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  • 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
  • The maximum number of Parter vertices of acyclic matrices
    Discret. Math. (IF 0.77) Pub Date : 2020-11-17
    Amélia Fonseca; Ângela Mestre; Ali Mohammadian; Cecília Perdigão; Maria Manuel Torres

    A vertex v of the underlying graph of a symmetric matrix A is called ‘Parter’ if the nullity of the matrix obtained from A by removing the row and column indexed by v is more than the nullity of A. Let A be a singular symmetric matrix with rank r whose underlying graph is a tree. It is known that the number of Parter vertices of A is at most r−1. We prove that when r is odd this number is at most r−2

    更新日期:2020-11-17
  • Improving High-Meets-Low technique to generate odd-variable resilient Boolean functions with currently best nonlinearity
    Discret. Math. (IF 0.77) Pub Date : 2020-11-16
    Yujuan Sun

    This paper studies the construction of resilient Boolean functions in odd variables with strictly almost optimal (SAO) nonlinearity. Based on High-Meets-Low technique, we present a new construction to obtain Boolean functions with higher resiliency order and currently best known nonlinearity. It is shown that this method can achieve a better tradeoff between resiliency and nonlinearity than those of

    更新日期:2020-11-17
  • Invariant spanning double rays in amenable groups
    Discret. Math. (IF 0.77) Pub Date : 2020-11-17
    Agelos Georgakopoulos; Florian Lehner

    A well-known result of Benjamini, Lyons, Peres, and Schramm states that if G is a finitely generated Cayley graph of a group Γ, then Γ is amenable if and only if G admits a Γ-invariant random spanning tree with at most two ends. We show that this is equivalent to the existence of a Γ-invariant random spanning double ray in a power of G.

    更新日期:2020-11-17
  • The perimeter generating function for nondirected diagonally convex polyominoes
    Discret. Math. (IF 0.77) Pub Date : 2020-11-13
    Svjetlan Feretić

    A polyomino is a finite, edge-connected set of cells in the plane. At the present time, an enumeration of all polyominoes is nowhere in sight. On the other hand, there are several subsets of polyominoes for which generating functions are known. For example, there exists extensive knowledge about column-convex polyominoes, a model introduced by Temperley in 1956. While studying column-convex polyominoes

    更新日期:2020-11-13
  • Addition chains, vector chains, and efficient computation
    Discret. Math. (IF 0.77) Pub Date : 2020-11-13
    Edward G. Thurber; Neill M. Clift

    The computational generation of minimal-length addition chains over the last sixty years or so has yielded many new results concerning addition chains. The computer search for such chains often involves a search tree which is traversed to produce addition chains. A depth first search of the search tree is greatly facilitated by pruning bounds which are sequences whose terms pair up with corresponding

    更新日期:2020-11-13
  • On Gallai’s conjecture for graphs with maximum degree 6
    Discret. Math. (IF 0.77) Pub Date : 2020-11-12
    Yanan Chu; Genghua Fan; Qinghai Liu

    Gallai’s conjecture asserts that a connected graph G on n vertices can be decomposed into ⌈n2⌉ paths. In this paper, we prove that a connected graph with maximum degree 6, in which the vertices of degree 6 form an independent set, can be decomposed into n2 paths, unless it is K3, K5 and K5−.

    更新日期:2020-11-12
  • Algorithms for q-ary error-correcting codes with limited magnitude and feedback
    Discret. Math. (IF 0.77) Pub Date : 2020-11-07
    Christian Deppe; Vladimir Lebedev

    Berlekamp and Zigangirov completely determined the capacity error function for binary error correcting codes with noiseless feedback. It is still an unsolved problem if the upper bound for the capacity error function in the non-binary case of Ahlswede, Lebedev, and Deppe is sharp. We consider wraparound channels with limited magnitude and noiseless feedback. We completely determine the capacity error

    更新日期:2020-11-09
  • Maximum sparse induced subgraphs of the binomial random graph with given number of edges
    Discret. Math. (IF 0.77) Pub Date : 2020-11-06
    Dmitry Kamaldinov; Arkadiy Skorkin; Maksim Zhukovskii

    We prove that a.a.s. the maximum size of an induced subtree of the binomial random graph G(n,p) is concentrated in 2 consecutive points. We also prove that, given a non-negative integer-valued function t(k)<εk2, under a certain smoothness condition on this function, a.a.s. the maximum size k of an induced subgraph with exactly t(k) edges of G(n,p) is concentrated in 2 consecutive points as well.

    更新日期:2020-11-06
  • Generalized asymptotic Sidon basis
    Discret. Math. (IF 0.77) Pub Date : 2020-11-04
    Sándor Z. Kiss; Csaba Sándor

    Let h,k≥2 be integers. We say a set A of positive integers is an asymptotic basis of order k if every large enough positive integer can be represented as the sum of k terms from A. A set of positive integers A is called a Bh[g] set if every positive integer can be represented as the sum of h terms from A in at most g different ways. In this paper we prove the existence of Bh[1] sets which are asymptotic

    更新日期:2020-11-04
  • On the number of distinct differences in an intersecting family
    Discret. Math. (IF 0.77) Pub Date : 2020-11-04
    Peter Frankl

    For a family F of subsets define F∖F={F∖F′:F,F′∈F}. The family F is called intersecting if F∩F′≠0̸ for all F,F′∈F. We prove best possible upper bounds for |F∖F| in the case that F is intersecting.

