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  • On the inclusion chromatic index of a graph
    J. Graph Theory (IF 0.922) Pub Date : 2020-10-21
    Jakub Przybyło; Jakub Kwaśny

    Let χ ′ ⊂ ( G ) be the least number of colours necessary to properly colour the edges of a graph G with minimum degree δ ≥ 2 so that the set of colours incident with any vertex is not contained in a set of colours incident to any of its neighbours. We provide an infinite family of examples of graphs G with χ ′ ⊂ ( G ) ≥ ( 1 + 1 δ − 1 ) Δ , where Δ is the maximum degree of G , and we conjecture that

    更新日期:2020-10-22
  • Flexibility of triangle‐free planar graphs
    J. Graph Theory (IF 0.922) Pub Date : 2020-10-21
    Zdeněk Dvořák; Tomáš Masařík; Jan Musílek; Ondřej Pangrác

    Let G be a planar graph with a list assignment L . Suppose a preferred color is given for some of the vertices. We prove that if G is triangle‐free and all lists have size at least four, then there exists an L ‐coloring respecting at least a constant fraction of the preferences.

    更新日期:2020-10-22
  • Good orientations of unions of edge‐disjoint spanning trees
    J. Graph Theory (IF 0.922) Pub Date : 2020-10-19
    Jørgen Bang‐Jensen; Stéphane Bessy; Jing Huang; Matthias Kriesell

    In this paper, we exhibit connections between the following subjects: Tree packing in graphs and digraphs (both behave completely different), the rigidity matroid of a graph, Henneberg moves on trees, the conjectures of Thomassen and Matthews and Sumner, and (s,t)‐orderings of digraphs. We do this by studying graphs which admit acyclic orientations that contain an out‐branching and in‐branching which

    更新日期:2020-10-19
  • Uniquely restricted matchings in subcubic graphs without short cycles
    J. Graph Theory (IF 0.922) Pub Date : 2020-10-19
    M. Fürst; D. Rautenbach

    A matching M in a graph G is uniquely restricted if no other matching in G covers the same set of vertices. We prove that any connected subcubic graph with n vertices and girth at least 5 contains a uniquely restricted matching of size at least ( n − 1 ) / 3 except for two exceptional cubic graphs of order 14 and 20.

    更新日期:2020-10-19
  • Minimal braces
    J. Graph Theory (IF 0.922) Pub Date : 2020-10-15
    Phelipe A. Fabres; Nishad Kothari; Marcelo H. de Carvalho

    McCuaig proved a generation theorem for braces, and used it as the principal induction tool to obtain a structural characterization of Pfaffian braces. A brace is minimal if deleting any edge results in a graph that is not a brace. From McCuaig's brace generation theorem, we derive our main theorem that may be viewed as an induction tool for minimal braces. As an application, we prove that a minimal

    更新日期:2020-10-17
  • Curvatures, graph products and Ricci flatness
    J. Graph Theory (IF 0.922) Pub Date : 2020-10-12
    David Cushing; Supanat Kamtue; Riikka Kangaslampi; Shiping Liu; Norbert Peyerimhoff

    In this paper, we compare Ollivier–Ricci curvature and Bakry–Émery curvature notions on combinatorial graphs and discuss connections to various types of Ricci flatness. We show that nonnegativity of Ollivier–Ricci curvature implies the nonnegativity of Bakry–Émery curvature under triangle‐freeness and an additional in‐degree condition. We also provide examples that both conditions of this result are

    更新日期:2020-10-13
  • On the maximum number of maximum independent sets in connected graphs
    J. Graph Theory (IF 0.922) Pub Date : 2020-10-11
    Elena Mohr; Dieter Rautenbach

    We characterize the connected graphs of given order n and given independence number α that maximize the number of maximum independent sets. For 3 ≤ α ≤ n ∕ 2 , there is a unique such graph that arises from the disjoint union of α cliques of orders ⌈ n α ⌉ and ⌊ n α ⌋ , which is the complement of a Turán graph, by selecting a vertex x in a largest clique and adding an edge between x and a vertex in

