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  • Janko sporadic group J2 as automorphism group of 3-designs
    Discret. Math. (IF 0.77) Pub Date : 2020-10-20
    Ali Reza Rahimipour; Hossein Moshtagh

    Presenting sporadic simple groups as automorphism group of designs and graphs is an exciting field in simple group theory. In this paper, we present five simple 3-designs with Janko sporadic group J2 automorphism. Also, we classify all block-transitive self-orthogonal designs and block-transitive t -designs for t≥2 on 100 points with block stabilizer of order greater than four that admit sporadic group

    更新日期:2020-10-20
  • A class of skew cyclic codes and application in quantum codes construction
    Discret. Math. (IF 0.77) Pub Date : 2020-10-20
    Hai Q. Dinh; Tushar Bag; Ashish K. Upadhyay; Ramakrishna Bandi; Roengchai Tansuchat

    Let p be an odd prime, and k an integer such that k∣(p−1). This paper studies quantum codes from skew cyclic codes over the ring R=Fq[u]〈uk+1−u〉, where q is a power of the prime p. We construct a set of orthogonal idempotents of the ring R and using the set, skew cyclic codes over the ring R are decomposed as direct sum of skew cyclic codes over Fq. We obtain a necessary and sufficient condition for

    更新日期:2020-10-20
  • Avoidance of split overlaps
    Discret. Math. (IF 0.77) Pub Date : 2020-10-20
    Daniel Gabric; Jeffrey Shallit; Xiao Feng Zhong

    We generalize Axel Thue’s familiar definition of overlaps in words, and observe that there are no infinite words avoiding split occurrences of these generalized overlaps. We give estimates for the length of the longest finite word that avoids split overlaps. Along the way we prove a useful theorem about repeated disjoint occurrences in words — an interesting natural variation on the classical de Bruijn

    更新日期:2020-10-20
  • The average number of spanning hypertrees in sparse uniform hypergraphs
    Discret. Math. (IF 0.77) Pub Date : 2020-10-16
    Haya S. Aldosari; Catherine Greenhill

    An r-uniform hypergraph H consists of a set of vertices V and a set of edges whose elements are r-subsets of V. We define a hypertree to be a connected hypergraph which contains no cycles. A hypertree spans a hypergraph H if it is a subhypergraph of H which contains all vertices of H. Greenhill et al. (2017) gave an asymptotic formula for the average number of spanning trees in graphs with given, sparse

    更新日期:2020-10-17
  • Singular graphs with dihedral group action
    Discret. Math. (IF 0.77) Pub Date : 2020-10-15
    Ali Sltan Ali AL-Tarimshawy; J. Siemons

    Let Γ be a simple undirected graph on a finite vertex set and let A be its adjacency matrix. Then Γ is singular if A is singular. The problem of characterizing singular graphs is easy to state but very difficult to resolve in any generality. In this paper we investigate the singularity of graphs for which the dihedral group acts transitively on vertices as a group of automorphisms.

    更新日期:2020-10-16
  • Total domination in plane triangulations
    Discret. Math. (IF 0.77) Pub Date : 2020-10-15
    Mercè Claverol; Alfredo García; Gregorio Hernández; Carmen Hernando; Montserrat Maureso; Mercè Mora; Javier Tejel

    A total dominating set of a graph G=(V,E) is a subset D of V such that every vertex in V is adjacent to at least one vertex in D. The total domination number of G, denoted by γt(G), is the minimum cardinality of a total dominating set of G. A near-triangulation is a biconnected planar graph that admits a plane embedding such that all of its faces are triangles except possibly the outer face. We show

    更新日期:2020-10-16
  • On the line graph of a graph with diameter 2
    Discret. Math. (IF 0.77) Pub Date : 2020-10-13
    Xiaoling Ma; Lan Lei; Hong-Jian Lai; Mingquan Zhan

    A graph G is pancyclic if it contains cycles of all possible lengths. A graph G is 1-hamiltonian if the removal of at most 1 vertices from G results in a hamiltonian graph. In Veldman (1988) Veldman showed that the line graph L(G) of a connected graph G with diameter at most 2 is hamiltonian. In this paper, we continue studying the line graph L(G) of a connected graph G with |E(G)|≥3 and diameter at

    更新日期:2020-10-13
  • On s-hamiltonicity of net-free line graphs
    Discret. Math. (IF 0.77) Pub Date : 2020-10-13
    Xiaoling Ma; Yang Wu; Mingquan Zhan; Hong-Jian Lai

