Square Heffter arrays are n×n arrays such that each row and each column contains k filled cells, each row and column sum is divisible by 2nk+1 and either x or −x appears in the array for each integer 1⩽x⩽nk. Archdeacon noted that a Heffter array, satisfying two additional conditions, yields a face 2-colourable embedding of the complete graph K2nk+1 on an orientable surface, where for each colour, the faces give a k-cycle system. Moreover, a cyclic permutation on the vertices acts as an automorphism of the embedding. These necessary conditions pertain to cyclic orderings of the entries in each row and each column of the Heffter array and are: (1) for each row and each column the sequential partial sums determined by the cyclic ordering must be distinct modulo 2nk+1; (2) the composition of the cyclic orderings of the rows and columns is equivalent to a single cycle permutation on the entries in the array. We construct Heffter arrays that satisfy condition (1) whenever (a) k≡0(mod4); or (b) n≡1(mod4) and k≡3(mod4); or (c) n≡0(mod4), k≡3(mod4) and n≫k. As corollaries to the above we obtain pairs of orthogonal k-cycle decompositions of K2nk+1.

The fork is the tree obtained from the claw K1,3 by subdividing one of its edges once, and the antifork is its complement graph. We give a complete description of all graphs that do not contain the fork or antifork as induced subgraphs.

It is known that for each k≥4, there exists an irreducible sign pattern with minimum rank k that does not allow diagonalizability. However, it is shown in this paper that every square sign pattern A with minimum rank 2 that has no zero line allows diagonalizability with rank 2 and also with rank equal to the maximum rank of the sign pattern. In particular, every irreducible sign pattern with minimum rank 2 allows diagonalizability. On the other hand, an example is given to show the existence of a square sign pattern with minimum rank 3 and no zero line that does not allow diagonalizability; however, the case for irreducible sign patterns with minimum rank 3 remains open. In addition, for a sign pattern that allows diagonalizability, the possible ranks of the diagonalizable real matrices with the specified sign pattern are shown to be lengths of certain composite cycles. Some results on sign patterns with minimum rank 2 are extended to sign pattern matrices whose maximal zero submatrices are “strongly disjoint” (that is, their row index sets as well as their column index sets are pairwise disjoint).

A cyclic ordering of the points in a Mendelsohn triple system of order v (or MTS (v)) is called a sequencing. A sequencing D is ℓ-good if there does not exist a triple (x,y,z) in the MTS (v) such that 1. the three points x,y, and z occur (cyclically) in that order in D; and 2. {x,y,z} is a subset of ℓ cyclically consecutive points of D. In this paper, we prove some upper bounds on ℓ for MTS (v) having ℓ-good sequencings and we prove that any MTS (v) with v≥7 has a 3-good sequencing. We also determine the optimal sequencings of every MTS (v) with v≤10.

In this paper we investigate the geometric properties of the configuration consisting of a subspace Γ and a canonical subgeometry Σ in PG(n−1,qn), with Γ∩Σ=0̸. The idea motivating is that such properties are reflected in the algebraic structure of the linear set which is projection of Σ from the vertex Γ. In particular we deal with the maximum scattered linear sets of the line PG(1,qn) found by Lunardon and Polverino (2001) and recently generalized by Sheekey (2016). Our aim is to characterize this family by means of the properties of the vertex of the projection as done by Csajbók and the first author of this paper for linear sets of pseudoregulus type. With reference to such properties, we construct new examples of scattered linear sets in PG(1,q6), yielding also to new examples of MRD-codes in Fq6×6 with left idealizer isomorphic to Fq6.

We generalize to sets with cardinality more than p a theorem of Rédei and Szőnyi on the number of directions determined by a subset U of the finite plane Fp2. A U-rich line is a line that meets U in at least #U∕p+1 points, while a U-poor line is one that meets U in at most #U∕p−1 points. The slopes of the U-rich and U-poor lines are called U-special directions. We show that either U is contained in the union of n=⌈#U∕p⌉ lines, or it determines “many” U-special directions. The core of our proof is a version of the polynomial method in which we study iterated partial derivatives of the Rédei polynomial to take into account the “multiplicity” of the directions determined by U.

A graph is said to be a bi-Cayley graph over a group H if it admits H as a semiregular automorphism group with two vertex-orbits. A bi-dicirculant is a bi-Cayley graph over a dicyclic group. In this paper, a classification of connected cubic bi-dicirculants is given. As byproducts, we show that every connected cubic vertex-transitive bi-dicirculant is Cayley, and up to isomorphism, there are two connected cubic edge-transitive bi-dicirculants, of which one has order 16, and the other has order 24.