    更新日期:2020-11-04
  • The m-Schröder paths and m-Schröder numbers
    Discret. Math. (IF 0.77) Pub Date : 2020-11-03
    Sheng-Liang Yang; Mei-yang Jiang

    In this paper, we introduce a variant of the m-Schröder paths, i.e. of those lattice paths from (0,0) to (mn,0) with up steps U=(1,1), down steps D(m)=(1,1−m) and horizontal-down steps H(m)=(2,2−m), which never go below the x-axis. The number of m-Schröder paths of length mn is called the nth m-Schröder number, while the number of m-Schröder paths of length mn with no horizontal-down steps ending on

    更新日期:2020-11-03
  • The cubic graphs with finite cyclic vertex connectivity larger than girth
    Discret. Math. (IF 0.77) Pub Date : 2020-11-01
    Jun Liang; Dingjun Lou; Zan-Bo Zhang

    Cyclic (vertex and edge) connectivity is an important concept in graphs. While cyclic edge connectivity (cλ) has been studied for many years, the study at cyclic vertex connectivity (cκ) is still at the initial stage. And cκ seems to be more complicated than cλ. We have got a sufficient condition that ν(G)≥2g(k−1) for cκ≠∞. On the other hand, if ν(G)<2g(k−1), then we have cκ=∞, or cκ≤(k−2)g, or (k−2)g

    更新日期:2020-11-02
  • On the cyclic coloring conjecture
    Discret. Math. (IF 0.77) Pub Date : 2020-11-02
    Stanislav Jendrol’; Roman Soták

    A cyclic coloring of a plane graph G is a coloring of its vertices such that vertices incident with the same face have distinct colors. The minimum number of colors in a cyclic coloring of a plane graph G is its cyclic chromatic number χc(G). Let Δ∗(G) be the maximum face degree of a graph G. In this note we show that to prove the Cyclic Coloring Conjecture of Borodin from 1984, saying that every connected

    更新日期:2020-11-02
  • Finite cubic graphs admitting a cyclic group of automorphism with at most three orbits on vertices
    Discret. Math. (IF 0.77) Pub Date : 2020-10-21
    Primož Potočnik; Micael Toledo

    The theory of voltage graphs has become a standard tool in the study of graphs admitting a semiregular group of automorphisms. We introduce the notion of a cyclic generalised voltage graph to extend the scope of this theory to graphs admitting a cyclic group of automorphisms that may not be semiregular. We use this new tool to classify all cubic graphs admitting a cyclic group of automorphisms with

    更新日期:2020-10-30
  • 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:2020-10-30
  • 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:2020-10-30
  • On the complexity of all (g,f)-factors problem
    Discret. Math. (IF 0.77) Pub Date : 2020-10-21
    Hongliang Lu; Wei Wang; Yaolin Jiang

    Let G be a graph with vertex set V and let g,f:V→Z+ be two functions such that g≤f. We say that G has all (g,f)-factors if G has an h-factor for every h:V→Z+ such that g(v)≤h(v)≤f(v) for every v∈V and ∑v∈Vh(v)≡0(mod2). Two decades ago, Niessen derived from Tutte’s f-factor theorem a similar characterization for the property of graphs having all (g,f)-factors and asked whether there is a polynomial

    更新日期:2020-10-30
  • 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:2020-10-30
  • Results on partial geometries with an abelian Singer group of rigid type
    Discret. Math. (IF 0.77) Pub Date : 2020-10-23
    Stefaan De Winter; Ellen Kamischke; Eric Neubert; Zeying Wang

    A partial geometry S admitting an abelian Singer group G is called of rigid type if all lines of S have a trivial stabilizer in G. In this paper, we show that if a partial geometry of rigid type has fewer than 1000000 points it must be the Van Lint–Schrijver geometry or be a hypothetical geometry with 1024 or 4096 or 194481 points, which provides evidence that partial geometries of rigid type are very

    更新日期:2020-10-30
  • Circumference of a graph and its distance dominating longest cycles
    Discret. Math. (IF 0.77) Pub Date : 2020-10-26
    Yibin Fang; Liming Xiong

    In this note, we prove the following: Let G be a k-connected graph (k≥2) with circumference c(G) and m a non-negative integer. Then • [(1)] Either c(G)≥(2m+2)k, or dG(v,C)≤m for any longest cycle C and any vertex v of G. • [(2)] Either c(G)≥(2m+3)k, or dG(e,C)≤m for any longest cycle C and any edge e of G. When m=0, C in (1) and (2) are well-known Hamiltonian cycle and dominating longest cycle, respectively

    更新日期:2020-10-30
  • Upper bounds on the signed edge domination number of a graph
    Discret. Math. (IF 0.77) Pub Date : 2020-10-23
    Fengming Dong; Jun Ge; Yan Yang

    A signed edge domination function (or SEDF) of a simple graph G=(V,E) is a function f:E→{1,−1} such that ∑e′∈N[e]f(e′)≥1 holds for each edge e∈E, where N[e] is the set of edges in G that share at least one endpoint with e. Let γs′(G) denote the minimum value of f(G) among all SEDFs f, where f(G)=∑e∈Ef(e). In 2005, Xu conjectured that γs′(G)≤n−1, where n is the order of G. This conjecture has been proved

    更新日期:2020-10-30
  • Central limit theorem for the prefix exchange distance under Ewens sampling formula
    Discret. Math. (IF 0.77) Pub Date : 2020-10-28
    Simona Grusea; Anthony Labarre

    The prefix exchange distance of a permutation is the length of its shortest factorisation into transpositions that all contain 1. Using a probabilistic approach, we obtain expressions for the mean and the variance, and prove the asymptotic normality of the distribution of this distance for a random permutation verifying the Ewens sampling formula. Analogous results in the uniform setting follow as

    更新日期:2020-10-30
  • 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
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