    For integers s1,s2,s3>0, let Ns1,s2,s3 denote the graph obtained by identifying each vertex of a K3 with an end vertex of three disjoint paths Ps1+1, Ps2+1, Ps3+1 of length s1,s2 and s3, respectively. We prove the following results. (i) Let N1={Ns1,s2,s3:s1>0,s1≥s2≥s3≥0 and s1+s2+s3≤6}. Then for any N∈N1, every N-free line graph L(G) with |V(L(G))|≥s+3 is s-hamiltonian if and only if κ(L(G))≥s+2. (ii)

    更新日期:2020-10-13
  • Bounds and constructions for multilength variable-weight optical orthogonal codes
    Discret. Math. (IF 0.77) Pub Date : 2020-10-08
    Hengming Zhao; Rongcun Qin

    Multilength variable-weight optical orthogonal codes (MLVWOOCs) are recently proposed for supporting multirate and integrated multimedia services in optical code division multiple access (OCDMA) networks. In this paper, we consider (N,M,W,1,Q;λ)-MLVWOOCs with λ=2 (the minimum nontrivial intercross-correlation). Some upper bounds on code size are obtained under certain restrictions, an (N,M,W,1,Q;λ)

    更新日期:2020-10-11
  • Cubic graphs with equal independence number and matching number
    Discret. Math. (IF 0.77) Pub Date : 2020-10-09
    Elena Mohr; Dieter Rautenbach

    Caro, Davila, and Pepper recently proved δ(G)α(G)≤Δ(G)μ(G) for every graph G with minimum degree δ(G), maximum degree Δ(G), independence number α(G), and matching number μ(G). Answering some problems they posed, we characterize the extremal graphs for δ(G)<Δ(G) as well as for δ(G)=Δ(G)=3.

    更新日期:2020-10-11
  • Reducing the domination number of graphs via edge contractions and vertex deletions
    Discret. Math. (IF 0.77) Pub Date : 2020-10-09
    Esther Galby; Paloma T. Lima; Bernard Ries

    In this work, we study the following problem: given a connected graph G, can we reduce the domination number of G by at least one using k edge contractions, for some fixed integer k>0? We show that for k=1 (resp. k=2), the problem is NP-hard (resp. coNP-hard). We further prove that for k=1, the problem is W[1]-hard parameterized by domination number plus the mim-width of the input graph, and that it

    更新日期:2020-10-11
  • Neighbor sum distinguishing total choosability of 1-planar graphs with maximum degree at least 24
    Discret. Math. (IF 0.77) Pub Date : 2020-10-10
    Lin Sun; Guanglong Yu; Xin Li

    For a simple graph G, a neighbor sum distinguishing total k-coloring of G is a mapping ϕ: V(G)∪E(G)→{1,2,…,k} such that no two adjacent or incident elements in V(G)∪E(G) receive the same color and wϕ(u)≠wϕ(v) for each edge uv∈E(G), where wϕ(v) (or wϕ(u)) denotes the sum of the color of v (or u) and the colors of all edges incident with v (or u). For each element x∈V(G)∪E(G), let L(x) be a list of integer

    更新日期:2020-10-11
  • A sign pattern with all diagonal entries nonzero whose minimal rank realizations are not diagonalizable over ℂ
    Discret. Math. (IF 0.77) Pub Date : 2020-10-08
    Yaroslav Shitov

    The rank of the 9 × 9 matrix Download : Download high-res image (50KB) Download : Download full-size image is 6. If we replace the ones by arbitrary non-zero numbers, we get a matrix B with rankB⩾6, and if rankB=6, the 6 × 6 principal minors of B vanish.

    更新日期:2020-10-08
  • Partitioning planar graphs without 4-cycles and 5-cycles into bounded degree forests
    Discret. Math. (IF 0.77) Pub Date : 2020-10-08
    Eun-Kyung Cho; Ilkyoo Choi; Boram Park

    In 1976, Steinberg conjectured that planar graphs without 4-cycles and 5-cycles are 3-colorable. This conjecture attracted numerous researchers for about 40 years, until it was recently disproved by Cohen-Addad et al. (2017). However, coloring planar graphs with restrictions on cycle lengths is still an active area of research, and the interest in this particular graph class remains. Let G be a planar

    更新日期:2020-10-08
  • Almost eulerian compatible spanning circuits in edge-colored graphs
    Discret. Math. (IF 0.77) Pub Date : 2020-10-08
    Zhiwei Guo; Hajo Broersma; Binlong Li; Shenggui Zhang