Subspace codes have attracted much attention in recent years due to their applications to error correction in random network coding. In this paper, we construct several kinds of large cyclic subspace codes via Sidon spaces and large subspace codes via unions of some Sidon spaces. Therefore, some known results are extended.

A digraph D=(V,A) has a good pair at a vertex r if D has a pair of arc-disjoint in- and out-branchings rooted at r. Let T be a digraph with t vertices u1,…,ut and let H1,…Ht be digraphs such that Hi has vertices ui,ji,1≤ji≤ni. Then the composition Q=T[H1,…,Ht] is a digraph with vertex set {ui,ji∣1≤i≤t,1≤ji≤ni} and arc set A(Q)=∪i=1tA(Hi)∪{uijiupqp∣uiup∈A(T),1≤ji≤ni,1≤qp≤np}. If T is strongly connected, then Q is called a strong composition and if T is semicomplete, i.e., there is at least one arc between every pair of vertices, then Q is called a semicomplete composition. We obtain the following result: every strong digraph composition Q in which ni≥2 for every 1≤i≤t, has a good pair at every vertex of Q. The condition of ni≥2 in this result cannot be relaxed. We characterize semicomplete compositions with a good pair, which generalizes the corresponding characterization by Bang-Jensen and Huang (1995) for quasi-transitive digraphs. As a result, we can decide in polynomial time whether a given semicomplete composition has a good pair rooted at a given vertex.

The property of non-catastrophicity of multidimensional convolutional codes is studied. In particular, an algebraic and system-theoretic characterization of non-catastrophicity is offered in the multidimensional setting, and the Massey–Sain classical criterion is extended to this setting.

An RE-m-coloring of a graph G is a vertex m-coloring of G, which is relaxed (every vertex shares the same color with at most one neighbor) and equitable (the sizes of all color classes differ by at most one). In this article, we prove that every planar graph with minimum degree at least 2 and girth at least 8 has an RE-m-coloring for each integer m≥4. We use the discharging method and Hall’s Theorem to simply the structures of counterexamples.

Many well-known Catalan-like sequences turn out to be Stieltjes moment sequences (Liang et al. (2016)). However, a Stieltjes moment sequence is in general not determinate; Liang et al. suggested a further analysis about whether these moment sequences are determinate and how to obtain the associated measures. In this paper we find necessary conditions for a Catalan-like sequence to be a Hausdorff moment sequence. As a consequence, we will see that many well-known counting coefficients, including the Catalan numbers, the Motzkin numbers, the central binomial coefficients, the central Delannoy numbers, are Hausdorff moment sequences. We can also identify the smallest interval including the support of the unique representing measure. Since Hausdorff moment sequences are determinate and a representing measure for above mentioned sequences are already known, we could almost complete the analysis raised by Liang et al. In addition, subsequences of Catalan-like number sequences are also considered; we will see a necessary and sufficient condition for subsequences of Stieltjes Catalan-like number sequences to be Stieltjes Catalan-like number sequences. We will also study a representing measure for a linear combination of consecutive terms in Catalan-like number sequences.

An upper bound on the diameter of the Stable Matching (Stable Marriage) polytope is known to be ⌊n2⌋ where n is the number of men (or women) involved in the matching. The current work complements that result by providing a lower bound and an algorithm computing it. It also presents a class of Stable Matching instances for which the lower bound coincides with the above-mentioned upper bound.

Let G=(V,E) be a graph and Γ an Abelian group both of order n. A Γ-distance magic labeling of G is a bijection ℓ:V→Γ for which there exists μ∈Γ such that ∑x∈N(v)ℓ(x)=μ for all v∈V, where N(v) is the neighborhood of v. Froncek showed that the Cartesian product Cm□Cn, m,n≥3 is a Zmn-distance magic graph if and only if mn is even. It is also known that if mn is even then Cm□Cn has Zα×A-magic labeling for any α≡0(modlcm(m,n)) and any Abelian group A of order mn∕α. However, the full characterization of group distance magic Cartesian product of two cycles is still unknown. In the paper we make progress towards the complete solution of this problem by proving some necessary conditions. We further prove that for n even the graph Cn□Cn has a Γ-distance magic labeling for any Abelian group Γ of order n2. Moreover we show that if m≠n, then there does not exist a (Z2)m+n-distance magic labeling of the Cartesian product C2m□C2n. We also give a necessary and sufficient condition for Cm□Cn with gcd(m,n)=1 to be Γ-distance magic.