    更新日期:2020-10-12
  • Clique‐convergence is undecidable for automatic graphs
    J. Graph Theory (IF 0.922) Pub Date : 2020-10-07
    C. Cedillo; M. A. Pizaña

    The clique operator transforms a graph G into its clique graph K ( G ) , which is the intersection graph of all the (maximal) cliques of G . Iterated clique graphs are then defined by K n ( G ) = K ( K n − 1 ( G ) ) , K 0 ( G ) = G . If there are some n ≠ m such that K n ( G ) ≅ K m ( G ) , then we say that G is clique‐convergent. The clique graph operator and iterated clique graphs have been studied

    更新日期:2020-10-07
  • A characterization of the subcubic graphs achieving equality in the Haxwell‐Scott lower bound for the matching number
    J. Graph Theory (IF 0.922) Pub Date : 2020-09-30
    Michael A. Henning; Zekhaya B. Shozi

    In 2004, Biedl et al proved that if G is a connected cubic graph of order n , then α ′ ( G ) ≥ 1 9 ( 4 n − 1 ) , where α ′ ( G ) is the matching number of G . The graphs achieving equality in this bound were characterized in 2010 by O and West. In 2017, Haxwell and Scott proved that if G is a connected subcubic graph, then α ′ ( G ) ≥ 4 9 n 3 ( G ) + 3 9 n 2 ( G ) + 2 9 n 1 ( G ) − 1 9 , where n i

    更新日期:2020-09-30
  • The maximum number of maximum dissociation sets in trees
    J. Graph Theory (IF 0.922) Pub Date : 2020-09-30
    Jianhua Tu; Zhipeng Zhang; Yongtang Shi

    A subset of vertices is a maximum independent set if no two of the vertices are adjacent and the subset has maximum cardinality. A subset of vertices is called a maximum dissociation set if it induces a subgraph with vertex degree at most 1, and the subset has maximum cardinality. Zito proved that the maximum number of maximum independent sets of a tree of order n is 2 n − 3 2 if n is odd, and 2 n

    更新日期:2020-09-30
  • A survey of χ‐boundedness
    J. Graph Theory (IF 0.922) Pub Date : 2020-08-24
    Alex Scott; Paul Seymour

    If a graph has bounded clique number and sufficiently large chromatic number, what can we say about its induced subgraphs? András Gyárfás made a number of challenging conjectures about this in the early 1980s, which have remained open until recently; but in the last few years there has been substantial progress. This is a survey of where we are now.

    更新日期:2020-09-26
  • Group connectivity under 3‐edge‐connectivity
    J. Graph Theory (IF 0.922) Pub Date : 2020-09-23
    Miaomiao Han; Jiaao Li; Xueliang Li; Meiling Wang

    Let S , T be two distinct finite Abelian groups with | S | = | T | . A fundamental theorem of Tutte shows that a graph admits a nowhere‐zero S ‐flow if and only if it admits a nowhere‐zero T ‐flow. Jaeger et al in 1992 introduced group connectivity as an extension of flow theory, and they asked whether such a relation holds for group connectivity analogy. It was negatively answered by Hušek et al in

    更新日期:2020-09-24
  • On the mean subtree order of graphs under edge addition
    J. Graph Theory (IF 0.922) Pub Date : 2020-09-23
    Ben Cameron; Lucas Mol

    For a graph G , the mean subtree order of G is the average order of a subtree of G . In this note, we provide counterexamples to a recent conjecture of Chin, Gordon, MacPhee, and Vincent, that for every connected graph G and every pair of distinct vertices u and v of G , the addition of the edge between u and v increases the mean subtree order. In fact, we show that the addition of a single edge between

    更新日期:2020-09-24
  • Graphs of bounded depth‐2 rank‐brittleness
    J. Graph Theory (IF 0.922) Pub Date : 2020-09-14
    O‐joung Kwon; Sang‐il Oum

