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  • Fractional revival and association schemes
    Discret. Math. (IF 0.77) Pub Date : 2020-07-01
    Ada Chan; Gabriel Coutinho; Christino Tamon; Luc Vinet; Hanmeng Zhan

    Fractional revival occurs between two vertices in a graph if a continuous-time quantum walk unitarily maps the characteristic vector of one vertex to a superposition of the characteristic vectors of the two vertices. This phenomenon is relevant in quantum information in particular for entanglement generation in spin networks. We study fractional revival in graphs whose adjacency matrices belong to

    更新日期:2020-07-01
  • Equitable 2-partitions of the Hamming graphs with the second eigenvalue
    Discret. Math. (IF 0.77) Pub Date : 2020-06-30
    Ivan Mogilnykh; Alexandr Valyuzhenich

    The eigenvalues of the Hamming graph H(n,q) are known to be λi(n,q)=(q−1)n−qi, 0≤i≤n. The characterization of equitable 2-partitions of the Hamming graphs H(n,q) with eigenvalue λ1(n,q) was obtained by Meyerowitz (2003). We study the equitable 2-partitions of H(n,q) with eigenvalue λ2(n,q). We show that these partitions are reduced to equitable 2-partitions of H(3,q) with eigenvalue λ2(3,q) with the

    更新日期:2020-06-30
  • Pattern-avoiding permutation powers
    Discret. Math. (IF 0.77) Pub Date : 2020-06-30
    Amanda Burcroff; Colin Defant

    Recently, Bóna and Smith defined strong pattern avoidance, saying that a permutation π strongly avoids a pattern τ if π and π2 both avoid τ. They conjectured that for every positive integer k, there is a permutation in Sk3 that strongly avoids 123⋯(k+1). We use the Robinson–Schensted–Knuth correspondence to settle this conjecture, showing that the number of such permutations is at least kk3∕2+O(k3∕logk)

    更新日期:2020-06-30
  • Restricted extension of sparse partial edge colorings of hypercubes
    Discret. Math. (IF 0.77) Pub Date : 2020-06-26
    Carl Johan Casselgren; Klas Markström; Lan Anh Pham

    We consider the following type of question: Given a partial proper d-edge coloring of the d-dimensional hypercube Qd, and lists of allowed colors for the non-colored edges of Qd, can we extend the partial coloring to a proper d-edge coloring using only colors from the lists? We prove that this question has a positive answer in the case when both the partial coloring and the color lists satisfy certain

    更新日期:2020-06-27
  • The geometry connectivity of hypergraphs
    Discret. Math. (IF 0.77) Pub Date : 2020-06-27
    Chunli Deng; Lizhu Sun; Changjiang Bu

    Let G be a k-uniform hypergraph, and LG be its Laplacian tensor. The β(G) denotes the maximum number of linearly independent nonnegative eigenvectors of LG corresponding to the eigenvalue 0. In this paper, β(G) is called the geometry connectivity of G. We show that the number of connected components of G equals the geometry connectivity β(G).

    更新日期:2020-06-27
  • A new bound on the acyclic edge chromatic number
    Discret. Math. (IF 0.77) Pub Date : 2020-06-27
    Paula M.S. Fialho; Bernardo N.B. de Lima; Aldo Procacci

    In this note we obtain a new bound for the acyclic edge chromatic number a′(G) of a graph G with maximum degree Δ proving that a′(G)≤3.569(Δ−1). To get this result we revisit and slightly modify the method described in Kirousis et al. (2017).

    更新日期:2020-06-27
  • Maximal sets of Hamilton cycles in Knr;λ1,λ2
    Discret. Math. (IF 0.77) Pub Date : 2020-06-10
    M. Demir; C.A. Rodger

    Let Knr;λ1,λ2 be the r-partite multigraph in which each part has size n, where two vertices in the same part or different parts are joined by exactly λ1 edges or λ2 edges, respectively. It is proved that there exists a maximal set of t edge-disjoint Hamilton cycles in Knr;λ1,λ2 for λ2n⌊r+34⌋≤t≤min{⌊λ2n2(r−1)2⌋,⌊λ1(n−1)+λ2n(r−1)2⌋}, the upper bound being best possible. The results proved make use of

