ones are no longer determined by their spectrum since they admit cospectral but non-isomorphic mates [1, 3]. Hence, the spectrum of the adjacency matrix of signed graphs has less control on the graph invariants. In view of the spectral moments we get the following proposition: Proposition 3.28. From the eigenvalues of a signed graph we obtain the following in- variants: • number of vertices and edges; • the difference between the number of positive and negative triangles ( 16 P 3i ); • the difference between the number of positive and negative closed walks of length p ( P pi ). Contrarily to unsigned graphs, from the spectrum we cannot decide any more whether the graph has some kind of signed regularity, or it is sign-symmetric. For the former, we note that the co-regular signed graph (C6,+) (it is a regular graph with net regular signature) is cospectral with P2 [ Q̃4 (cf. Figure 4). For the latter, we observe that the signed graphs A1 and A2 are cospectral but A1 is not sign-symmetric while A2 is sign- symmetric. u u u u u u u u u u u u @ @ @ @ @ @ @ @ @ @ @ @ Figure 4: The cospectral pair (C6,+) and P2 [ Q̃4. 3.7 Operations on signed graphs In graph theory we can find several operations and operators acting on graphs. For example, we have the complement of a graph, the line graph, the subdivision graph and several kind of products as the cartesian product, and so on. Most of them have been ported to the level of signed graph, in a way that the resulting underlying graph is the same obtained from the theory of unsigned graphs, while the signatures are given in order to preserve the balance property, signed regularities, and in many cases also the corresponding spectra. However, there are a few operations and operators which do not yet have a, satisfactory, ‘signed’ variant. One operator that is missing in the signed graph theory is the complement of a signed graph. The complement of signed graph should be a signed graph whose underlying graph is the usual complement, however the signature has not been defined in a satisfactory way yet. What we can ask from the signature of the complement of a signed graph? One could expect some nice features on the spectrum, as for the Laplacian, so that the spectra of the two signed graphs and ̄ are complementary to the spectrum of the obtained complete graph. F. Belardo et al.: Open problems in the spectral theory of signed graphs 17 Problem 3.29. Given a graph = (G,), define the complement ̄ = (Ḡ, ̄) such that there are nice (spectral) properties derived from the complete signed graph [ ̄. In terms of operators, in the literature we have nice definitions for subdivision and line graphs of signed graphs [6, 83]. The signed total graph has been recently considered and defined in [7]. From the product viewpoint, most standard signed graph products have been defined and considered in [29] and the more general NEPS (or, Cvetković product) of signed graphs have been there considered. In [66] the lexicographic product was also considered, but the given definition is not stable under the equivalence switching classes. However, there are some graph products which do not have a signed variant yet. As an example, we mention here the wreath product and the co-normal product. 3.8 Seidel matrices The Seidel matrix of a graph on n vertices is the adjacency matrix of a signed complete graph Kn in which the edges of are negative (1) and the edges not in are positive (+1). More formally, the Seidel matrix S() equals Jn In 2A(). Zaslavsky [83] confesses that This fact inspired my work on adjacency matrices of signed graphs. Seidel matrices were introduced by van Lint and Seidel [75] and studied by many peo- ple due to their interesting properties and connections to equiangular lines, two-graphs, strongly regular graphs, mutually unbiased bases and so on (see [10, Section 10.6] and [4, 37, 64] for example). The connection between Seidel matrices and equiangular lines is perhaps best summarized in [10, p. 161]: To find large sets of equiangular lines, one has to find large graphs where the smallest Seidel eigenvalue has large multiplicity. Let d be a natural number and Rd denote the Euclidean d-dimensional space with the usual inner product h·, ·i. A set of n 1 lines (represented by unit vectors) v1, . . . , vn 2 Rd is called equiangular if there is a constant ↵ > 0 such that hvi, vji = ±↵ for any 1  i < j  n. For given ↵, let N↵(d) be the maximum n with this property. The Gram matrix G of the vectors v1, . . . , vn is the n ⇥ n matrix whose (i, j)-th entry equals hvi, vji. The matrix S := (G I)/↵ is a symmetric matrix with 0 diagonal and ±1 entries off-diagonal. It is therefore the Seidel matrix of some graph and contains all the relevant parameters of the equiangular line system. The multiplicity of the smallest eigenvalue 1/↵ of S is the smallest dimension d where the line system can be embedded into Rd. Lemmens and Seidel [53] (see also [4, 37, 49, 54, 59] for more details) showed that N1/3(d) = 2d 2 for d sufficiently large and made the following conjecture. Conjecture 3.30. If 23  d  185, N1/5(d) = 276. If d 185, then N1/5(d) = b3(d 1)/2c. The fact that N1/5(d) = b3(d 1)/2c for d sufficiently large was proved by Neumaier [59] and Greaves, Koolen, Munemasa and Szöllősi [37]. Recently, Lin and Yu [54] made progress in this conjecture by proving some claims from Lemmens and Seidel [53]. Note that these results can be reformulated in terms of Seidel matrices with smallest eigenvalue 5. Seidel and Tsaranov [65] classified the Seidel matrices with smallest eigenvalue 3. 