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  • On the neighborhood complex of s→-stable Kneser graphs
    Discret. Math. (IF 0.77) Pub Date : 2021-01-23
    Hamid Reza Daneshpajouh; József Osztényi

    In 2002, Björner and de Longueville showed the neighborhood complex of the 2-stable Kneser graph KG(n,k)2−stab has the same homotopy type as the (n−2k)-sphere. A short time ago, an analogous result about the homotopy type of the neighborhood complex of almost s-stable Kneser graph has been announced by the second author. Combining this result with the famous Lovász’s topological lower bound on the

    更新日期:2021-01-24
  • No sporadic almost simple group acts primitively on the points of a generalised quadrangle
    Discret. Math. (IF 0.77) Pub Date : 2021-01-23
    John Bamberg; James Evans

    A generalised quadrangle is a point–line incidence geometry G such that: (i) any two points lie on at most one line, and (ii) given a line L and a point p not incident with L, there is a unique point on L collinear with p. They are a specific case of the generalised polygons introduced by Tits (1959), and these structures and their automorphism groups are of some importance in finite geometry. An integral

    更新日期:2021-01-24
  • A complete solution to the existence of normal bimagic squares of even order
    Discret. Math. (IF 0.77) Pub Date : 2021-01-23
    Fengchu Pan; Wen Li; Guangzhou Chen; Bangying Xin

    Magic square is an ancient and important subject in combinatorial mathematics, and many kinds of magic square are studied and concerned by many scholars, in particular, the existence of normal bimagic squares has been studied for over one hundred years. In this paper, the existence of the normal bimagic squares of even order is investigated, the ideas of general row (column) bimagic rectangles and

    更新日期:2021-01-24
  • On Tuza’s conjecture for triangulations and graphs with small treewidth
    Discret. Math. (IF 0.77) Pub Date : 2021-01-23
    Fábio Botler; Cristina G. Fernandes; Juan Gutiérrez

    Tuza (1981) conjectured that the size τ(G) of a minimum set of edges that intersects every triangle of a graph G is at most twice the size ν(G) of a maximum set of edge-disjoint triangles of G. In this paper we present three results regarding Tuza’s Conjecture. We verify it for graphs with treewidth at most 6; we show that τ(G)≤32ν(G) for every planar triangulation G different from K4; and that τ(G)≤95ν(G)+15

    更新日期:2021-01-24
  • The relation between composability and splittability of permutation classes
    Discret. Math. (IF 0.77) Pub Date : 2021-01-22
    Rachel Yun Zhang

    A permutation class C is said to be splittable if there exist two proper subclasses A,B⊊C such that any σ∈C can be red–blue colored so that the red (respectively, blue) subsequence of σ is order isomorphic to an element of A (respectively, B). The class C is said to be composable if there exists some number of proper subclasses A1,…,Ak⊊C such that any σ∈C can be written as α1∘⋯∘αk for some αi∈Ai. We

    更新日期:2021-01-24
  • Classification by girth of three-dimensional algebraically defined monomial graphs over the real numbers
    Discret. Math. (IF 0.77) Pub Date : 2021-01-21
    Alex Kodess; Brian G. Kronenthal; Diego Manzano-Ruiz; Ethan Noe

    For positive integers s,t,u,v, we define a bipartite graph ΓR(XsYt,XuYv) where each partite set is a copy of R3, and a vertex (a1,a2,a3) in the first partite set is adjacent to a vertex [x1,x2,x3] in the second partite set if and only if a2+x2=a1sx1tanda3+x3=a1ux1v.In this paper, we classify all such graphs according to girth.

    更新日期:2021-01-22
  • Self-conjugate (s,s+d,s+2d)-core partitions and free Motzkin paths
    Discret. Math. (IF 0.77) Pub Date : 2021-01-20
    Sherry H.F. Yan; Danna Yan; Hao Zhou

    Simultaneous core partitions have been extensively exploited after Anderson’s work on the enumeration of (s,t)-core partitions. Ford, Mai and Sze established a bijection between self-conjugate (s,t)-core partitions and lattice paths in the ⌊s2⌋×⌊t2⌋ rectangle consisting of north and east steps, thereby showing that the number of such partitions is given by ⌊s2⌋+⌊t2⌋⌊s2⌋ for relatively prime integers

    更新日期:2021-01-21
  • Brick partition problems in three dimensions
    Discret. Math. (IF 0.77) Pub Date : 2021-01-19
    Ilkyoo Choi; Minseong Kim; Kiwon Seo