    Let G be a (not necessarily properly) edge-colored graph. A compatible spanning circuit in G is a closed trail containing all vertices of G in which any two consecutively traversed edges have distinct colors. As two extremal cases, the existence of compatible (i.e., properly edge-colored) Hamilton cycles and compatible Euler tours have been studied extensively. More recently, sufficient conditions

    更新日期:2020-10-08
  • Pushable chromatic number of graphs with degree constraints
    Discret. Math. (IF 0.77) Pub Date : 2020-10-07
    Julien Bensmail; Sandip Das; Soumen Nandi; Soumyajit Paul; Théo Pierron; Sagnik Sen; Éric Sopena

    Pushable homomorphisms and the pushable chromatic number χp of oriented graphs were introduced by Klostermeyer and MacGillivray in 2004. They notably observed that, for any oriented graph G⃗, we have χp(G⃗)≤χo(G⃗)≤2χp(G⃗), where χo(G⃗) denotes the oriented chromatic number of G⃗. This stands as the first general bounds on χp. This parameter was further studied in later works. This work is dedicated

    更新日期:2020-10-07
  • Rainbow vertex-pancyclicity of strongly edge-colored graphs
    Discret. Math. (IF 0.77) Pub Date : 2020-10-06
    Maoqun Wang; Jianguo Qian

    A strongly edge-colored graph is an edge-colored graph in which every path of length 3 is rainbow or, equivalently, every monochromatic subgraph is an induced matching. In this paper, we show that every strongly edge-colored graph with n vertices and minimum degree δ≥2n∕3 is rainbow vertex-pancyclic. This extends a recent result given by Cheng et al., that is, every strongly edge-colored graph with

    更新日期:2020-10-06
  • A universal partition result for infinite homogeneous Kn-free and related graphs
    Discret. Math. (IF 0.77) Pub Date : 2020-09-30
    Andres Aranda; Claude Laflamme; Daniel T. Soukup; Robert Woodrow

    We study new partition properties of infinite Kn-free graphs. First, we investigate the number bpi(G,m) introduced by A. Aranda et al. (denoted there by r(G,m)) : the minimal r so that for any partition of G into r classes of equal size, there exists an independent set which meets at least m classes in size |G|. In the case of Henson’s countable universal Kn-free graph Hn, we express bpi(Hnm) by well-known

    更新日期:2020-10-02
  • Hamiltonicity and restricted degree conditions on induced subgraphs in claw-free graphs
    Discret. Math. (IF 0.77) Pub Date : 2020-09-29
    Zhi-Hong Chen

    For a graph S, a vertex with degree one in S is an end-vertex of S and an edge is a pendant edge if one of its ends is an end-vertex. A net N is a graph obtained from a K3 by adding a pendant edge at each vertex of the K3. A P6 is a path on 6 vertices. In this paper, we solve two conjectures and prove that for 2-connected claw-free graphs H and for a fixed graph S∈{N,P6}, if the degree at each end-vertex

    更新日期:2020-09-29
  • Lewis Carroll and the Red Hot Potato: A graph theoretic approach to a linear algebraic identity
    Discret. Math. (IF 0.77) Pub Date : 2020-09-29
    Melanie Fraser

    The Lewis Carroll Identity expresses the determinant of a matrix in terms of subdeterminants obtained by deleting one row and column or a pair of rows and columns. Using the matrix tree theorem, we can convert this into an equivalent identity involving sums over pairs of forests. Unlike the Lewis Carroll Identity, the Forest Identity involves no minus signs. In 2011, Vlasev and Yeats (2012) suggested

    更新日期:2020-09-29
  • Paths with many shortcuts in tournaments
    Discret. Math. (IF 0.77) Pub Date : 2020-09-29
    Raphael Yuster

    A shortcut of a directed path v1v2⋯vn is an edge vivj with j>i+1. If j=i+2 the shortcut is a hop. If all hops are present, the path is called hop complete so the path and its hops form a square of a path. We prove that every tournament with n≥4 vertices has a Hamiltonian path with at least (4n−10)∕7 hops, and has a hop complete path of order at least n0.295. A spanning binary tree of a tournament is

    更新日期:2020-09-29
  • Designing DNA codes from reversible self-dual codes over GF(4)
    Discret. Math. (IF 0.77) Pub Date : 2020-09-28
    Hyun Jin Kim; Whan-Hyuk Choi; Yoonjin Lee