    We characterize classes of graphs closed under taking vertex‐minors and having no P n and no disjoint union of n copies of the 1‐subdivision of K 1 , n for some n . Our characterization is described in terms of a tree of radius 2 whose leaves are labeled by the vertices of a graph G , and the width is measured by the maximum possible cut‐rank of a partition of V ( G ) induced by splitting an internal

    更新日期:2020-09-15
  • Size reconstructibility of graphs
    J. Graph Theory (IF 0.922) Pub Date : 2020-08-05
    Carla Groenland; Hannah Guggiari; Alex Scott

    The deck of a graph G is given by the multiset of (unlabeled) subgraphs { G − v : v ∈ V ( G ) } . The subgraphs G − v are referred to as the cards of G . Brown and Fenner recently showed that, for n ≥ 29 , the number of edges of a graph G can be computed from any deck missing 2 cards. We show that, for sufficiently large n , the number of edges can be computed from any deck missing at most 1 20 n cards

    更新日期:2020-08-06
  • Proving a conjecture on chromatic polynomials by counting the number of acyclic orientations
    J. Graph Theory (IF 0.922) Pub Date : 2020-07-31
    Fengming Dong; Jun Ge; Helin Gong; Bo Ning; Zhangdong Ouyang; Eng Guan Tay

    The chromatic polynomial P ( G , x ) of a graph G of order n can be expressed as ∑ i = 1 n ( − 1 ) n − i a i x i , where a i is interpreted as the number of broken‐cycle‐free spanning subgraphs of G with exactly i components. The parameter ϵ ( G ) = ∑ i = 1 n ( n − i ) a i ∕ ∑ i = 1 n a i is the mean size of a broken‐cycle‐free spanning subgraph of G . In this article, we confirm and strengthen a conjecture

    更新日期:2020-07-31
  • Hamiltonian cycles in planar cubic graphs with facial 2‐factors, and a new partial solution of Barnette's Conjecture
    J. Graph Theory (IF 0.922) Pub Date : 2020-07-18
    Behrooz Bagheri Gh; Tomas Feder; Herbert Fleischner; Carlos Subi

    We study the existence of hamiltonian cycles in plane cubic graphs G having a facial 2‐factor Q . Thus hamiltonicity in G is transformed into the existence of a (quasi) spanning tree of faces in the contraction G ∕ Q . In particular, we study the case where G is the leapfrog extension (called vertex envelope of a plane cubic graph G 0 . As a consequence we prove hamiltonicity in the leapfrog extension

    更新日期:2020-07-18
  • Zip product of graphs and crossing numbers
    J. Graph Theory (IF 0.922) Pub Date : 2020-07-14
    Zhangdong Ouyang; Yuanqiu Huang; Fengming Dong; Eng Guan Tay

    D. Bokal proved that the crossing number is additive for the zip product under the condition of having two coherent bundles in the zipped graphs. This property is very effective when dealing with the crossing numbers of (capped) Cartesian product of trees with graphs containing a dominating vertex. In this paper, we first prove that the crossing number is still additive for the zip product under a

    更新日期:2020-07-15
  • Arc‐transitive maps with underlying Rose Window graphs
    J. Graph Theory (IF 0.922) Pub Date : 2020-07-14
    Isabel Hubard; Alejandra Ramos‐Rivera; Primož Šparl

    In the late 1990s, Graver and Watkins initiated the study of all edge‐transitive maps. Recently, Gareth Jones revisited the study of such maps and suggested classifying the maps in terms of either their automorphism groups or their underlying graphs. A natural step towards classifying edge‐transitive maps is to study the arc‐transitive ones. In this paper, we investigate the connection of a class of

    更新日期:2020-07-15
  • Minimum weighted clique cover on claw‐free perfect graphs
    J. Graph Theory (IF 0.922) Pub Date : 2020-07-10
    Flavia Bonomo; Gianpaolo Oriolo; Claudia Snels