    A d-dimensional brick is a set I1×⋯×Id where each Ii is an interval. Given a brick B, a brick partition of B is a partition of B into bricks. A brick partition Pd of a d-dimensional brick is k-piercing if every axis–parallel line intersects at least k bricks in Pd. Bucic et al. (2019) explicitly asked the minimum size p(d,k) of a k-piercing brick partition of a d-dimensional brick. The answer is known

    更新日期:2021-01-20
  • A combinatorial identity with applications to forest graphs
    Discret. Math. (IF 0.77) Pub Date : 2021-01-16
    Tony Dorlas; Alexei Rebenko; Baptiste Savoie

    We give an elementary proof of an interesting combinatorial identity which is of particular interest in graph theory and its applications. Two applications to enumeration of forests with closed-form expressions are given.

    更新日期:2021-01-18
  • Extremal generalized hexagons and triple intersection numbers
    Discret. Math. (IF 0.77) Pub Date : 2021-01-16
    Bart De Bruyn; Frédéric Verduyn

    A triple intersection number in a finite generalized hexagon S of order (s,t) is a number of the form |Γi(x)∩Γj(y)∩Γk(ω)|, where x, y are two points and ω is either a point z or a line L. The fact that S is extremal, i.e. satisfies t=s3>1, implies that certain combinatorial properties regarding triple intersection numbers hold. The earliest result in this direction is due to Haemers who showed that

    更新日期:2021-01-18
  • Genus, thickness and crossing number of graphs encoding the generating properties of finite groups
    Discret. Math. (IF 0.77) Pub Date : 2021-01-14
    Cristina Acciarri; Andrea Lucchini

    Assume that G is a finite group and let a and b be non-negative integers. We define an undirected graph Γa,b(G) whose vertices correspond to the elements of Ga∪Gb and in which two tuples (x1,…,xa) and (y1,…,yb) are adjacent if and only if 〈x1,…,xa,y1,…,yb〉=G. Our aim is to estimate the genus, the thickness and the crossing number of the graph Γa,b(G) when a and b are positive integers, giving explicit

    更新日期:2021-01-14
  • Asymptotic bounds on total domination in regular graphs
    Discret. Math. (IF 0.77) Pub Date : 2021-01-14
    Carlos Hoppen; Giovane Mansan

    We obtain new upper bounds on the size of a minimum total dominating set for random regular graphs and for regular graphs with large girth. In particular, they imply that an upper bound conjectured by Thomassé and Yeo (2007) holds asymptotically almost surely for 5-regular graphs and holds for all 5-regular graphs with sufficiently large girth. Our bounds are obtained through the analysis of a local

    更新日期:2021-01-14
  • Three-class association schemes from partial geometric designs
    Discret. Math. (IF 0.77) Pub Date : 2021-01-12
    Jing Qu; Jianguo Lei

    In this paper, we show that three-class association schemes can be constructed from certain partial geometric designs. First, we show that partial geometric designs with exactly three indices: r−n, λ1 and λ2 give rise to three-class association schemes, using the method of Beker and Haemers (1980). Second, we describe parameter sets of certain partial geometric designs that also give rise to three-class

    更新日期:2021-01-13
  • Spectral characterization of the complete graph removing a path: Completing the proof of Cámara–Haemers Conjecture
    Discret. Math. (IF 0.77) Pub Date : 2021-01-11
    Muhuo Liu; Xiaofeng Gu

    A graph G is A−DS if every A-cospectral graph of G is isomorphic to G. Denote by Kn∖Pk the graph obtained from the complete graph Kn with n vertices by deleting all edges of a path Pk with k vertices. In 2014, Cámara and Haemers conjectured that Kn∖Pk is A−DS for every 2≤k≤n. The conjecture has been confirmed for k=n (Doob and Haemers, 2002), 2≤k≤6 (Cámara and Haemers, 2014), 7≤k≤9 (Mao et al., 2019)

    更新日期:2021-01-12
  • New constructions of large cyclic subspace codes and Sidon spaces
    Discret. Math. (IF 0.77) Pub Date : 2021-01-08
    Tao Feng; Ye Wang

    Let n be a positive integer with a factor k such that n≥3k. Let q be a prime power, and let Gq(n,k) be the set of all k-dimensional Fq-subspaces of the field Fqn. In this paper, we construct cyclic subspace codes in Gq(n,k) with minimum distance 2k−2 and size (⌈n2k⌉−1)⋅(qn−1)qkq−1. In the case n=3k, their sizes differ from the sphere-packing bound for subspace codes by a factor of 1q−1 asymptotically