    We present an explicit method for designing DNA codes from reversible self-dual codes over the finite field GF(4) of order 4. We use Euclidean reversible self-dual codes (Euclidean RSD codes for short) since they have some advantages in designing DNA codes; hence, in this work RSD codes mean Euclidean RSD codes. We first work on a construction method for reversible self-dual codes over GF(4), and we

    更新日期:2020-09-28
  • A new family of Hadamard matrices of order 4(2q2+1)
    Discret. Math. (IF 0.77) Pub Date : 2020-09-28
    Ka Hin Leung; Koji Momihara; Qing Xiang

    Let q be a prime power of the form q=12c2+4c+3 for c an integer. In this paper we construct a difference family with parameters (2q2;q2,q2,q2,q2−1;2q2−2) in Z2×(Fq2,+). As a consequence, by applying the Wallis–Whiteman array, we obtain Hadamard matrices of order 4(2q2+1) for the aforementioned q’s.

    更新日期:2020-09-28
  • Peakless Motzkin paths with marked level steps at fixed height
    Discret. Math. (IF 0.77) Pub Date : 2020-09-28
    Naiomi Cameron; Everett Sullivan

    A Motzkin path is a path starting and ending on the x-axis which uses up, down and level steps and never goes below the x-axis. We consider Motzkin paths where any number of level steps on the x-axis are allowed to be marked. It is known that the number of such paths corresponds to a matrix which turns out to be a pseudo-involution in the Riordan group. We provide a combinatorial proof of this result

    更新日期:2020-09-28
  • Graded alphabets, circular codes, free Lie algebras and comma-free codes
    Discret. Math. (IF 0.77) Pub Date : 2020-09-25
    Dominique Perrin; Christophe Reutenauer

    We show how the use of graded alphabets allows one to provide simpler proofs of some results on free monoids and free Lie algebras. We first generalize to graded alphabets the characterization of the length distributions of circular codes. We also show that the existence of a circular code with a given distribution of degrees is equivalent to the existence of an embedding of Lie algebras. We finally

    更新日期:2020-09-26
  • Convex lattice polygons with all lattice points visible
    Discret. Math. (IF 0.77) Pub Date : 2020-09-25
    Ralph Morrison; Ayush Kumar Tewari

    Two lattice points are visible to one another if there exist no other lattice points on the line segment connecting them. In this paper we study convex lattice polygons that contain a lattice point such that all other lattice points in the polygon are visible from it. We completely classify such polygons, show that there are finitely many of lattice width greater than 2, and computationally enumerate

    更新日期:2020-09-25
  • Ramsey numbers and bipartite Ramsey numbers via quasi-random graphs
    Discret. Math. (IF 0.77) Pub Date : 2020-09-25
    Meng Liu; Yusheng Li

    In this paper we show that r(C4,Kt,t)≥Ω(t3∕2logt) via quasi-random graphs giving a polylogarithmic improvement over the currently best lower bound, which implies r(C4,Kt)≥Ω(t3∕2logt) and br(C4,Kt,t)≥Ω(t3∕2logt), where br(C4,Kt,t) is the bipartite Ramsey number of C4 and Kt,t. This builds on a recent breakthrough of Mubayi and Verstraëte (2019) reducing off-diagonal Ramsey numbers to the existence of

    更新日期:2020-09-25
  • Local orientation-preserving symmetry preserving operations on polyhedra
    Discret. Math. (IF 0.77) Pub Date : 2020-09-21
    Pieter Goetschalckx; Kris Coolsaet; Nico Van Cleemput

    Unifying approaches by amongst others Archimedes, Kepler, Goldberg, Caspar and Klug, Coxeter, and Conway, and extending on a previous formalization of the concept of local symmetry preserving (lsp) operations, we introduce a formal definition of local operations on plane graphs that preserve orientation-preserving symmetries, but not necessarily orientation-reversing symmetries. This operations include

    更新日期:2020-09-22
  • A best possible result for the square of a 2-block to be hamiltonian
    Discret. Math. (IF 0.77) Pub Date : 2020-09-18
    Jan Ekstein; Herbert Fleischner

    It is shown that for any choice of four different vertices x1,…,x4 in a 2-block G of order p>3, there is a hamiltonian cycle in G2 containing four different edges xiyi of E(G) for certain vertices yi, i=1,2,3,4. This result is best possible.