    更新日期:2020-06-26
  • Efficient universal cycle constructions for weak orders
    Discret. Math. (IF 0.77) Pub Date : 2020-06-25
    Joe Sawada; Dennis Wong

    A weak order is a way n competitors can rank in an event, where ties are allowed. A weak order can also be thought of as a relation on {1,2,…,n} that is transitive and complete. We provide the first efficient algorithms to construct universal cycles for weak orders by considering both rank and height representations. Each algorithm constructs the universal cycles in O(n) time per symbol using O(n)

    更新日期:2020-06-25
  • Most binary matrices have no small defining set
    Discret. Math. (IF 0.77) Pub Date : 2020-06-25
    Carly Bodkin; Anita Liebenau; Ian M. Wanless

    Consider a matrix M chosen uniformly at random from a class of m×n matrices of zeros and ones with prescribed row and column sums. A partially filled matrix D is a defining set for M if M is the unique member of its class that contains the entries in D. The size of a defining set is the number of filled entries. A critical set is a defining set for which the removal of any entry stops it being a defining

    更新日期:2020-06-25
  • An improved Moore bound and some new optimal families of mixed Abelian Cayley graphs
    Discret. Math. (IF 0.77) Pub Date : 2020-06-25
    C. Dalfó; M.A. Fiol; N. López; J. Ryan

    We consider the case in which mixed graphs (with both directed and undirected edges) are Cayley graphs of Abelian groups. In this case, some Moore bounds were derived for the maximum number of vertices that such graphs can attain. We first show these bounds can be improved if we know more details about the order of some elements of the generating set. Based on these improvements, we present some new

    更新日期:2020-06-25
  • Generating Gray codes for weak orders in constant amortized time
    Discret. Math. (IF 0.77) Pub Date : 2020-06-23
    Marsden Jacques; Dennis Wong; Kyounga Woo

    A weak order is a way to rank n objects where ties are allowed. In this paper, we develop a generic framework to generate Gray codes for weak orders. We then describe a simple algorithm based on the framework that generates cyclic 2-Gray codes for weak orders in constant amortized time per string. This is the first known cyclic 2-Gray codes for weak orders. The framework can easily be modified to generate

    更新日期:2020-06-24
  • Recurrence along directions in multidimensional words
    Discret. Math. (IF 0.77) Pub Date : 2020-06-24
    Émilie Charlier; Svetlana Puzynina; Élise Vandomme

    In this paper we introduce and study new notions of uniform recurrence in multidimensional words. A d-dimensional word is called uniformly recurrent if for all (s1,…,sd)∈Nd there exists n∈N such that each block of size (n,…,n) contains the prefix of size (s1,…,sd). We are interested in a modification of this property. Namely, we ask that for each rational direction (q1,…,qd), each rectangular prefix

    更新日期:2020-06-24
  • On Haar digraphical representations of groups
    Discret. Math. (IF 0.77) Pub Date : 2020-06-23
    Jia-Li Du; Yan-Quan Feng; Pablo Spiga

    In this paper we extend the notion of digraphical regular representations in the context of Haar digraphs. Given a group G, a Haar digraph Γ over G is a bipartite digraph having a bipartition {X,Y} such that G is a group of automorphisms of Γ acting regularly on X and on Y. We say that G admits a Haar digraphical representation (HDR for short), if there exists a Haar digraph over G such that its automorphism

    更新日期:2020-06-23
  • On Voloshin colorings in 3-hypergraph designs
    Discret. Math. (IF 0.77) Pub Date : 2020-06-20
    Paola Bonacini; Lucia Marino

    Let P(3)(2,4) be the hypergraph having vertex set {1,2,3,4} and edge set {{1,2,3},{1,2,4}}. In this paper we consider vertex colorings of P(3)(2,4)-designs in such a way any block is neither monochromatic nor polychromatic. We find bounds for the upper and lower chromatic numbers, showing also that these bounds are sharp. Indeed, for any admissible v there exists a P(3)(2,4)-design of order v having

    更新日期:2020-06-22
  • The proportion of trees that are linear
    Discret. Math. (IF 0.77) Pub Date : 2020-06-18
    Tanay Wakhare; Eric Wityk; Charles R. Johnson

    We study several enumeration problems connected to linear trees, a broad class which includes stars, paths, generalized stars, and caterpillars. We provide generating functions for counting the number of linear trees on n vertices, characterize the asymptotic growth rate of the number of nonisomorphic linear trees, and show that the distribution of k-linear trees on n vertices follows a central limit

    更新日期:2020-06-18
  • Not all partial cubes are Θ-graceful
    Discret. Math. (IF 0.77) Pub Date : 2020-06-17
    Nathann Cohen; Matjaž Kovše

    It is shown that the graph obtained by merging two vertices of two 4-cycles is not a Θ-graceful partial cube, thus answering in the negative a question by Brešar and Klavžar from Brešar and Klavžar (2006), who asked whether every partial cube is Θ-graceful.