    更新日期:2021-01-10
  • Long cycles through specified vertices
    Discret. Math. (IF 0.77) Pub Date : 2021-01-09
    Shuchao Li; Xuechao Li; Bing Wei

    Let σk(G)=min{∑i=1kdG(vi):{v1,v2,…,vk}is an independent set ofG} for a graph G and an integer k≥1. In 1991, Egawa et al. (1991) generalized a famous result on the existence of a hamiltonian cycle given by Ore (1960). In this paper, we will enhance the result of Egawa et al.. by providing the following result: Let G be a k-connected graph (k≥2) and let S be an any vertex set with |S|≤k. Then G has a

    更新日期:2021-01-10
  • On dominating set polyhedra of circular interval graphs
    Discret. Math. (IF 0.77) Pub Date : 2021-01-09
    Silvia Bianchi; Graciela Nasini; Paola Tolomei; Luis Miguel Torres

    Clique-node and closed neighborhood matrices of circular interval graphs are circular matrices. The stable set polytope and the dominating set polytope on these graphs are therefore closely related to the set packing polytope and the set covering polyhedron on circular matrices. Eisenbrand et al. (2008) take advantage of this relationship to propose a complete linear description of the stable set polytope

    更新日期:2021-01-10
  • Near MDS codes from oval polynomials
    Discret. Math. (IF 0.77) Pub Date : 2021-01-08
    Qiuyan Wang; Ziling Heng

    A linear code with parameters of the form [n,k,n−k+1] is referred to as an MDS (maximum distance separable) code. A linear code with parameters of the form [n,k,n−k] is said to be almost MDS (i.e., almost maximum distance separable) or AMDS for short. A code is said to be near maximum distance separable (in short, near MDS or NMDS) if both the code and its dual are almost maximum distance separable

    更新日期:2021-01-08
  • A note on a Caro–Wei bound for the bipartite independence number in graphs
    Discret. Math. (IF 0.77) Pub Date : 2021-01-07
    Shimon Kogan

    A bi-hole of size t in a bipartite graph G is a copy of Kt,t in the bipartite complement of G. Given an n×n bipartite graph G, let β(G) be the largest k for which G has a bi-hole of size k. We prove that β(G)≥12⋅∑v∈V(G)1d(v)+1.Furthermore, we prove the following generalization of the result above. Given an n×n bipartite graph G, let βd(G) be the largest k for which G has a k×k induced d-degenerate

    更新日期:2021-01-07
  • On the spectral radius of graphs without a star forest
    Discret. Math. (IF 0.77) Pub Date : 2021-01-06
    Ming-Zhu Chen; A-Ming Liu; Xiao-Dong Zhang

    Let F=∪i=1kSdi be the union of pairwise vertex-disjoint k stars of order d1+1,…,dk+1, respectively, where k≥2 and d1≥⋯≥dk≥1. In this paper, we present two sharp upper bounds for the spectral radius of F-free (bipartite) graphs and characterize all corresponding extremal graphs. Moreover, the minimum least eigenvalue of the adjacency matrix of an F-free graph and all extremal graphs are obtained.

    更新日期:2021-01-06
  • Infinite family of nonschurian separable association schemes
    Discret. Math. (IF 0.77) Pub Date : 2021-01-06
    Grigory Ryabov

    An infinite family of nonschurian separable association schemes is constructed.

    更新日期:2021-01-06
  • Disjoint directed cycles with specified lengths in directed bipartite graphs
    Discret. Math. (IF 0.77) Pub Date : 2021-01-06
    Yunshu Gao; Hong Wang; Qingsong Zou

    Let D=(V1,V2;A) be a directed bipartite graph with |V1|=|V2|=n≥2. Let a(v) denote the degree of v in D for all v∈V(D). Suppose that a(x)+a(y)≥3n for all x∈V1 and y∈V2. Then for any k positive integers n1,…,nk with n1+⋯+nk≤n, D contains k disjoint directed cycles of lengths 2n1, 2n2, …, 2nk, respectively, unless n is even and D is isomorphic to an exceptional directed bipartite graph. This confirms

    更新日期:2021-01-06
  • On partial parallel classes in partial Steiner triple systems
    Discret. Math. (IF 0.77) Pub Date : 2021-01-06
    Douglas R. Stinson