18 Art Discrete Appl. Math. 1 (2018) #P2.10 Neumann (cf. [53, Theorem 3.4]) proved that if N↵(d) 2d, then 1/↵ is an odd integer. Bukh [12] proved that N↵(d)  c↵d, where c↵ is a constant depending only on ↵. Balla, Dräxler, Keevash and Sudakov [4] improved this bound and showed that for d sufficiently large and ↵ 6= 1/3, N↵(d)  1.93d. Jiang and Polyanskii [49] further improved these results and showed that if ↵ /2 {1/3, 1/5, 1/(1 + 2 p 2)}, then N↵(d)  1.49d for d sufficiently large. When 1/↵ is an odd integer, Glazyrin and Yu [32] obtained a general bound N↵(d)  2↵2/3 + 4/7 d+ 2 for all n. Bukh [12] and also, Balla, Dräxler, Keevash and Sudakov [4] conjecture the following. Conjecture 3.31. If r 2 is an integer, then N 1 2r1 (d) = r(n1)r1 +O(1) for n sufficiently large. When 1/↵ is not a totally real algebraic integer, then N↵(d) = d. Jiang and Polyanskii [49] studied the set T = {↵ | ↵ 2 (0, 1), lim supd!1 N↵(d)/d > 1} and showed that the closure of T contains the closed interval [0, 1/ pp 5 + 2] using results of Shearer [67] on the spectral radius of unsigned graphs. Seidel matrices with two distinct eigenvalues are equivalent to regular two-graphs and correspond to equality in the relative bound (see [10, Section 10.3] or [37] for example). It is natural to study the combinatorial and spectral properties of Seidel matrices with three distinct eigenvalues, especially since for various large systems of equiangular lines, the respective Seidel matrices have this property. Recent work in this direction has been done by Greaves, Koolen, Munemasa and Szöllősi [37] who determined several properties of such Seidel matrices and raised the following interesting problem. Problem 3.32. Find a combinatorial interpretation of Seidel matrices with three distinct eigenvalues. A classification for the class of Seidel matrices with exactly three distinct eigenvalues of order less than 23 was obtained by Szöllősi and Östergård [69]. Several parameter sets for which existence is not known were also compiled in [37]. Greaves [38] studied Seidel matrices with three distinct eigenvalues, observed that there is only one Seidel matrix of order at most 12 having three distinct eigenvalues, but its switching class does not contain any regular graphs. In [38], he also showed that if the Seidel matrix S of a graph has three distinct eigenvalues of which at least one is simple, then the switching class of contains a strongly regular graph. The following question was posed in [38]. Problem 3.33. Do there exist any Seidel matrices of order at least 14 with precisely three distinct eigenvalues whose switching class does not contain a regular graph? The switching class of conference graph and isolated vertex has two distinct eigenval- ues. If these two eigenvalues are not rational, then the switching class does not contain a regular graph. So we suspect that there must be infinitely many graphs whose Seidel ma- trix has exactly three distinct eigenvalues and its switching graph does not contain a regular graph. A related problem also appears in [38]. Problem 3.34. Does every Seidel matrix with precisely three distinct rational eigenvalues contain a regular graph in its switching class? The Seidel energy S() of a graph is the sum of absolute values of the eigenvalues of the Seidel matrix S of . This parameter was introduced by Haemers [41] who proved that F. Belardo et al.: Open problems in the spectral theory of signed graphs 19 S()  n p n 1 for any graph of order n with equality if and only S is a conference matrix. Haemers [41] also conjectured that the complete graphs on n vertices (and the graphs switching equivalent to them) minimize the Seidel energy. Conjecture 3.35. If is a graph on n vertices, then S() S(Kn) = 2(n 1). Ghorbani [31] proved the Haemers’ conjecture in the case det(S) n 1 and very recently, Akbari, Einollahzadeh, Karkhaneei and Nematollah [2] finished the proof of the conjecture. Ghorbani [31, p. 194] also conjectured that the fraction of graphs on n vertices with | detS| < n 1 goes to 0 as n tends to infinity. This conjecture was also recently proved by Rizzolo [63]. It is known that if has even order, then its Seidel matrix S is full-rank. If a graph has odd order n, then rank(S) n 1. There are examples such C5 for example where rank(S) = n 1. Haemers [40] posed the following problem which is still open to our knowledge. Problem 3.36. If rank(S) = n 1, then there exists an eigenvector of S corresponding to 0 that has only ±1 entries? 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