    更新日期:2020-09-20
  • Laplacian state transfer in total graphs
    Discret. Math. (IF 0.77) Pub Date : 2020-09-19
    Xiaogang Liu; Qiang Wang

    The total graph of a graph G, denoted by T(G), is defined to be the graph whose vertices are the vertices and edges of G, with two vertices of T(G) adjacent if and only if the corresponding elements of G are adjacent or incident. In this paper, we investigate the existence of Laplacian perfect state transfer and Laplacian pretty good state transfer in the total graph of an r-regular graph, where r≥2

    更新日期:2020-09-20
  • Permutation polynomials with Carlitz rank 2
    Discret. Math. (IF 0.77) Pub Date : 2020-09-19
    José Alves Oliveira; F.E. Brochero Martínez

    Let Fq denote the finite field with q elements. The Carlitz rank of a permutation polynomial is an important measure of complexity of a polynomial. In this paper we find a sharp lower bound for the weight of any permutation polynomial with Carlitz rank 2, improving the bound found by Gómez-Pérez, Ostafe and Topuzoğlu in that case.

    更新日期:2020-09-20
  • Mass formulae for Euclidean self-orthogonal and self-dual codes over finite commutative chain rings
    Discret. Math. (IF 0.77) Pub Date : 2020-09-16
    Monika Yadav; Anuradha Sharma

    In this paper, we obtain explicit mass formulae for all Euclidean self-orthogonal and self-dual codes of an arbitrary length over finite commutative chain rings of odd characteristic. With the help of these mass formulae, we classify all Euclidean self-orthogonal and self-dual codes of lengths 2,3,4,5 over the chain ring F5[u]∕ and of lengths 2,3,4 over the chain ring F7[u]∕.

    更新日期:2020-09-16
  • Duality respecting representations and compatible complexity measures for gammoids
    Discret. Math. (IF 0.77) Pub Date : 2020-09-16
    Immanuel Albrecht

    We show that every gammoid has special digraph representations, such that a representation of the dual of the gammoid may be easily obtained by reversing all arcs. In an informal sense, the duality notion of a poset applied to the digraph of a special representation of a gammoid commutes with the operation of forming the dual of that gammoid. We use these special representations in order to define

    更新日期:2020-09-16
  • Choosability with union separation of triangle-free planar graphs
    Discret. Math. (IF 0.77) Pub Date : 2020-09-12
    Jianfeng Hou; Hongguo Zhu

    For a graph G and a positive integer k, a k-list assignment of G is a function L on the vertices of G such that for each vertex v∈V(G), |L(v)|≥k. Let s be a nonnegative integer. Then L is a (k,k+s)-list assignment of G if |L(u)∪L(v)|≥k+s for each edge uv. If for each (k,k+s)-list assignment L of G, G admits a proper coloring φ such that φ(v)∈L(v) for each v∈V(G), then we say G is (k,k+s)-choosable

    更新日期:2020-09-12
  • Modular knight distance in graphs and applications on the n-queens problem
    Discret. Math. (IF 0.77) Pub Date : 2020-09-12
    O. Kolossoski; L.C. Matioli; E.M.R. Torrealba; J.G. Silva

    In this paper we introduce modular knight distance on a chessboard and use it to study the geometry of a class of solutions of the (modular) n-queens problem. Such geometry is related to the movement of a knight in classic chess, therefore we benefit from defining modular knight distance prior to the study. The concept of the distance is incorporated to graph theory with the notion of the knight graph

    更新日期:2020-09-12
  • Zero-free intervals of chromatic polynomials of hypergraphs
    Discret. Math. (IF 0.77) Pub Date : 2020-09-09
    Ruixue Zhang; Fengming Dong

    In this paper, we prove that (−∞,0) is a zero-free interval for chromatic polynomials of a family L0 of hypergraphs and (0,1) is a zero-free interval for chromatic polynomials of a subfamily L0′ of L0 of hypergraphs. These results extend known results on zero-free intervals of chromatic polynomials of graphs and hypergraphs.

    更新日期:2020-09-10
  • The rigidity of the graphs of homology spheres minus one edge
    Discret. Math. (IF 0.77) Pub Date : 2020-09-09
    Hailun Zheng

    We prove that for d−1≥3, the graph of any homology (d−1)-sphere Δ with g2(Δ)≥1 and without missing (d−1)-faces is generically redundantly d-rigid. This confirms a conjecture of Nevo and Novinsky.