    更新日期:2020-06-17
  • The layer number of α-evenly distributed point sets
    Discret. Math. (IF 0.77) Pub Date : 2020-06-16
    Ilkyoo Choi; Weonyoung Joo; Minki Kim

    For a finite point set in Rd, we consider a peeling process where the vertices of the convex hull are removed at each step. The layer number L(X) of a given point set X is defined as the number of steps of the peeling process in order to delete all points in X. It is known that if X is a set of random points in Rd, then the expectation of L(X) is Θ(|X|2∕(d+1)), and recently it was shown that if X is

    更新日期:2020-06-16
  • On the existence of dense substructures in finite groups
    Discret. Math. (IF 0.77) Pub Date : 2020-06-13
    Ching Wong

    Let k≥6. We prove that any large enough finite group G contains k elements which span quadratically many triples of the form (a,b,ab)∈S×G, given any dense set S⊆G×G. The quadratic bound is asymptotically optimal. In particular, this provides an elementary proof of a conjecture of Brown, Erdős and Sós, in finite groups and when k is large.

    更新日期:2020-06-13
  • Constructing cospectral bipartite graphs
    Discret. Math. (IF 0.77) Pub Date : 2020-06-13
    Yizhe Ji; Shicai Gong; Wei Wang

    Given a bipartite graph G=(V,E) with bipartition V=A∪B, where |A|=|B|. We present a new construction of pairs of cospectral bipartite graphs by “rotating” G in two different ways. More precisely, let l and k be two positive integers with l≠k. Take l copies of A (resp. B) and k copies of B (resp. A), say A1,A2,…,Al and B1,B2,…,Bk (resp. B1′,B2′,…,Bl′ and A1′,A2′,…,Ak′). Let Γ (resp. Γ′) be the graph

    更新日期:2020-06-13
  • Vertex-faithful regular polyhedra
    Discret. Math. (IF 0.77) Pub Date : 2020-06-12
    Gabe Cunningham; Mark Mixer

    We study the abstract regular polyhedra with automorphism groups that act faithfully on their vertices, and show that each non-flat abstract regular polyhedron covers a “vertex-faithful” polyhedron with the same number of vertices. We then use this result and earlier work on flat polyhedra to study abstract regular polyhedra based on the size of their vertex set. In particular, we classify all regular

    更新日期:2020-06-12
  • Completing the spectrum for three designs related to the graph K4−e
    Discret. Math. (IF 0.77) Pub Date : 2020-06-12
    C. Shen; Y. Zheng; H. Cao

    In this paper, we study three designs related to the graph K4−e. We complete the determination of the spectrum of uniform frames, maximum resolvable packings and minimum resolvable coverings of K4−e.

    更新日期:2020-06-12
  • Minimal ordered Ramsey graphs
    Discret. Math. (IF 0.77) Pub Date : 2020-06-12
    Jonathan Rollin

    An ordered graph is a graph equipped with a linear ordering of its vertex set. A pair of ordered graphs is Ramsey finite if it has only finitely many minimal ordered Ramsey graphs and Ramsey infinite otherwise. Here an ordered graph F is an ordered Ramsey graph of a pair (H,H′) of ordered graphs if for any coloring of the edges of F in colors red and blue there is either a copy of H with all edges

    更新日期:2020-06-12
  • Remarks on the distribution of colors in Gallai colorings
    Discret. Math. (IF 0.77) Pub Date : 2020-06-11
    Joseph Feffer; Yaoying Fu; Jun Yan

    A Gallai coloring of a complete graph Kn is an edge coloring without triangles colored with three different colors. A sequence e1≥⋯≥ek of positive integers is an (n,k)-sequence if ∑i=1kei=n2. An (n,k)-sequence is a G-sequence if there is a Gallai coloring of Kn with k colors such that there are ei edges of color i for all i,1≤i≤k. Gyárfás, Pálvölgyi, Patkós and Wales proved that for any integer k≥3