    The first combinatorial algorithm for the minimum weighted clique cover (MWCC) in a claw‐free perfect graph G due to Hsu and Nemhauser dates back to 1984. It is essentially a “dual” algorithm as it relies on any algorithm for the maximum weighted stable set (MWSS) problem in claw‐free graphs and, taking into account the best‐known complexity for the latter problem, its complexity is O ( ∣ V ( G ) ∣

    更新日期:2020-07-10
  • Proper‐walk connection number of graphs
    J. Graph Theory (IF 0.922) Pub Date : 2020-06-30
    Jørgen Bang‐Jensen; Thomas Bellitto; Anders Yeo

    This paper studies the problem of proper‐walk connection number: given an undirected connected graph, our aim is to colour its edges with as few colours as possible so that there exists a properly coloured walk between every pair of vertices of the graph, that is, a walk that does not use consecutively two edges of the same colour. The problem was already solved on several classes of graphs but still

    更新日期:2020-07-01
  • Multiple list coloring of 3‐choice critical graphs
    J. Graph Theory (IF 0.922) Pub Date : 2020-07-01
    Rongxing Xu; Xuding Zhu

    A graph G is called 3‐choice critical if G is not 2‐choosable but any proper subgraph is 2‐choosable. A characterization of 3‐choice critical graphs was given by Voigt in 1998. Voigt conjectured that if G is a bipartite 3‐choice critical graph, then G is ( 4 m , 2 m ) ‐choosable for every integer m . This conjecture was disproved by Meng et al. in 2017. They showed that if G = Θ r , s , t where r

    更新日期:2020-07-01
  • Hamiltonicity, pancyclicity, and full cycle extendability in multipartite tournaments
    J. Graph Theory (IF 0.922) Pub Date : 2020-06-27
    Zan‐Bo Zhang; Xiaoyan Zhang; Gregory Gutin; Dingjun Lou

    A digraph D with n vertices is Hamiltonian (pancyclic and vertex‐pancyclic, respectively) if D contains a Hamilton cycle (a cycle of every length 3 , 4 , … , n , for every vertex v ∈ V ( D ) , a cycle of every length 3 , 4 , … , n through v , respectively.) It is well‐known that a strongly connected tournament is Hamiltonian, pancyclic, and vertex pancyclic. A digraph D is cycle extendable if for every

    更新日期:2020-06-28
  • List Ramsey numbers
    J. Graph Theory (IF 0.922) Pub Date : 2020-06-27
    Noga Alon; Matija Bucić; Tom Kalvari; Eden Kuperwasser; Tibor Szabó

    We introduce a list‐coloring extension of classical Ramsey numbers. We investigate when the two Ramsey numbers are equal, and in general, how far apart they can be from each other. We find graph sequences where the two are equal and where they are far apart. For ℓ ‐uniform cliques we prove that the list Ramsey number is bounded by an exponential function, while it is well known that the Ramsey number

    更新日期:2020-06-28
  • k‐quasi‐transitive digraphs of large diameter
    J. Graph Theory (IF 0.922) Pub Date : 2020-06-27
    Jesús Alva‐Samos; César Hernández‐Cruz

    Given an integer k with k ≥ 2 , a digraph D = ( V D , A D ) is k ‐quasi‐transitive if for every u v ‐directed path of length k in D , we have ( u , v ) ∈ A D or ( v , u ) ∈ A D (or both). In this study, we prove that if k is an odd integer, k ≥ 5 , then every strong k ‐quasi‐transitive digraph of diameter at least k + 2 admits a partition of its vertex set V D = ( V 1 , V 2 ) such that D [ V 1 ] is

    更新日期:2020-06-28
  • Order plus size of τ‐critical graphs
    J. Graph Theory (IF 0.922) Pub Date : 2020-06-26
    András Gyárfás; Jenő Lehel

    Let G = ( V , E ) be a τ ‐critical graph with τ ( G ) = t . Erdős and Gallai proved that ∣ V ∣ ≤ 2 t and the bound ∣ E ∣ ≤ t + 1 2 was obtained by Erdős, Hajnal, and Moon. We give here the sharp combined bound ∣ E ∣ + ∣ V ∣ ≤ t + 2 2 and find all graphs with equality.