    For an integer ρ such that 1≤ρ≤v∕3, define β(ρ,v) to be the maximum number of blocks in any partial Steiner triple system on v points in which the maximum partial parallel class has size ρ. We obtain lower bounds on β(ρ,v) by giving explicit constructions, and upper bounds on β(ρ,v) result from counting arguments. We show that β(ρ,v)∈Θ(v) if ρ is a constant, and β(ρ,v)∈Θ(v2) if ρ=v∕c, where c is a

    更新日期:2021-01-06
  • Decomposability and co-modular indices of tournaments
    Discret. Math. (IF 0.77) Pub Date : 2021-01-01
    Houmem Belkhechine; Cherifa Ben Salha

    Given a tournament T, a module of T is a subset X of V(T) such that for x,y∈X and v∈V(T)∖X, (x,v)∈A(T) if and only if (y,v)∈A(T). The trivial modules of T are 0̸, {u} (u∈V(T)) and V(T). The tournament T is indecomposable if all its modules are trivial; otherwise it is decomposable. The decomposability index of T, denoted by δ(T), is the smallest number of arcs of T that must be reversed to make T indecomposable

    更新日期:2021-01-01
  • A note on the maximum size of Berge-C4-free hypergraphs
    Discret. Math. (IF 0.77) Pub Date : 2020-12-31
    Beka Ergemlidze

    In this paper, we consider maximum possible value for the sum of cardinalities of hyperedges of a hypergraph without a Berge 4-cycle. We significantly improve the previous upper bound provided by Gerbner and Palmer. Furthermore, we provide a construction that slightly improves the previous lower bound.

    更新日期:2020-12-31
  • A Fan-type condition for graphs to be k-leaf-connected
    Discret. Math. (IF 0.77) Pub Date : 2020-12-29
    Shun-ichi Maezawa; Ryota Matsubara; Haruhide Matsuda

    For k≥2, a graph G is said to be k-leaf connected if |G|>k and for each subset S of V(G) with |S|=k, G has a spanning tree T with precisely S as the set of endvertices of T. This property is a general concept of Hamiltonian-connected. This paper gives a Fan-type condition for graphs to be k-leaf-connected.

    更新日期:2020-12-29
  • Graphs whose Kronecker covers are bipartite Kneser graphs
    Discret. Math. (IF 0.77) Pub Date : 2020-12-29
    Takahiro Matsushita

    We show that there are k simple graphs whose Kronecker covers are isomorphic to the bipartite Kneser graph H(n,k), and determine the automorphism groups of these graphs. Using the neighborhood complexes of graphs, we also show that the chromatic numbers of these graphs coincide with χ(K(n,k))=n−2k+2.

    更新日期:2020-12-29
  • Implications in rainbow forbidden subgraphs
    Discret. Math. (IF 0.77) Pub Date : 2020-12-26
    Qing Cui; Qinghai Liu; Colton Magnant; Akira Saito

    An edge-colored graph G is rainbow if each edge receives a different color. For a connected graph H, G is rainbow H-free if G does not contain a rainbow subgraph which is isomorphic to H. By definition, if H′ is a connected subgraph of H, every rainbow H′-free graph is rainbow H-free. In this note, we consider a kind of reverse implication, where H′ is a connected proper supergraph and every rainbow

    更新日期:2020-12-26
  • Upper bounds on positional Paris–Harrington games
    Discret. Math. (IF 0.77) Pub Date : 2020-12-21
    Lorenzo Carlucci; Massimo Lauria

    We give upper bounds for a positional game – in the sense of Beck – based on the Paris–Harrington Theorem for bi-colorings of graphs and uniform hypergraphs of arbitrary dimension. The bounds for the positional game show a striking difference when compared to the bounds for the combinatorial principle itself. Our results confirm a phenomenon already observed by Beck and others: the upper bounds for

    更新日期:2020-12-21
  • Generalized Gardiner–Praeger graphs and their symmetries
    Discret. Math. (IF 0.77) Pub Date : 2020-12-21
    Štefko Miklavič; Primož Šparl; Stephen E. Wilson

    A subgroup of the automorphism group of a graph acts half-arc-transitively on the graph if it acts transitively on the vertex-set and on the edge-set of the graph but not on the arc-set of the graph. If the full automorphism group of the graph acts half-arc-transitively, the graph is said to be half-arc-transitive. In 1994 Gardiner and Praeger introduced two families of tetravalent arc-transitive graphs