    更新日期:2020-06-11
  • Computational results and new bounds for the circular flow number of snarks
    Discret. Math. (IF 0.77) Pub Date : 2020-06-11
    Jan Goedgebeur; Davide Mattiolo; Giuseppe Mazzuoccolo

    It is well known that the circular flow number of a bridgeless cubic graph can be computed in terms of certain partitions of its vertex set with prescribed properties. In the present paper, we first study some of these properties that turn out to be useful in order to make an efficient and practical implementation of an algorithm for the computation of the circular flow number of a bridgeless cubic

    更新日期:2020-06-11
  • Spectral extrema of graphs: Forbidden hexagon
    Discret. Math. (IF 0.77) Pub Date : 2020-06-10
    Mingqing Zhai; Huiqiu Lin

    To determine the Turán numbers of even cycles is a central problem of extremal graph theory. Even for C6, the Turán number is still open. Till now, the best known upper bound is given by Füredi, Naor and Verstraëte [On the Turán number for the hexagon, Advances in Math.]. In 2010, Nikiforov posed a spectral version of extremal graph theory problem: what is the maximum spectral radius ρ of an H-free

    更新日期:2020-06-10
  • Colourings of oriented connected cubic graphs
    Discret. Math. (IF 0.77) Pub Date : 2020-06-10
    Christopher Duffy

    In this note we show every orientation of a connected cubic graph admits an oriented 8-colouring. This lowers the best-known upper bound for the chromatic number of the family of orientations of connected cubic graphs. We further show that every such oriented graph admits a 2-dipath 7-colouring. These results imply that either the oriented chromatic number for the family of orientations of connected

    更新日期:2020-06-10
  • Circular embeddability of isometric words
    Discret. Math. (IF 0.77) Pub Date : 2020-06-10
    Jianxin Wei; Yujun Yang; Guangfu Wang

    Let f be a binary word and n≥1. Then the generalized Lucas cube Qn(f↽) is the graph obtained from the n-cube Qn by removing all vertices that have a circulation containing f as a factor. Ilić, Klavžar and Rho solved the question for which f and n, Qn(f↽) is an isometric subgraph of Qn for all binary words of length at most five. This question is further studied in this paper. For an isometric word

    更新日期:2020-06-10
  • Inducibility of directed paths
    Discret. Math. (IF 0.77) Pub Date : 2020-06-10
    Ilkyoo Choi; Bernard Lidický; Florian Pfender

    A long standing open problem in extremal graph theory is to describe all graphs that maximize the number of induced copies of a path on four vertices. The character of the problem changes in the setting of oriented graphs, and becomes more tractable. Here we resolve this problem in the setting of oriented graphs without transitive triangles.

    更新日期:2020-06-10
  • A probabilistic variant of Sperner ’s theorem and of maximal r-cover free families
    Discret. Math. (IF 0.77) Pub Date : 2020-06-09
    Noga Alon; Shoni Gilboa; Shay Gueron

    A family of sets is called r-cover free if no set in the family is contained in the union of r (or less) other sets in the family. A 1-cover free family is simply an antichain with respect to set inclusion. Thus, Sperner’s classical result determines the maximal cardinality of a 1-cover free family of subsets of an n-element set. Estimating the maximal cardinality of an r-cover free family of subsets

    更新日期:2020-06-09
  • On varieties defined by large sets of quadrics and their application to error-correcting codes
    Discret. Math. (IF 0.77) Pub Date : 2020-06-06
    Simeon Ball; Valentina Pepe

    Let U be a (k−12−1)-dimensional subspace of quadratic forms defined on Fk with the property that U does not contain any reducible quadratic form. Let V(U) be the points of PG(k−1,F) which are zeros of all quadratic forms in U. We will prove that if there is a group G which fixes U and no line of PG(k−1,F) and V(U) spans PG(k−1,F) then any hyperplane of PG(k−1,F) is incident with at most k points of

    更新日期:2020-06-06
  • Signless Laplacian spectral radius and fractional matchings in graphs
    Discret. Math. (IF 0.77) Pub Date : 2020-06-05
    Yingui Pan; Jianping Li; Wei Zhao