    更新日期:2020-09-10
  • Inverse problems for certain subsequence sums in integers
    Discret. Math. (IF 0.77) Pub Date : 2020-09-08
    Jagannath Bhanja; Ram Krishna Pandey

    Let A be a nonempty finite set of k integers. Given a subset B of A, the sum of all elements of B is called the subset sum of B and we denote it by s(B). For a nonnegative integer α (≤k), let Σα(A)≔{s(B):B⊂A,|B|≥α}.Now, let A=(a1,…,a1︸r1copies,a2,…,a2︸r2copies,…,ak,…,ak︸rkcopies) be a finite sequence of integers with k distinct terms, where ri≥1 for i=1,2,…,k. Given a subsequence B of A, the sum of

    更新日期:2020-09-10
  • Long shortest vectors in low dimensional lattices
    Discret. Math. (IF 0.77) Pub Date : 2020-09-08
    Florian Pausinger

    For coprime integers N,a,b,c, with 0

    更新日期:2020-09-10
  • Monochromatic stars in rainbow K3-free and S3+-free colorings
    Discret. Math. (IF 0.77) Pub Date : 2020-09-08
    Xihe Li; Ligong Wang

    Given two graphs G and H, we consider the Ramsey-type problem of finding the minimum integer n (denoted by egrk(G:H)) such that n2≥k and for every N≥n, every rainbow G-free k-coloring (using exactly k colors) of the complete graph KN contains a monochromatic copy of H. In this paper, we determine egrk(K3:K1,t) for all integers t≥1 and k≥3 completely. Let S3+ be the unique graph on four vertices consisting

    更新日期:2020-09-08
  • Vertex-disjoint rainbow triangles in edge-colored graphs
    Discret. Math. (IF 0.77) Pub Date : 2020-09-08
    Jie Hu; Hao Li; Donglei Yang

    Let G be an edge-colored graph of order n. The minimum color degree of G, denoted by δc(G), is the largest integer k such that for every vertex v, there are at least k distinct colors on edges incident to v. We say that an edge-colored graph is rainbow if all its edges have different colors. In this paper, we consider vertex-disjoint rainbow triangles in edge-colored graphs. Li (2013) showed that if

    更新日期:2020-09-08
  • Edge colorings of graphs without monochromatic stars
    Discret. Math. (IF 0.77) Pub Date : 2020-09-08
    Lucas Colucci; Ervin Győri; Abhishek Methuku

    In this note, we improve on results of Hoppen, Kohayakawa and Lefmann about the maximum number of edge colorings without monochromatic copies of a star of a fixed size that a graph on n vertices may admit. Our results rely on an improved application of an entropy inequality of Shearer.

    更新日期:2020-09-08
  • Exponential lower bounds on the generalized Erdős–Ginzburg–Ziv constant
    Discret. Math. (IF 0.77) Pub Date : 2020-09-04
    Jared Bitz; Sarah Griffith; Xiaoyu He

    For a finite abelian group G, the generalized Erdős–Ginzburg–Ziv constant sk(G) is the smallest m such that a sequence of m elements in G always contains a k-element subsequence which sums to zero. If n=exp(G) is the exponent of G, the previously best known bounds for skn(Cnr) were linear in n and r when k≥2. Via a probabilistic argument, we produce the exponential lower bound s2n(Cnr)>n2[1.25+o(1)]rfor

    更新日期:2020-09-05
  • Hamiltonian cycles in 4-connected plane triangulations with few 4-separators
    Discret. Math. (IF 0.77) Pub Date : 2020-09-04
    On-Hei Solomon Lo

    Hakimi, Schmeichel and Thomassen showed in 1979 that every 4-connected triangulation on n vertices has at least n∕log2n hamiltonian cycles, and conjectured that the sharp lower bound is 2(n−2)(n−4). Recently, Brinkmann, Souffriau and Van Cleemput gave an improved lower bound 125(n−2). In this paper we show that every 4-connected triangulation with O(logn) 4-separators has Ω(n2∕log2n) hamiltonian cycles

    更新日期:2020-09-05
  • Digraphs of directed treewidth one
    Discret. Math. (IF 0.77) Pub Date : 2020-09-04
    Sebastian Wiederrecht

    We characterise digraphs of directed treewidth one in terms of forbidden butterfly minors and in terms of the structure of the cycle hypergraph to a digraph D, i.e. the hypergraph on vertex set V(D) whose hyperedges correspond to the vertex sets of directed cycles in D.