    更新日期:2020-12-21
  • New bounds on the Ramsey number r(Im,Ln)
    Discret. Math. (IF 0.77) Pub Date : 2020-12-21
    Ferdinand Ihringer; Deepak Rajendraprasad; Thilo Weinert

    We investigate the Ramsey number r(Im,Ln) which is the smallest natural number k such that every oriented graph on k vertices contains either an independent set of size m or a transitive tournament on n vertices. Continuing research by Larson and Mitchell and earlier work by Bermond we establish two new upper bounds for r(Im,L3) which are paramount in proving r(I4,L3)=15<23=r(I5,L3) and r(Im,L3)=Θ(m2∕logm)

    更新日期:2020-12-21
  • Turán and Ramsey numbers in linear triple systems
    Discret. Math. (IF 0.77) Pub Date : 2020-12-15
    András Gyárfás; Gábor N. Sárközy

    In this paper we study Turán and Ramsey numbers in linear triple systems, defined as 3-uniform hypergraphs in which any two triples intersect in at most one vertex. A famous result of Ruzsa and Szemerédi is that for any fixed c>0 and large enough n the following Turán-type theorem holds. If a linear triple system on n vertices has at least cn2 edges then it contains a triangle: three pairwise intersecting

    更新日期:2020-12-15
  • The extremal spectral radii of the arithmetical structures on paths
    Discret. Math. (IF 0.77) Pub Date : 2020-12-14
    Dijian Wang; Yaoping Hou

    An arithmetical structure on a finite, connected graph G is a pair of vectors (d,r) with positive integer entries for which (diag(d)−A(G))rT=0, where diag(d)=diag(d1,d2,…,dn), A(G) is the adjacency matrix of G and the entries of r have no common factor. In this paper, we will study the spectral radii of arithmetical structures on the path Pn and determine the arithmetical structures with the minimal

    更新日期:2020-12-14
  • On the smallest trees with the same restricted U-polynomial and the rooted U-polynomial
    Discret. Math. (IF 0.77) Pub Date : 2020-12-14
    José Aliste-Prieto; Anna de Mier; José Zamora

    In this article, we construct explicit examples of pairs of non-isomorphic trees with the same restricted U-polynomial for every k; by this we mean that the polynomials agree on terms with degree at most k+1. The main tool for this construction is a generalization of the U-polynomial to rooted graphs, which we introduce and study in this article. Most notably we show that rooted trees can be reconstructed

    更新日期:2020-12-14
  • The Namer–Claimer game
    Discret. Math. (IF 0.77) Pub Date : 2020-12-14
    Ben Barber

    In each round of the Namer–Claimer game, Namer names a distance d, then Claimer claims a subset of [n] that does not contain two points that differ by d. The game ends once sets covering [n] have been claimed. We show that the length of this game is Θ(loglogn) with optimal play from each side. This parameter is an online analogue of the previously studied upper chromatic number for the family of ‘distance-d’

    更新日期:2020-12-14
  • Value functions and two classes of binary codes
    Discret. Math. (IF 0.77) Pub Date : 2020-12-14
    Zihui Liu

    We state that the minimum and the maximum support weight of subcodes of a code are useful in several aspects. Based on the above findings, we will determine part of the above mentioned parameters for two classes of binary codes. We also present the separation property and calculate the trellis complexity of a class of binary codes.

    更新日期:2020-12-14
  • Proximity and remoteness in directed and undirected graphs
    Discret. Math. (IF 0.77) Pub Date : 2020-12-10
    Jiangdong Ai; Stefanie Gerke; Gregory Gutin; Sonwabile Mafunda

    Let D be a strongly connected digraph. The average distance σ̄(v) of a vertex v of D is the arithmetic mean of the distances from v to all other vertices of D. The remoteness ρ(D) and proximity π(D) of D are the maximum and the minimum of the average distances of the vertices of D, respectively. We obtain sharp upper and lower bounds on π(D) and ρ(D) as a function of the order n of D and describe the

    更新日期:2020-12-10
  • Automorphisms of the cube nd
    Discret. Math. (IF 0.77) Pub Date : 2020-12-09
    Pavel Dvořák; Tomáš Valla

    Consider a hypergraph nd where the vertices are points of the d-dimensional cube [n]d and the edges are all sets of n points such that they are in one line. We study the structure of the group of automorphisms of nd, i.e., permutations of points of [n]d preserving the edges. In this paper we provide a complete characterization. Moreover, we consider the Colored Cube Isomorphism problem of deciding

    更新日期:2020-12-09
  • On dominating pair degree conditions for hamiltonicity in balanced bipartite digraphs
    Discret. Math. (IF 0.77) Pub Date : 2020-12-07
    Janusz Adamus

    We prove several new sufficient conditions for hamiltonicity and bipancyclicity in balanced bipartite digraphs, in terms of sums of degrees over dominating or dominated pairs of vertices.