    A fractional matching of a graph G is a function f giving each edge a number in [0,1] so that ∑e∈Γ(v)f(e)≤1 for each vertex v∈V(G), where Γ(v) is the set of edges incident to v. The fractional matching number of G, written α∗′(G), is the maximum value of ∑e∈E(G)f(e) over all fractional matchings. In this paper, we investigate the relations between the fractional matching number and the signless Laplacian

    更新日期:2020-06-05
  • Adjacent vertex distinguishing total coloring of planar graphs with maximum degree 8
    Discret. Math. (IF 0.77) Pub Date : 2020-06-05
    Yulin Chang; Jie Hu; Guanghui Wang; Xiaowei Yu

    A (proper) total-k-coloring ϕ:V(G)∪E(G)→{1,2,…,k} is called adjacent vertex distinguishing if Cϕ(u)≠Cϕ(v) for each edge uv∈E(G), where Cϕ(u) is the set of the color of u and the colors of all edges incident with u. We use χa′′(G) to denote the smallest value k in such a coloring of G. Zhang et al. (2005) first introduced this coloring and conjectured that χa′′(G)≤Δ(G)+3 for every simple graph G. It

    更新日期:2020-06-05
  • Some q-series identities extending work of Andrews, Crippa, and Simon on sums of divisors functions
    Discret. Math. (IF 0.77) Pub Date : 2020-06-05
    Kathrin Bringmann; Chris Jennings-Shaffer

    In this article we extend a theorem of Andrews, Crippa, and Simon on the asymptotic behavior of polynomials defined by a general class of recursive equations. Here the polynomials are in the variable q, and the recursive definition at step n introduces a polynomial in n. Our extension replaces the polynomial in n with either an exponential or periodic function of n.

    更新日期:2020-06-05
  • A note on packing two copies of a tree into a graph with small maximum degree
    Discret. Math. (IF 0.77) Pub Date : 2020-06-04
    Agnieszka Görlich; Mariusz Woźniak

    Let T be a non-star tree of order n and maximum degree Δ. Hong Wang showed that there is a packing σ such that the degree of every vertex v in T⊕σ(T) does not exceed Δ(T)+2. In this note we improve this result in two directions. First, we show that with only one exception, there exists a packing σ such that Δ(T⊕σ(T))≤Δ(T)+1. Moreover, this bound is, in a sense, uniform.

    更新日期:2020-06-04
  • Bounded degree complexes of forests
    Discret. Math. (IF 0.77) Pub Date : 2020-06-02
    Anurag Singh

    Given an arbitrary sequence of non-negative integers λ→=(λ1,…,λn) and a graph G with vertex set {v1,…,vn}, the bounded degree complex, denoted as BDλ→(G), is a simplicial complex whose faces are the subsets H⊆E(G) such that for each i∈{1,…,n}, the degree of vertex vi in the induced subgraph G[H] is at most λi. When λi=k for all i, the bounded degree complex BDλ→(G) is called the k-matching complex

    更新日期:2020-06-02
  • Turán problems for vertex-disjoint cliques in multi-partite hypergraphs
    Discret. Math. (IF 0.77) Pub Date : 2020-06-01
    Erica L.L. Liu; Jian Wang

    For two s-uniform hypergraphs H and F, the Turán number exs(H,F) is the maximum number of edges in an F-free subgraph of H. Let s,r,k,n1,…,nr be integers satisfying 2≤s≤r and n1≤n2≤⋯≤nr. De Silva, Heysse and Young determined ex2(Kn1,…,nr,kK2) and De Silva, Heysse, Kapilow, Schenfisch and Young determined ex2(Kn1,…,nr,kKr). In this paper, as a generalization of these results, we consider three Turán-type

    更新日期:2020-06-01
  • An equivalent formulation of chromatic quasi-polynomials
    Discret. Math. (IF 0.77) Pub Date : 2020-06-01
    Tan Nhat Tran

    Given a central integral arrangement, the reduction of the arrangement modulo a positive integer q gives rise to a subgroup arrangement in Zqℓ. Kamiya et al. (2008) introduced the notion of characteristic quasi-polynomial, which enumerates the cardinality of the complement of this subgroup arrangement. Chen and Wang (2012) found a similar but more general setting that replacing the integral arrangement

    更新日期:2020-06-01
  • On eigenvalue multiplicity in signed graphs
    Discret. Math. (IF 0.77) Pub Date : 2020-05-28
    Farzaneh Ramezani; Peter Rowlinson; Zoran Stanić

    Given a signed graph Σ with n vertices, let μ be an eigenvalue of Σ, and let t be the codimension of the corresponding eigenspace. We prove that n≤t+23whenever μ∉{0,1,−1}. We show that this bound is sharp by providing examples of signed graphs in which it is attained. We also discuss particular cases in which the bound can be decreased.