A subset C of the vertex set of a graph Γ is called a perfect code in Γ if every vertex of Γ is at distance no more than 1 to exactly one vertex of C. A subset C of a group G is called a perfect code of G if C is a perfect code in some Cayley graph of G. In this paper we give sufficient and necessary conditions for a subgroup H of a finite group G to be a perfect code of G. Based on this, we determine the finite groups that have no nontrivial subgroup as a perfect code, which answers a question by Ma, Walls, Wang and Zhou.

Let q be a prime-power, and n≥3 an odd integer. We determine the structure of the Weierstrass semigroup H(P) where P is an arbitrary Fq2-rational point of GK2,n where GK2,n stands for the Fq2n-maximal curve of Beelen and Montanucci. We prove that these points are Weierstrass points, and we compute the Frobenius dimension of GK2,n. Using these results, we also show that GK2,n is isomorphic to the Güneri–Garcìa–Stichtenoth only for n=3. Furthermore, AG codes and AG quantum codes from the GK2,n are constructed and discussed. In some cases, they have better parameters compared with those of the known linear codes.

Boolean functions used in symmetric-key cryptosystems must have high second-order nonlinearity to withstand several known attacks and some potential attacks which may exist but are not yet efficient and might be improved in the future. The second-order nonlinearity of Boolean functions also plays an important role in coding theory, since the maximal second-order nonlinearity of all Boolean functions in n variables equals the covering radius of the Reed–Muller code with length 2n and order r. It is well-known that providing a tight lower bound on the second-order nonlinearity of a general Boolean function with high algebraic degree is a hard task, excepting a few special classes of Boolean functions. In this paper, we improve the lower bounds on the second-order nonlinearity of three classes of Boolean functions of the form fi(x)=Tr1n(xdi) in n variables for i=1,2 and 3, where Tr1n denotes the absolute trace mapping from F2n to F2 and di’s are of the form (1) d1=2m+1+3 and n=2m, (2) d2=2m+2m+12+1, n=2m with odd m, and (3) d3=22r+2r+1+1 and n=4r with even r.

The strong fractional choice number of a graph G is the infimum of those real numbers r such that G is (⌈rm⌉,m)-choosable for every positive integer m. The strong fractional choice number of a family G of graphs is the supremum of the strong fractional choice number of graphs in G. We denote by Qk the class of series–parallel graphs with girth at least k. This paper proves that the strong fractional choice number of Qk equals 2+1⌊(k+1)∕4⌋.

This paper proves that if G is a planar graph without 4-cycles and l-cycles for some l∈{5,6,7}, then there exists a matching M such that AT(G−M)≤3. This implies that every planar graph without 4-cycles and l-cycles for some l∈{5,6,7} is 1-defective 3-paintable.

The Tamari lattice, defined on Catalan objects such as binary trees and Dyck paths, is a well-studied poset in combinatorics. It is thus natural to try to extend it to other families of lattice paths. In this article, we fathom such a possibility by defining and studying an analogy of the Tamari lattice on Motzkin paths. While our generalization is not a lattice, each of its connected components is isomorphic to an interval in the classical Tamari lattice. With this structural result, we proceed to the enumeration of components and intervals in the poset of Motzkin paths we defined. We also extend the structural and enumerative results to Schröder paths. We conclude by a discussion on the relation between our work and that of Baril and Pallo (2014).

Inspired by the Kruskal–Katona theorem a minimization problem is studied, where the role of the shadow is replaced by the image of the action of a certain subset of the monoid of increasing functions. One of our main results shows that compressed sets are a solution to this problem. Several applications to simplicial complexes are discussed.

Pairs of complementary sequences such as Golay pairs have zero sum autocorrelation at all non-trivial phases. Several generalizations are known where conditions on either the autocorrelation function, or the entries of the sequences are altered. We aim to unify most of these ideas by introducing autocorrelation functions that apply to any sequences with entries in a set equipped with a ring-like structure which is closed under multiplication and contains multiplicative inverses. Depending on the elements of the chosen set, the resulting complementary pairs may be used to construct a variety of combinatorial structures such as Hadamard matrices, complex generalized weighing matrices, and signed group weighing matrices. We may also construct quasi-cyclic and quasi-constacyclic linear codes which over finite fields of order less than 5 are also Hermitian self-orthogonal. As the literature on binary and ternary Golay sequences is already quite deep, one intention of this paper is to survey and assimilate work on more general pairs of complementary sequences and related constructions of combinatorial objects, and to combine the ideas into a single theoretical framework.