    更新日期:2020-12-07
  • Enumeration of row-increasing tableaux of two-row skew shapes
    Discret. Math. (IF 0.77) Pub Date : 2020-12-05
    Xiaomei Chen

    In this paper, we firstly extend a result of Bonin, Shapiro and Simion by giving the distribution of the major index over generalized Schröder paths. Then by providing a bijection between generalized Schröder paths and row-increasing tableaux of skew shapes with two rows, we obtain the distribution of the major index and the amajor index over these tableaux, which extends a result of Du, Fan and Zhao

    更新日期:2020-12-05
  • Structural domination and coloring of some (P7,C7)-free graphs
    Discret. Math. (IF 0.77) Pub Date : 2020-12-05
    S.A. Choudum; T. Karthick; Manoj M. Belavadi

    We show that every connected induced subgraph of a graph G is dominated by an induced connected split graph if and only if G is C-free, where C is a set of six graphs which includes P7 and C7, and each containing an induced P5. A similar characterization is shown for the class of graphs which are dominated by an induced connected complete split graph. Motivated by these results, we study structural

    更新日期:2020-12-05
  • Connectivity keeping caterpillars and spiders in 2-connected graphs
    Discret. Math. (IF 0.77) Pub Date : 2020-12-05
    Yanmei Hong; Qinghai Liu; Changhong Lu; Qingjie Ye

    Mader (2010) conjectured that for any tree T of order m, every k-connected graph G with minimum degree at least ⌊3k2⌋+m−1 contains a subtree T′≅T such that G−V(T′) is k-connected. A caterpillar is a tree in which a single path is incident to every edge. The conjecture has been proved when k=1 and for some special caterpillars when k=2. A spider is a tree with at most one vertex with degree more than

    更新日期:2020-12-05
  • On quasi-strongly regular graphs with parameters (n,k,a;k−1,c2)
    Discret. Math. (IF 0.77) Pub Date : 2020-12-04
    Jiayi Xie; Dongdong Jia; Gengsheng Zhang

    Quasi-strongly regular graphs are a combinatorial generalization of strongly regular graphs. A quasi-strongly regular graph of grade 2 with parameters (n,k,a;c1,c2) is a k-regular graph on n vertices such that any two adjacent vertices have a common neighbours, any two non-adjacent vertices have c1 or c2 common neighbours, and for each ci (i=1,2), there exists a pair of non-adjacent vertices sharing

    更新日期:2020-12-04
  • On the Tu-Zeng permutation trinomial of type (1∕4,3∕4)
    Discret. Math. (IF 0.77) Pub Date : 2020-12-04
    Xiang-Dong Hou

    Let q be a power of 2. Recently, Tu and Zeng considered trinomials of the form f(X)=X+aX(1∕4)q2(q−1)+bX(3∕4)q2(q−1), where a,b∈Fq2∗. They proved that f is a permutation polynomial of Fq2 if b=a2−q and X3+X+a−1−q has no root in Fq. In this paper, we show that the above sufficient condition is also necessary.

    更新日期:2020-12-04
  • Striking patterns in natural magic squares’ associated electrostatic potentials: Matrices of the 4th and 5th order
    Discret. Math. (IF 0.77) Pub Date : 2020-12-02
    Peyman Fahimi; Cyrus Ahmadi Toussi; Walter Trump; Javad Haddadnia; Chérif F. Matta

    A magic square is a square matrix whereby the sum of any row, column, or any one of the two principal diagonals is equal. A surrogate of this abstract mathematical construct, introduced in 2012 by Fahimi and Jaleh, is the “electrostatic potential (ESP)” that results from treating the matrix elements of the magic square as electric charges. The overarching idea is to characterize patterns associated

    更新日期:2020-12-03
  • On (s,t)-supereulerian graphs with linear degree bounds
    Discret. Math. (IF 0.77) Pub Date : 2020-12-03
    Lan Lei; Wei Xiong; Yikang Xie; Mingquan Zhan; Hong-Jian Lai