    更新日期:2020-05-28
  • Graphs without two vertex-disjoint S-cycles
    Discret. Math. (IF 0.77) Pub Date : 2020-05-27
    Minjeong Kang; O-joung Kwon; Myounghwan Lee

    Lovász (1965) characterized graphs without two vertex-disjoint cycles, which implies that such graphs have at most three vertices hitting all cycles. In this paper, we ask whether such a small hitting set exists for S-cycles, when a graph has no two vertex-disjoint S-cycles. For a graph G and a vertex set S of G, an S-cycle is a cycle containing a vertex of S. We provide an example G on 21 vertices

    更新日期:2020-05-27
  • Combinatorial reciprocity for the chromatic polynomial and the chromatic symmetric function
    Discret. Math. (IF 0.77) Pub Date : 2020-05-27
    Olivier Bernardi; Philippe Nadeau

    Let G be a graph, and let χG be its chromatic polynomial. For any non-negative integers i,j, we give an interpretation for the evaluation χG(i)(−j) in terms of acyclic orientations. This recovers the classical interpretations due to Stanley and to Greene and Zaslavsky respectively in the cases i=0 and j=0. We also give symmetric function refinements of our interpretations, and some extensions. The

    更新日期:2020-05-27
  • Ambiguous reconstructions of hv-convex polyominoes
    Discret. Math. (IF 0.77) Pub Date : 2020-05-27
    Paolo Dulio; Andrea Frosini; Silvia M.C. Pagani; Simone Rinaldi

    The core problem of discrete tomography, i.e., the faithful reconstruction of an unknown discrete object from projections along a given set of directions, is ill-posed in general. When further constraints are imposed on the object or on the employed directions, uniqueness of the reconstruction can be obtained. It is the case, for example, of convex lattice sets in Z2, for which a theorem by Gardner

    更新日期:2020-05-27
  • On the number of Laplacian eigenvalues of trees less than the average degree
    Discret. Math. (IF 0.77) Pub Date : 2020-05-23
    Changhyon Sin

    Laplacian eigenvalues of trees have been studied by many researchers. They are closely related to the diameter, domination number, matching number and Laplacian energy of trees. The goal of this paper is to prove a conjecture on the number of Laplacian eigenvalues of trees that are less than the average degree.

    更新日期:2020-05-23
  • Another simple proof of Graham and Pollak’s theorem
    Discret. Math. (IF 0.77) Pub Date : 2020-05-23
    Zhibin Du; Jean Yeh

    Let T be a tree with n vertices, and Dn be the distance matrix of T. Graham and Pollak (1971) discovered an elegant formula for the determinant of Dn: det(Dn)=−(n−1)(−2)n−2. It is surprising that it depends only on the order of T, not on the specific structure of T. By virtue of the classical Dodgson’s determinant-evaluation rule, Yan and Yeh (2006) presented a simple proof of the formula above. In

    更新日期:2020-05-23
  • Hyperplane sections of determinantal varieties over finite fields and linear codes
    Discret. Math. (IF 0.77) Pub Date : 2020-05-22
    Peter Beelen; Sudhir R. Ghorpade

    We determine the number of Fq-rational points of hyperplane sections of classical determinantal varieties defined by the vanishing of minors of a fixed size of a generic matrix, and identify the hyperplane sections giving the maximum number of Fq-rational points. Further we consider similar questions for sections by linear subvarieties of a fixed codimension in the ambient projective space. This is

    更新日期:2020-05-22
  • On symmetries of edge and vertex colourings of graphs
    Discret. Math. (IF 0.77) Pub Date : 2020-05-22
    Florian Lehner; Simon M. Smith