Recently, Beck posed two conjectures on the difference between the number of (respectively, distinct) parts in the odd partitions of n and the number of (respectively, distinct) parts in the distinct partitions of n. These two conjectures were first confirmed by Andrews using generating functions, and then generalized by Fu and Tang, and Yang in different ways. Motivated by Yang’s work, we present two more generalized results and prove them both analytically and combinatorially.

The cluster of a crossing in a graph drawing in the plane is the set of the four end-vertices of its two crossed edges. Two crossings are independent if their clusters do not intersect. In this paper, we prove that every plane graph with independent crossings has an equitable partition into m induced forests for any m≥8. Moreover, we decrease this lower bound 8 for m to 6, 5, 4 and 3 if we additionally assume that the girth of the considering graph is at least 4, 5, 6 and 26, respectively.

In 1996, Bang-Jensen, Gutin, and Li proposed the following conjecture: If D is a strong digraph of order n where n≥2 with the property that d(x)+d(y)≥2n−1 for every pair of dominated non-adjacent vertices {x,y}, then D is hamiltonian. In this paper, we give an infinite family of counterexamples to this conjecture. In the same paper, they showed that for the above x,y, if they satisfy the condition either d(x)≥n, d(y)≥n−1 or d(x)≥n−1, d(y)≥n, then D is hamiltonian. It is natural to ask if there is an integer k≥1 such that every strong digraph of order n satisfying either d(x)≥n+k, d(y)≥n−1−k, or d(x)≥n−1−k, d(y)≥n+k, for every pair of dominated non-adjacent vertices {x,y}, is hamiltonian. In this paper, we show that k must be at most n−5 and prove that every strong digraph with k=n−4 satisfying the above condition is hamiltonian, except for one digraph on 5 vertices.

We give a bijection between the set of ordinary partitions and that of self-conjugate partitions with some restrictions. Also, we show the relationship between hook lengths of a self-conjugate partition and its corresponding partition via the bijection. As a corollary, we give new combinatorial interpretations for the Catalan number and the Motzkin number in terms of self-conjugate simultaneous core partitions.

We prove that for a simple graph G with |V(G)|≥32, if min{δ(G),δ(Gc)}≥4, then either G or its complementary graph Gc has flow index strictly less than 3. This is proved by a newly developed closure operation, which may be useful in studying further flow index problems. In particular, our result supports a recent conjecture of Li et al. (2018), and improves a result of Hou et al. (2012) on nowhere-zero 3-flows.

Let Gn denote the group C2n×C2n, where Ck is the cyclic group of order k. We give an algorithm for enumerating the regular nontrivial partial difference sets (PDS) in Gn. We use our algorithm to obtain all of these PDS in Gn for 2≤n≤9, and we obtain partial results for n=10 and n=11. Most of these PDS are new. For n≤4 we also identify group-inequivalent PDS. Our approach involves constructing tree diagrams and canonical colorings of these diagrams. Both the total number and the number of group-inequivalent PDS in Gn appear to grow super-exponentially in n. For n=9, a typical canonical coloring represents in excess of 10146 group-inequivalent PDS, and there are precisely 2520 reversible Hadamard difference sets.

For codes over Zm, pomset metric is a generalization to poset metric, in some sense. We define pomset weight enumerator of a code C and establish MacWilliams type identities for linear codes with respect to certain pomsets. The identities for a particular type of linear codes (those that can be described as direct sum of linear codes) are established by considering direct and ordinal sum of pomsets on them. By induction, this result is extended for more than two pomsets. Moreover, MacWilliams type identities for any linear code with chain pomset is derived which paved a way to arrive at those identities for direct sum of codes with disjoint union of chains.

Strong difference families of special types are introduced to produce new relative difference families from the point of view of both asymptotic existences and concrete examples. As applications, group divisible designs of type 30u with block size 6 are discussed, r-rotational balanced incomplete block designs with block size 6 are derived for r∈{6,10}, and several classes of optimal optical orthogonal codes with weight 5, 6, 7, or 8 are obtained.