    For integers s≥0 and t≥0, a graph G is (s,t)-supereulerian if for any disjoint edge sets X,Y⊆E(G) with |X|≤s and |Y|≤t, G has a spanning closed trail that contains X and avoids Y. Pulleyblank in [J. Graph Theory, 3 (1979) 309-310] showed that determining whether a graph is (0,0)-supereulerian, even when restricted to planar graphs, is NP-complete. Settling an open problem of Bauer, Catlin in [J. Graph

    更新日期:2020-12-03
  • Induced designs and fixed points
    Discret. Math. (IF 0.77) Pub Date : 2020-12-03
    Jianfu Chen; Shenglin Zhou

    Let D be a 2-(v,k,1) design and H be an automorphism group of D. This paper studies the induced design on the fixed points of H, denoted by FixH. We extend the work of A.R. Camina and J. Siemons and we also state that if there is an induced design on FixH, then |FixH|≤r in most cases, where r is the number of blocks containing a given point. Moreover, for a given k0, there exist only finitely many

    更新日期:2020-12-03
  • Generalised dual Seidel switching and Deza graphs with strongly regular children
    Discret. Math. (IF 0.77) Pub Date : 2020-12-01
    Vladislav V. Kabanov; Elena V. Konstantinova; Leonid Shalaginov

    A Deza graph G with parameters (n,k,b,a) is a k-regular graph with n vertices such that any two distinct vertices have b or a common neighbours, where b⩾a. The children GA and GB of a Deza graph G are defined on the vertex set of G such that every two distinct vertices are adjacent in GA or GB if and only if they have a or b common neighbours, respectively. In this paper we present a general approach

    更新日期:2020-12-02
  • Euclidean matchings and minimality of hyperplane arrangements
    Discret. Math. (IF 0.77) Pub Date : 2020-12-02
    Davide Lofano; Giovanni Paolini

    We construct a new class of maximal acyclic matchings on the Salvetti complex of a locally finite hyperplane arrangement. Using discrete Morse theory, we then obtain an explicit proof of the minimality of the complement. Our construction provides interesting insights also in the well-studied case of finite arrangements, and gives a nice geometric description of the Betti numbers of the complement.

    更新日期:2020-12-02
  • Bounding and approximating minimum maximal matchings in regular graphs
    Discret. Math. (IF 0.77) Pub Date : 2020-12-01
    Julien Baste; Maximilian Fürst; Michael A. Henning; Elena Mohr; Dieter Rautenbach

    The edge domination number γe(G) of a graph G is the minimum size of a maximal matching in G. It is well known that this parameter is computationally very hard, and several approximation algorithms and heuristics have been studied. In the present paper, we provide best possible upper bounds on γe(G) for regular and non-regular graphs G in terms of their order and maximum degree. Furthermore, we discuss

    更新日期:2020-12-01
  • Eigenfunctions and minimum 1-perfect bitrades in the Hamming graph
    Discret. Math. (IF 0.77) Pub Date : 2020-11-27
    Alexandr Valyuzhenich

    The Hamming graph H(n,q) is the graph whose vertices are the words of length n over the alphabet {0,1,…,q−1}, where two vertices are adjacent if they differ in exactly one coordinate. The adjacency matrix of H(n,q) has n+1 distinct eigenvalues n(q−1)−q⋅i with corresponding eigenspaces Ui(n,q) for 0≤i≤n. In this work we study functions belonging to a direct sum Ui(n,q)⊕Ui+1(n,q)⊕⋯⊕Uj(n,q) for 0≤i≤j≤n

    更新日期:2020-11-27
  • On linear sets of minimum size
    Discret. Math. (IF 0.77) Pub Date : 2020-11-27
    Dibyayoti Jena; Geertrui Van de Voorde

    An Fq-linear set of rank k, k≤h, on a projective line PG(1,qh), containing at least one point of weight one, has size at least qk−1+1 (see De Beule and Van De Voorde (2019)). The classical example of such a set is given by a club. In this paper, we construct a broad family of linear sets meeting this lower bound, where we are able to prescribe the weight of the heaviest point to any value between k∕2

    更新日期:2020-11-27
  • Star-critical Ramsey numbers involving graphs with long suspended paths
    Discret. Math. (IF 0.77) Pub Date : 2020-11-24
    Ye Wang; Yusheng Li; Yan Li