    Let c and c′ be edge or vertex colourings of a graph G. The stabiliser of c is the set of automorphisms of G that preserve the colouring. We say that c′ is less symmetric than c if the stabiliser of c′ is contained in the stabiliser of c. We show that if G is not a bicentred tree, then for every vertex colouring of G there is a less symmetric edge colouring with the same number of colours. On the other

    更新日期:2020-05-22
  • Flag-transitive 2-(v,k,λ) symmetric designs with λ≥(k,λ)2 and alternating socle
    Discret. Math. (IF 0.77) Pub Date : 2020-05-22
    Yajie Wang; Shenglin Zhou

    Let D be 2-(v,k,λ) symmetric designs, let G≤Aut(D) be flag-transitive. In this paper, all symmetric designs D with λ≥(k,λ)2 and Soc(G)=An for n≥5 are classified.

    更新日期:2020-05-22
  • Pattern distributions in Dyck paths with a first return decomposition constrained by height
    Discret. Math. (IF 0.77) Pub Date : 2020-05-22
    Jean-Luc Baril; Richard Genestier; Sergey Kirgizov

    We provide generating functions for the popularity and the distribution of patterns of length at most three over the set of Dyck paths having a first return decomposition constrained by height.

    更新日期:2020-05-22
  • On simple 3-designs having 2-resolutions
    Discret. Math. (IF 0.77) Pub Date : 2020-05-21
    Tran van Trung

    Simple nontrivial s-resolvable t-designs for t≥3 and 1

    更新日期:2020-05-21
  • MRD codes with maximum idealizers
    Discret. Math. (IF 0.77) Pub Date : 2020-05-20
    Bence Csajbók; Giuseppe Marino; Olga Polverino; Yue Zhou

    Left and right idealizers are important invariants of linear rank-distance codes. In the case of maximum rank-distance (MRD for short) codes in Fqn×n the idealizers have been proved to be isomorphic to finite fields of size at most qn. Up to now, the only known MRD codes with maximum left and right idealizers are generalized Gabidulin codes, which were first constructed in 1978 by Delsarte and later

    更新日期:2020-05-20
  • The least H-eigenvalue of signless Laplacian of non-odd-bipartite hypergraphs
    Discret. Math. (IF 0.77) Pub Date : 2020-05-20
    Jiang-Chao Wan; Yi-Zheng Fan; Yi Wang

    Let G be a connected non-odd-bipartite hypergraph with even uniformity. The least H-eigenvalue of the signless Laplacian tensor of G is simply called the least eigenvalue of G and the corresponding H-eigenvectors are called the first eigenvectors of G. In this paper we give some numerical and structural properties about the first eigenvectors of G which contains an odd-bipartite branch, and investigate

    更新日期:2020-05-20
  • Number of distinguishing colorings and partitions
    Discret. Math. (IF 0.77) Pub Date : 2020-05-20
    Bahman Ahmadi; Fatemeh Alinaghipour; Mohammad Hadi Shekarriz

    A vertex coloring of a graph G is called distinguishing (or symmetry breaking) if no non-identity automorphism of G preserves it, and the distinguishing number, shown by D(G), is the smallest number of colors required for such a coloring. This paper is about counting non-equivalent distinguishing colorings of graphs with k colors. A parameter, namely Φk(G), which is the number of non-equivalent distinguishing

    更新日期:2020-05-20
  • On the cover Ramsey number of Berge hypergraphs
    Discret. Math. (IF 0.77) Pub Date : 2020-05-20
    Linyuan Lu; Zhiyu Wang

    For a fixed set of positive integers R, we say H is an R-uniform hypergraph, or R-graph, if the cardinality of each edge belongs to R. An R-graph H is covering if every vertex pair of H is contained in some hyperedge. For a graph G=(V,E), a hypergraph H is called a Berge-G, denoted by BG, if there is an injection i:V(G)→V(H) and a bijection f:E(G)→E(H) such that for all e=uv∈E(G), we have {i(u),i(v)}⊆f(e)

    更新日期:2020-05-20
  • Polarized relations at singulars over successors
    Discret. Math. (IF 0.77) Pub Date : 2020-05-20
    Shimon Garti

    Erdős, Hajnal and Rado asked whether ℵω1ℵ2→ℵω1ℵ02 and whether ℵω1ℵ2→ℵω1ℵ12. We shall prove that both relations are independent over ZFC. We shall also prove that μℵ2→μℵ22 is independent over ZF for some μ>cf(μ)=ω1.