Let p be a prime, s, m be positive integers, λ be a nonzero element of the finite field Fpm. The b-distance generalizes the Hamming distance (b=1), and the symbol-pair distance (b=2). While the Hamming and symbol-pair distances of all λ-constacyclic codes of length ps are completely determined, the general b-distance of such codes was left opened. In this paper, we provide a new technique to establish the b-distance of all λ-constacyclic codes of length ps, where 1≤b≤⌊p2⌋. As an application, all MDS b-symbol constacyclic codes of length ps over Fpm are obtained.

We construct a non-commutative, non-cocommutative, graded bialgebra Π with a basis indexed by the permutations in all finite symmetric groups. Unlike the formally similar Malvenuto–Poirier–Reutenauer Hopf algebra, this bialgebra does not have finite graded dimension. After giving formulas for the product and coproduct, we show that there is a natural morphism from Π to the algebra of quasi-symmetric functions, under which the image of a permutation is its associated Stanley symmetric function. As an application, we use this morphism to derive some new enumerative identities. We also describe analogues of Π for the other classical types. In these cases, the relevant objects are module coalgebras rather than bialgebras, but there are again natural morphisms to the quasi-symmetric functions, under which the image of a signed permutation is the corresponding Stanley symmetric function of type B, C, or D.

We consider the problem of embedding a symmetric configuration with block size 3 in an orientable surface in such a way that the blocks of the configuration form triangular faces and there is only one extra large face. We develop a sufficient condition for such an embedding to exist given any orientation of the configuration, and show that this condition is satisfied for all configurations on up to 19 points. We also show that there exists a configuration on 21 points which is not embeddable in any orientation. As a by-product, we give a revised table of numbers of configurations, correcting the published figure for 19 points. We give a number of open questions about embeddability of configurations on larger numbers of points.

For any prime number p, positive integers m,k,n, where n satisfies gcd(p,n)=1, and for any non-zero element λ0 of the finite field Fpm of cardinality pm, we prove that any λ0pk-constacyclic code of length pkn over the finite field Fpm is monomially equivalent to a matrix-product code of a nested sequence of pk λ0-constacyclic codes with length n over Fpm. As an application, we completely determine the Hamming distances of all negacyclic codes of length 7⋅2l over F7 for any integer l≥3.

A strong edge-coloring of a graph G=(V,E) is a partition of its edge set E into induced matchings. The edge weight of a graph G is defined to be max{dG(u)+dG(v)|e=uv∈E(G)}. We study graphs with edge weight at most 7. We show that 1) every graph with edge weight at most 6 has a strong edge-coloring using at most 10 colors; and 2) every graph with edge weight at most 7 has a strong edge-coloring using at most 15 colors.

Let F be a finite field, an algebraically closed field, or the field of real numbers. Consider the vector space V=F3⊗F3 of 3 × 3 matrices over F, and let G≤PGL(V) be the setwise stabiliser of the corresponding Segre variety S3,3(F) in the projective space PG(V). The G-orbits of lines in PG(V) were determined by the first author and Sheekey as part of their classification of tensors in F2⊗V in [15]. Here we solve the related problem of classifying those line orbits that may be represented by symmetric matrices, or equivalently, of classifying the line orbits in the F-span of the Veronese variety V3(F)⊂S3,3(F) under the natural action of K=PGL(3,F). Interestingly, several of the G-orbits that have symmetric representatives split under the action of K, and in many cases this splitting depends on the characteristic of F. Although our main focus is on the case where F is a finite field, our methods (which are mostly geometric) are easily adapted to include the case where F is an algebraically closed field, or the field of real numbers. The corresponding orbit sizes and stabiliser subgroups of K are also determined in the case where F is a finite field, and connections are drawn with old work of Jordan and Dickson on the classification of pencils of conics in PG(2,F), or equivalently, of pairs of ternary quadratic forms over F.

Based on the algorithmic proof of Lovász local lemma due to Moser and Tardos, Dujmović et al. (2016) initiated the use of the so-called entropy compression method for graph coloring problems. Then, using the same approach Esperet and Parreau (2013) proved new upper bounds for several chromatic numbers, and explained how that approach could be used for many different coloring problems. Here, we follow this line of research for the particular case of acyclic coloring: we show that every graph with maximum degree Δ has acyclic chromatic number at most 32Δ43+O(Δ).