    更新日期:2020-09-05
  • Majorization, degree sequence and Aα-spectral characterization of graphs
    Discret. Math. (IF 0.77) Pub Date : 2020-09-02
    Huiqiu Lin; Liwen Zhang; Jie Xue

    Let π=(d1,d2,…,dn) and π′=(d1′,d2′,…,dn′) be two non-increasing degree sequences. We say π is majorizated by π′, denoted by π≺π′, if and only if ∑i=1ndi=∑i=1ndi′ and ∑i=1jdi≤∑i=1jdi′ for j=1,2,…,n−1. By using majorization, we show that some graphs are determined by their Aα-spectra. Firstly, we show that the lollipop graph is determined by its Aα-spectrum for 0<α<1, which generalizes the result of

    更新日期:2020-09-03
  • On the anti-Ramsey numbers of linear forests
    Discret. Math. (IF 0.77) Pub Date : 2020-09-03
    Tian-Ying Xie; Long-Tu Yuan

    For a fixed graph F, the anti-Ramsey number, AR(n,F), is the maximum number of colors in an edge-coloring of Kn which does not contain a rainbow copy of F. In this paper, we determine the exact value of anti-Ramsey numbers of linear forests for sufficiently large n, and show the extremal edge-colored graphs. This answers a question of Fang, Győri, Lu and Xiao.

    更新日期:2020-09-03
  • Spanning eulerian subdigraphs avoiding k prescribed arcs in tournaments
    Discret. Math. (IF 0.77) Pub Date : 2020-09-03
    Jørgen Bang-Jensen; Hugues Déprés; Anders Yeo

    Fraise and Thomassen (1987) proved that every (k+1)-strong tournament has a hamiltonian cycle which avoids any prescribed set of k arcs. Bang-Jensen, Havet and Yeo showed in Bang-Jensen et al. (2019) that a number of results concerning vertex-connectivity and hamiltonian cycles in tournaments have analogues when we replace vertex connectivity by arc-connectivity and hamiltonian cycles by spanning eulerian

    更新日期:2020-09-03
  • On degree sum conditions for directed path-factors with a specified number of paths
    Discret. Math. (IF 0.77) Pub Date : 2020-09-03
    Shuya Chiba; Eishi Mishio; Pierre Montalbano

    A directed path-factor of a digraph is a spanning subdigraph consisting of a union of vertex-disjoint directed paths in the digraph. In this paper, we give the following result: If D is a digraph of order n≥(2ℓ−1)k−1, and if dD+(u)+dD−(v)≥n−k for every two distinct vertices u and v with (u,v)∉A(D), then D has a directed path-factor with exactly k directed paths of order at least ℓ(≥2). To show this

    更新日期:2020-09-03
  • On existence of perfect bitrades in Hamming graphs
    Discret. Math. (IF 0.77) Pub Date : 2020-09-02
    I.Yu. Mogilnykh; F.I. Solov’eva

    A pair (T0,T1) of disjoint sets of vertices of a graph G is called a perfect bitrade in G if any ball of radius 1 in G contains exactly one vertex in T0 and T1 or none simultaneously. The volume of a perfect bitrade (T0,T1) is the size of T0. If C0 and C1 are distinct perfect codes with minimum distance 3 in G then (C0∖C1,C1∖C0) is a perfect bitrade. For any q≥3, r≥1 we construct perfect bitrades of

    更新日期:2020-09-02
  • Analytic properties of combinatorial triangles related to Motzkin numbers
    Discret. Math. (IF 0.77) Pub Date : 2020-09-01
    Xi Chen; Yi Wang; Sai-Nan Zheng

    The Motzkin numbers count the number of lattice paths which go from (0,0) to (n,0) using steps (1,1),(1,0) and (1,−1) and never go below the x-axis. Let Mn,k be the number of such paths with exactly k horizontal steps. We investigate the analytic properties of various combinatorial triangles related to the Motzkin triangle [Mn,k]n,k≥0, including their total positivity, the real-rootedness and interlacing

    更新日期:2020-09-02
  • Vertex-disjoint cycles in local tournaments
    Discret. Math. (IF 0.77) Pub Date : 2020-08-28
    Ruijuan Li; Juanjuan Liang; Xinhong Zhang; Yubao Guo

    Let r be a positive integer. The Bermond–Thomassen conjecture states that, a digraph of minimum out-degree at least 2r−1 contains r vertex-disjoint directed cycles. A digraph D is called a local tournament if for every vertex x of D, both the out-neighbours and the in-neighbours of x induce tournaments. Note that tournaments form the subclass of local tournaments. In this paper, we verify that the

    更新日期:2020-08-29
  • On clique immersions in line graphs
    Discret. Math. (IF 0.77) Pub Date : 2020-08-28
    Michael Guyer; Jessica McDonald