    For graphs F, G and H, let F→(G,H) signify that any red/blue edge coloring of F contains a red G or a blue H. The star-critical Ramsey number RS(G,H)=max{n|Kr∖K1,n→(G,H)}, where r=R(G,H). In this note, it is shown that RS(G,Hn)=n+O(1) for fixed G and H as n→∞, where Hn is a graph of order n obtained from H by lengthening an edge to a path, each internal vertex of which has degree two. In addition,

    更新日期:2020-11-25
  • Structure of the largest subgraphs of Gn,p with a given matching number
    Discret. Math. (IF 0.77) Pub Date : 2020-11-24
    Abigail Raz

    This paper examines the structure of the largest subgraphs of the Erdős–Rényi random graph, Gn,p, with a given matching number. This extends a result of Erdős and Gallai who, in 1959, gave a classification of the structures of the largest subgraphs of Kn with a given matching number. We show that their result extends to Gn,p with high probability when p≥8lnnn or p≪1n, but that it does not extend (again

    更新日期:2020-11-25
  • A lower bound for the discriminant of polynomials related to Chebyshev polynomials
    Discret. Math. (IF 0.77) Pub Date : 2020-11-24
    Slobodan Filipovski

    In this paper we give a lower bound for the discriminant of the polynomials {Gk,i(x)} defined by Gk,0(x)=1,Gk,1(x)=x+1 and Gk,i+2(x)=xGk,i+1(x)−(k−1)Gk,i(x)fori≥0. These polynomials are closely related to the Chebyshev polynomials of the second kind. We derive our result by using orthogonality properties of the polynomials {Fk,i(x)} defined by Fk,0(x)=1,Fk,1(x)=x,Fk,2(x)=x2−k and Fk,i+2(x)=xFk,i+1(x)−(k−1)Fk

    更新日期:2020-11-25
  • Minimal linear codes from weakly regular plateaued balanced functions
    Discret. Math. (IF 0.77) Pub Date : 2020-11-23
    Ahmet Sınak

    Minimal linear codes have diverse applications in many areas such as secret sharing schemes and secure two-party computation. There are several construction methods for these codes, one of which is based on functions over finite fields. In this paper, to construct minimal codes with few weights, we make use of weakly regular plateaued balanced functions over Fp, where p is an odd prime, in the second

    更新日期:2020-11-23
  • Degree sums and dominating cycles
    Discret. Math. (IF 0.77) Pub Date : 2020-11-21
    Yueyu Wu; Yaojun Chen; T.C. Edwin Cheng

    A cycle C of a graph G is dominating if any vertex of V(G)∖V(C) has at least one neighbor on C and V(G)∖V(C) is an independent set. Let G be a k-connected graph of order n≥3 with k≥2. In this paper, we show that every longest cycle of G is dominating if the degree sums is more than (k+1)(n+1)∕3 for any k+1 pairwise nonadjacent vertices, and the lower bound is sharp, which generalizes the results due

    更新日期:2020-11-22
  • On the strong chromatic number of a random 3-uniform hypergraph
    Discret. Math. (IF 0.77) Pub Date : 2020-11-20
    Arseniy E. Balobanov; Dmitry A. Shabanov

    This paper deals with estimating the threshold for the strong r-colorability of a random 3-uniform hypergraph in the binomial model H(n,3,p). A vertex coloring is said to be strong for a hypergraph if every two vertices sharing a common edge are colored with distinct colors. It is known that the threshold corresponds to the sparse case, when the expected number of edges is a linear function of n, pn3=cn

    更新日期:2020-11-21
  • The r-signed Birkhoff transform
    Discret. Math. (IF 0.77) Pub Date : 2020-11-18
    Richard Ehrenborg

    We introduce the r-signed Birkhoff transform of a distributive lattice, extending Hsiao’s notion of the signed Birkhoff transform. We show how to compute the ab-index of the r-signed Birkhoff transform from the ab-index of the distributive lattice, generalizing work of Billera, Ehrenborg and Readdy, by extending their ω map to ωr. We also obtain new expressions for the ab-index of the r-cubical lattice

    更新日期:2020-11-18
  • Threshold behavior of bootstrap percolation
    Discret. Math. (IF 0.77) Pub Date : 2020-11-18
    Ahad N. Zehmakan

    Consider a graph G and an initial random configuration, where each node is black with probability p and white otherwise, independently. In discrete-time rounds, each node becomes black if it has at least r black neighbors and white otherwise. We prove that this basic process exhibits a threshold behavior with two phase transitions when the underlying graph is a d-dimensional torus and identify the

    更新日期:2020-11-18
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