A directed triple system of order v (or, DTS (v)) is a decomposition of the complete directed graph Kv→ into transitive triples. A v-good sequencing of a DTS (v) is a permutation of the points of the design, say [x1⋯xv], such that, for every triple (x,y,z) in the design, it is not the case that x=xi, y=xj and z=xk with i

Let G be a connected simple graph with at least one edge. The hypergraph H=H(G) with the same vertex set as G whose hyper-edges are the maximal cliques of G is called the clique-hypergraph of G. A list-assignment of G is a function L which assigns to each vertex v∈V(G) a set L(v) (called the list of v). A k-list-assignment of G is a list-assignment L such that L(v) has at least k elements for every v∈V(G). For a given list assignment L, a list-coloring of H(G) is a function c:V(G)→∪vL(v) such that c(v)∈L(v) for every v∈V(G) and no hyper-edge of H(G) is monochromatic. A list-coloring of H(G) is strong if no 3-cycle of G is monochromatic. H(G) is (strongly) k-choosable if, for every k-list assignment L, there exists a (strong) list-coloring of H(G). Mohar and Sˇkrekovski proved that the clique-hypergraphs of planar graphs are strongly 4-choosable (Electr. J. Combin. 6 (1999), #R26). In this paper we give a short proof of the result and present a linear time algorithm for the strong list-4-coloring of H(G) if G is a planar graph. In addition, we prove that H(G) is strongly 4-choosable if G is a K5-minor-free graph, which is a generalization of their result.

Divisible design digraphs which can be obtained as Cayley digraphs are studied. A characterization of divisible design Cayley digraphs in terms of the generating sets is given. Further, we give several constructions of divisible design Cayley digraphs and classify divisible design Cayley digraphs on v≤27 vertices.

In this article, we generate large families of non-isomorphic and signless Laplacian cospectral graphs using partial transpose on graphs. Our constructions are significantly powerful. More than 70% of non-isomorphic signless-Laplacian cospectral graphs can be generated with partial transpose when number of vertices is ≤8. We have also produced numerous examples of non-isomorphic signless Laplacian cospectral graphs.

We say that a set system F⊆2[n] shatters a set S⊆[n] if every possible subset of S appears as the intersection of S with some element of F, and we denote by Sh(F) the family of sets shattered by F. According to the Sauer–Shelah lemma we know that in general, every set system F shatters at least |F| sets, and we call a set system shattering-extremal if |Sh(F)|=|F|. In Mészáros (2010) and Rónyai and Mészáros (2011), among other things, an algebraic characterization of shattering-extremality was given, which offered the possibility to generalize the notion to general finite point sets. Here we extend the results obtained for set systems to this more general setting, and as an application, strengthen a result of Li, Zhang and Dong from Li et al. (2012).

A segment of a tree T is a path whose end vertices have degree 1 or at least 3, while all internal vertices have degree 2. The lengths of all the segments of T form its segment sequence, in analogy to the degree sequence. For a connected graph G=(V(G),E(G)), the cover cost (resp. reverse cover cost) of a vertex u in G is defined as CCG(u)=∑v∈V(G)Huv (resp. RCG(u)=∑v∈V(G)Hvu), where Huv is the expected hitting time for random walk beginning at u to visit v. In this paper, the unique tree with the minimum cover cost and minimum reverse cover cost among all trees with given segment sequence are characterized. Furthermore, the unique tree with the maximal reverse cover cost among all trees with given segment sequence are also identified.

In this paper, we connect two types of representations of a permutation σ of the finite field Fq. One type is algebraic, in which the permutation is represented as the composition of degree-one polynomials and k copies of xq−2, for some prescribed value of k. The other type is combinatorial, in which the permutation is represented as the composition of a degree-one rational function followed by the product of k 2-cycles on P1(Fq)≔Fq∪{∞}, where each 2-cycle moves ∞. We show that, after modding out by obvious equivalences amongst the algebraic representations, then for each k there is a bijection between the algebraic representations of σ and the combinatorial representations of σ. We also prove analogous results for permutations of P1(Fq). One consequence is a new characterization of the notion of Carlitz rank of a permutation on Fq, which we use elsewhere to provide an explicit formula for the Carlitz rank. Another consequence involves a classical theorem of Carlitz, which says that if q>2 then the group of permutations of Fq is generated by the permutations induced by degree-one polynomials and xq−2. Our bijection provides a new perspective from which the two proofs of this result in the literature can be seen to arise naturally, without requiring the clever tricks that previously appeared to be needed in order to discover those proofs.