    We prove that if L(G) immerses Kt then L(mG) immerses Kmt, where mG is the graph obtained from G by replacing each edge in G with a parallel edge of multiplicity m. This implies that when G is a simple graph, L(mG) satisfies a conjecture of Abu-Khzam and Langston. We also show that when G is a line graph, G has a Kt-immersion iff G has a Kt-minor whenever t≤4, but this equivalence fails in both directions

    更新日期:2020-08-29
  • Cyclic derangement polynomials of the wreath product Cr≀Sn
    Discret. Math. (IF 0.77) Pub Date : 2020-08-28
    Lily Li Liu; Mengmeng Dong

    A classical problem in enumerative combinatorics is to count the number of derangements, i.e., permutations with no fixed point. In this paper, we study the cyclic derangement polynomials, which count the number of derangements in the wreath product Cr≀Sn. We first establish the relation between the cyclic Eulerian polynomials and the cyclic derangement polynomials. Then we show that the coefficients

    更新日期:2020-08-28
  • Antimagic orientations of graphs with large maximum degree
    Discret. Math. (IF 0.77) Pub Date : 2020-08-28
    Donglei Yang; Joshua Carlson; Andrew Owens; K.E. Perry; Inne Singgih; Zi-Xia Song; Fangfang Zhang; Xiaohong Zhang

    Given a digraph D with m arcs, a bijection τ:A(D)→{1,2,…,m} is an antimagic labeling of D if no two vertices in D have the same vertex-sum, where the vertex-sum of a vertex u in D under τ is the sum of labels of all arcs entering u minus the sum of labels of all arcs leaving u. We say (D,τ) is an antimagic orientation of a graph G if D is an orientation of G and τ is an antimagic labeling of D. Motivated

    更新日期:2020-08-28
  • An Erdős–Ko–Rado theorem for unions of length 2 paths
    Discret. Math. (IF 0.77) Pub Date : 2020-08-28
    Carl Feghali; Glenn Hurlbert; Vikram Kamat

    A family of sets is intersecting if any two sets in the family intersect. Given a graph G and an integer r≥1, let I(r)(G) denote the family of independent sets of size r of G. For a vertex v of G, the family of independent sets of size r that contain v is called an r-star. Then G is said to be r-EKR if no intersecting subfamily of I(r)(G) is bigger than the largest r-star. Let n be a positive integer

    更新日期:2020-08-28
  • Constructions of several classes of linear codes with a few weights
    Discret. Math. (IF 0.77) Pub Date : 2020-08-28
    Hongwei Liu; Jianxin Ouyang; Xiaoqiang Wang

    Recently, linear codes with a few weights have been constructed and extensively studied due to their applications in secret sharing, authentication codes, association schemes, and strongly regular graphs. In this paper, we construct several classes of linear codes with a few weights over Fp, where p is an odd prime. The weight distributions of these constructed codes are also settled by applications

    更新日期:2020-08-28
  • A note on purely imaginary independence roots
    Discret. Math. (IF 0.77) Pub Date : 2020-08-28
    Jason I. Brown; Ben Cameron

    The independence polynomial of a graph is the generating polynomial for the number of independent sets of each cardinality and its roots are called independence roots. We investigate here purely imaginary independence roots. We show that for all k≥4, there are connected graphs with independence number k and purely imaginary independence roots. We also show that every graph is an induced subgraph of

    更新日期:2020-08-28
  • Properly colored C4’s in edge-colored graphs
    Discret. Math. (IF 0.77) Pub Date : 2020-08-26
    Chuandong Xu; Colton Magnant; Shenggui Zhang

    When many colors appear in edge-colored graphs, it is only natural to expect rainbow subgraphs to appear. This anti-Ramsey problem has been studied thoroughly and yet there remain many gaps in the literature. Expanding upon classical and recent results forcing rainbow triangles to appear, we consider similar conditions which force the existence of a properly colored copy of C4.

    更新日期:2020-08-27
  • Complementary Schur asymptotics for partitions
    Discret. Math. (IF 0.77) Pub Date : 2020-08-26
    Jaroslav Hančl

    We construct partition ideals with highly oscillating growth function. Partition ideal X is a set of integer partitions closed under the subpartition relation. Its growth function p(n,X) counts the number of partitions of n that are in X. We present a partition ideal with growth function being infinitely many times both zero and close to the partition function p(n) of all partitions of n. Our main

    更新日期:2020-08-26
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