    更新日期:2020-06-26
  • Short rainbow cycles in graphs and matroids
    J. Graph Theory (IF 0.922) Pub Date : 2020-06-23
    Matt DeVos; Matthew Drescher; Daryl Funk; Sebastián González Hermosillo de la Maza; Krystal Guo; Tony Huynh; Bojan Mohar; Amanda Montejano

    Let G be a simple n ‐vertex graph and c be a coloring of E ( G ) with n colors, where each color class has size at least 2. We prove that ( G , c ) contains a rainbow cycle of length at most ⌈ n 2 ⌉ , which is best possible. Our result settles a special case of a strengthening of the Caccetta‐Häggkvist conjecture, due to Aharoni. We also show that the matroid generalization of our main result also

    更新日期:2020-06-23
  • On the existence and the enumeration of bipartite regular representations of Cayley graphs over abelian groups
    J. Graph Theory (IF 0.922) Pub Date : 2020-06-23
    Jia‐Li Du; Yan‐Quan Feng; Pablo Spiga

    In this paper, we are interested in the asymptotic enumeration of bipartite Cayley digraphs and Cayley graphs over abelian groups. Let A be an abelian group and let ι be the automorphism of A defined by a ι = a − 1 , for every a ∈ A . A Cayley graph Cay ( A , S ) is said to have an automorphism group as small as possible if Aut ( Cay ( A , S ) ) = 〈 A , ι 〉 . In this paper, we show that, except for

    更新日期:2020-06-23
  • Families in posets minimizing the number of comparable pairs
    J. Graph Theory (IF 0.922) Pub Date : 2020-06-23
    József Balogh; Šárka Petříčková; Adam Zsolt Wagner

    Given a graded poset P we say a family F ⊆ P is centered if it is obtained by ‘taking sets as close to the middle layer as possible.’ A poset P is said to have the centeredness property if for any M , among all families of size M in P , centered families contain the minimum number of comparable pairs. Kleitman showed that the Boolean lattice { 0 , 1 } n has the centeredness property. It was conjectured

    更新日期:2020-06-23
  • Proof of a conjecture on the nullity of a graph
    J. Graph Theory (IF 0.922) Pub Date : 2020-05-20
    Long Wang; Xianya Geng

    Let G be a finite undirected graph without loops and multiple edges. The nullity of G , written as η ( G ) , is defined to be the multiplicity of 0 as an eigenvalue of its adjacency matrix. The left problem of establishing an upper bound for an arbitrary graph in terms of order and maximum degree was recently solved by Zhou et al. Zhou et al proved that η ( G ) ≤ Δ − 1 Δ n for an arbitrary graph G

    更新日期:2020-05-20
  • Generating simple near‐bipartite bricks
    J. Graph Theory (IF 0.922) Pub Date : 2020-05-20
    Nishad Kothari; Marcelo H. de Carvalho

    A brick is a 3‐connected graph such that the graph obtained from it by deleting any two distinct vertices has a perfect matching. A brick G is near‐bipartite if it has a pair of edges α and β such that G − { α , β } is bipartite and matching covered; examples are K 4 and the triangular prism C 6 ¯ . The significance of near‐bipartite bricks arises from the theory of ear decompositions of matching covered

    更新日期:2020-05-20
  • Partitions of hypergraphs under variable degeneracy constraints
    J. Graph Theory (IF 0.922) Pub Date : 2020-05-05
    Thomas Schweser; Michael Stiebitz

    The paper deals with partitions of hypergraphs into induced subhypergraphs satisfying constraints on their degeneracy. Our hypergraphs may have multiple edges, but no loops. Given a hypergraph H and a sequence f = ( f 1 , f 2 , … , f p ) of p ≥ 1 vertex functions f i : V ( H ) → N 0 such that f 1 ( v ) + f 2 ( v ) + ⋯ + f p ( v ) ≥ d H ( v ) for all v ∈ V ( H ) , we want to find a sequence ( H 1 ,