For a graph G=(V,E) and positive integer p, the exact distance-p graph G[♮p] is the graph with vertex set V and with an edge between vertices x and y if and only if x and y have distance p. Recently, there has been an effort to obtain bounds on the chromatic number χ(G[♮p]) of exact distance-p graphs for G from certain classes of graphs. In particular, if a graph G has tree-width t, it has been shown that χ(G[♮p])∈O(pt−1) for odd p, and χ(G[♮p])∈O(ptΔ(G)) for even p. We show that if G is chordal and has tree-width t, then χ(G[♮p])∈O(pt2) for odd p, and χ(G[♮p])∈O(pt2Δ(G)) for even p. If we could show that for every graph H of tree-width t there is a chordal graph G of tree-width t which contains H as an isometric subgraph (i.e., a distance preserving subgraph), then our results would extend to all graphs of tree-width t. While we cannot do this, we show that for every graph H of genus g there is a graph G which is a triangulation of genus g and contains H as an isometric subgraph.

We study the asymptotic number of certain monotonically labeled increasing trees arising from a generalized evolution process. The main difference between the presented model and the classical model of binary increasing trees is that the same label can appear in distinct branches of the tree. In the course of the analysis we develop a method to extract asymptotic information on the coefficients of purely formal power series. The method is based on an approximate Borel transform (or, more generally, Mittag-Leffler transform) which enables us to quickly guess the exponential growth rate. With this guess the sequence is then rescaled and a singularity analysis of the generating function of the scaled counting sequence yields accurate asymptotics. The actual analysis is based on differential equations and a Tauberian argument. The counting problem for trees of size n exhibits interesting asymptotics involving powers of n with irrational exponents.

In this paper, we investigate free Hermitian self-dual codes whose generator matrices are of the form [I,A+vB] over the ring F2+vF2={0,1,v,1+v} with v2=v. We use the double-circulant, the bordered double-circulant and the symmetric construction methods to obtain free Hermitian self-dual codes of even length. By describing a new shortening method over this ring, we are able to obtain Hermitian self-dual codes of odd length. Using these methods, we also obtain a number of extremal codes. We tabulate the Hermitian self-dual codes with the highest minimum weights of lengths up to 50.

For a finite group G, a positive integer λ, and subsets X,Y of G, write λG=XY if the products xy (x∈X,y∈Y), cover G precisely λ times. Such a subset Y is called a code with respect to X, and when λ=1 it is a perfect code in the Cayley graph Cay (G, X). In this paper we present various families of examples of such codes, with X closed under conjugation and Y a subgroup, in symmetric groups, and also in special linear groups SL2(q). We also propose conjectures about the existence of some much wider families.

DP-coloring (also known as correspondence coloring) is a generalization of list coloring, introduced by Dvořák and Postle in 2017. It is well-known that there are non-4-choosable planar graphs. Much attention has recently been put on sufficient conditions for planar graphs to be DP- 4-colorable. In particular, for each k∈{3,4,5,6}, every planar graph without k-cycles is DP-4-colorable. In this paper, we prove that every planar graph without 7-cycles and butterflies is DP-4-colorable. Our proof can be easily modified to prove other sufficient conditions that forbid clusters formed by many triangles.

It was proved in Feng et al. (2015) that a cubic symmetric graph with a solvable automorphism group is either a Cayley graph or a 2-regular graph of type 22, that is, a graph with no automorphism of order 2 interchanging two adjacent vertices. In this paper an infinite family of non-Cayley cubic 2-regular graphs of type 22 with a solvable automorphism group is constructed, and the smallest graph has order 6174. This answers a question posed by Estélyi and Pisanski in 2016. Moreover, it includes a subfamily of graphs which are connected 2-regular covers of the Pappus graph with covering transformation group Zp3, and these graphs were missed in Oh (2009).

A set S of vertices in a graph G is a dominating set if every vertex not in S is adjacent to a vertex in S. If, in addition, S is an independent set, then S is an independent dominating set. The domination number γ(G) of G is the minimum cardinality of a dominating set in G, while the independent domination number i(G) of G is the minimum cardinality of an independent dominating set in G. It is known (Goddard et al., 2012) that if G is a connected 3-regular graph, then i(G)∕γ(G)≤3∕2, with equality if and only if G=K3,3. In this paper, we extend this result to graphs of larger regularity and show that if k∈{4,5,6} and G is a connected k-regular graph, then i(G)∕γ(G)≤k∕2, with equality if and only if G=Kk,k.

The main aim of this paper is to study 2-(v,4,λ) designs admitting a block-transitive automorphism group G. We indeed determine all such designs when G is point-imprimitive or point-primitive group of product type.