    更新日期:2020-05-20
  • Maker–Breaker domination game
    Discret. Math. (IF 0.77) Pub Date : 2020-05-19
    Eric Duchêne; Valentin Gledel; Aline Parreau; Gabriel Renault

    We introduce the Maker–Breaker domination game, a two player game on a graph. At his turn, the first player, Dominator, selects a vertex in order to dominate the graph while the other player, Staller, forbids a vertex to Dominator in order to prevent him to reach his goal. Both players play alternately without missing their turn. This game is a particular instance of the so-called Maker–Breaker games

    更新日期:2020-05-19
  • On quaternary Goppa codes
    Discret. Math. (IF 0.77) Pub Date : 2020-05-19
    Markel Epelde; Xabier Larrucea; Ignacio F. Rúa

    In Goppa (1970), V. D. Goppa presents a new family of linear codes over finite fields. This family, now known as Goppa codes, has been studied for nearly 50 years, including efficient decoding algorithms (Patterson, 1975) and its applications in cryptography (McEliece, 1978). Moreover, in Andrade et al. (2005), de Andrade and Palazzo present a generalization of Goppa codes to finite rings. Motivated

    更新日期:2020-05-19
  • Distinct distances in planar point sets with forbidden 4-point patterns
    Discret. Math. (IF 0.77) Pub Date : 2020-05-19
    Adrian Dumitrescu

    A point set in a metric space is said to have property Φ(4,5) if every 4 elements determine at least 5 distinct distances. According to an old conjecture of Erdős (1986 or earlier), a set of n points in the Euclidean plane satisfying this restriction determines Ω(n2) distinct distances. This property (restriction) is shown to be equivalent to forbidding eight 4-element patterns, πi, i=1,…,8 (described

    更新日期:2020-05-19
  • On certain properties of partitions of Zm with the same representation function
    Discret. Math. (IF 0.77) Pub Date : 2020-05-19
    Shi-Qiang Chen; Xiao-Hui Yan

    For a given set S⊆Zm and n¯∈Zm, RS(n¯) is defined as the number of solutions of the equation n¯=s¯+s′¯ with unordered pair (s¯,s′¯)∈S2 and s¯≠s′¯. In this paper, we prove that if m is an even positive integer and not a power of 2 then there exist two distinct sets A,B with A∪B=Zm, |A∩B|=2, B≠A+m∕2¯ such that RA(n¯)=RB(n¯) for all n¯∈Zm. If m is a power of 2, A∪B=Zm, |A∩B|=2, then RA(n¯)=RB(n¯) for

    更新日期:2020-05-19
  • Restricted permutations refined by number of crossings and nestings
    Discret. Math. (IF 0.77) Pub Date : 2020-05-19
    Paul M. Rakotomamonjy

    Let st={st1,…,stk} be a set of k statistics on permutations with k≥1. We say that two given subsets of permutations T and T′ are st-Wilf-equivalent if the joint distributions of all statistics in st over the sets of T-avoiding permutations Sn(T) and T′-avoiding permutations Sn(T′) are the same. The main purpose of this paper is the (cr,nes)-Wilf-equivalence classes for all single patterns in S3, where

    更新日期:2020-05-19
  • Eulerian and even-face ribbon graph minors
    Discret. Math. (IF 0.77) Pub Date : 2020-05-19
    Metrose Metsidik; Xian’an Jin

    In this paper, we introduce Eulerian and even-face ribbon graph minors. These minors preserve Eulerian and even-face properties of ribbon graphs, respectively. We then characterize Eulerian, even-face, plane Eulerian and plane even-face ribbon graphs using these minors.

    更新日期:2020-05-19
  • Sets of class [q+1,2q+1,3q+1]3 in PG(4,q)
    Discret. Math. (IF 0.77) Pub Date : 2020-05-19
    S.G. Barwick; Wen-Ai Jackson

    The article gives a combinatorial characterisation of a ruled cubic surface in PG(4,q). In PG(4,q), q odd, q≥9, a set K of q2+2q+1 points is a ruled cubic surface if and only if K has class [q+1,2q+1,3q+1]3 such that every plane that contains four points of K contains at least q+1 points of K.

    更新日期:2020-05-19
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