The prism over a graph G is the cartesian product G□K2. It is known that the property of having a Hamiltonian prism (prism-Hamiltonicity) is stronger than that of having a 2-walk (spanning closed walk using every vertex at most twice) and weaker than that of having a Hamilton path. For a graph G, it is known that α(G)≤2κ(G), where α(G) is the independence number and κ(G) is the connectivity, implies existence of a 2-walk in G, and the bound is sharp. West asked for a bound on α(G) in terms of κ(G) guaranteeing prism-Hamiltonicity. In this paper we answer this question and prove that α(G)≤2κ(G) implies the stronger condition, prism-Hamiltonicity of G.

In this note, we give a simple extension map from partitions of subsets of [n] to partitions of [n+1], which sends δ-distant k-crossings to (δ+1)-distant k-crossings (and similarly for nestings). This map provides a combinatorial proof of the fact that the numbers of enhanced, classical, and 2-distant k-noncrossing partitions are each related to the next via the binomial transform. Our work resolves a recent conjecture of Zhicong Lin and generalizes earlier reduction identities for partitions.

We consider realizations of a graph in the plane such that the distances between adjacent vertices satisfy the constraints given by an edge labeling. If there are infinitely many such realizations, counted modulo rigid motions, the labeling is called flexible. The existence of a flexible labeling, possibly non-generic, has been characterized combinatorially by the existence of a so called NAC-coloring. Nevertheless, the corresponding realizations are often non-injective. In this paper, we focus on flexible labelings with infinitely many injective realizations. We provide a necessary combinatorial condition on existence of such a labeling based also on NAC-colorings of the graph. By introducing new tools for the construction of such labelings, we show that the necessary condition is also sufficient up to 8 vertices, but this is not true in general for more vertices.

A Steiner quadruple system of order v (SQS(v)) is said to be have an almost spanning block design and denoted by 1-AFSQS(v) if it contains a subdesign S(2,4,v−1). In 1992, Hartman and Phelps posed a problem: Show that there exists a 1-AFSQS(v) for each v≡2(mod12). In this paper, we prove that the necessary condition for the existence of a 1-AFSQS(v) is also sufficient with a definite exception v=14 and possible exceptions v∈{86,206,374,398}.

The weak r-coloring numbers wcolr(G) of a graph G were introduced by the first two authors as a generalization of the usual coloring number col(G), and have since found interesting theoretical and algorithmic applications. This has motivated researchers to establish strong bounds on these parameters for various classes of graphs. Let Gp denote the pth power of G. We show that, all integers p>0 and Δ≥3 and graphs G with Δ(G)≤Δ satisfy col(Gp)∈O(p⋅wcol⌈p∕2⌉(G)(Δ−1)⌊p∕2⌋); for fixed tree width or fixed genus the ratio between this upper bound and worst case lower bounds is polynomial in p. For the square of graphs G, we also show that, if the maximum average degree 2k−2

The Poincaré polynomial of the complement of an arrangements in a non compact group G is a specialization of the G-Tutte polynomial associated with the arrangement. In this article we show two unimodular elliptic arrangements (built up from two graphs) with the same Tutte polynomial, having different Betti numbers.

The 2-domination number γ2(G) of a graph G is the minimum cardinality of a set S⊆V(G) such that every vertex from V(G)∖S is adjacent to at least two vertices in S. The annihilation number a(G) is the largest integer k such that the sum of the first k terms of the non-decreasing degree sequence of G is at most the number of its edges. It was conjectured that γ2(G)≤a(G)+1 holds for every connected graph G. The conjecture was earlier confirmed, in particular, for graphs of minimum degree 3, for trees, and for block graphs. In this paper, we disprove the conjecture by proving that the 2-domination number can be arbitrarily larger than the annihilation number. On the positive side we prove the conjectured bound for a large subclass of bipartite, connected cacti, thus generalizing a result of Jakovac from Jakovac (2019).

Bachet’s game is a variant of the game of Nim. There are n objects in one pile. Two players take turns to remove any positive number of objects not exceeding some fixed number m. The player who takes the last object loses. We consider a variant of Bachet’s game in which each move is a lottery over set {1,2,…,m}. The outcome of a lottery is the number of objects that player takes from the pile. We show that under some nondegenericity assumptions on the set of available lotteries the probability that the first player wins in subgame perfect Nash equilibrium converges to 1∕2 as n tends to infinity.