    更新日期:2020-05-05
  • Finding any given 2‐factor in sparse pseudorandom graphs efficiently
    J. Graph Theory (IF 0.922) Pub Date : 2020-05-05
    Jie Han; Yoshiharu Kohayakawa; Patrick Morris; Yury Person

    Given an n ‐vertex pseudorandom graph G and an n ‐vertex graph H with maximum degree at most two, we wish to find a copy of H in G , that is, an embedding φ : V ( H ) → V ( G ) so that φ ( u ) φ ( v ) ∈ E ( G ) for all u v ∈ E ( H ) . Particular instances of this problem include finding a triangle‐factor and finding a Hamilton cycle in G . Here, we provide a deterministic polynomial time algorithm

    更新日期:2020-05-05
  • Strong cliques in vertex‐transitive graphs
    J. Graph Theory (IF 0.922) Pub Date : 2020-05-03
    Ademir Hujdurović

    A clique (resp, independent set) in a graph is strong if it intersects every maximal independent set (resp, every maximal clique). A graph is clique intersect stable set (CIS) if all of its maximal cliques are strong and localizable if it admits a partition of its vertex set into strong cliques. In this paper we prove that a clique C in a vertex‐transitive graph Γ is strong if and only if ∣ C ∣ ∣ I

    更新日期:2020-05-03
  • Lines in bipartite graphs and in 2‐metric spaces
    J. Graph Theory (IF 0.922) Pub Date : 2020-05-03
    Martín Matamala; José Zamora

    The line generated by two distinct points, x and y , in a finite metric space M = ( V , d ) , is the set of points given by { z ∈ V : d ( x , y ) = | d ( x , z ) + d ( z , y ) | or d ( x , y ) = | d ( x , z ) − d ( z , y ) | } .

    更新日期:2020-05-03
  • Coloring graphs with no induced five‐vertex path or gem
    J. Graph Theory (IF 0.922) Pub Date : 2020-04-30
    Maria Chudnovsky; T. Karthick; Peter Maceli; Frédéric Maffray

    For a graph G , let χ ( G ) and ω ( G ) , respectively, denote the chromatic number and clique number of G . We give an explicit structural description of ( P 5 , gem)‐free graphs, and show that every such graph G satisfies χ ( G ) ≤ ⌈ 5 ω ( G ) 4 ⌉ . Moreover, this bound is best possible. Here a gem is the graph that consists of an induced four‐vertex path plus a vertex which is adjacent to all the

    更新日期:2020-04-30
  • Fractional chromatic number of a random subgraph
    J. Graph Theory (IF 0.922) Pub Date : 2020-04-22
    Bojan Mohar; Hehui Wu

    It is well known that a random subgraph of the complete graph K n has chromatic number Θ ( n ∕ log n ) w.h.p. Boris Bukh asked whether the same holds for a random subgraph of any n ‐chromatic graph, at least in expectation. In this paper it is shown that for every graph, whose fractional chromatic number is at least n , the fractional chromatic number of its random subgraph is at least n ∕ ( 8 log

    更新日期:2020-04-22
  • Proper orientation number of triangle‐free bridgeless outerplanar graphs
    J. Graph Theory (IF 0.922) Pub Date : 2020-04-20
    Jiangdong Ai; Stefanie Gerke; Gregory Gutin; Yongtang Shi; Zhenyu Taoqiu

    An orientation of G is a digraph obtained from G by replacing each edge by exactly one of two possible arcs with the same endpoints. We call an orientation proper if neighboring vertices have different in‐degrees. The proper orientation number of a graph G , denoted by χ → ( G ) , is the minimum maximum in‐degree of a proper orientation of G . Araujo et al asked whether there is a constant c such that

    更新日期:2020-04-20
  • A characterization of nonfeasible sets in matching covered graphs
    J. Graph Theory (IF 0.922) Pub Date : 2020-04-16
    Qinghai Liu; Qing Cui; Xing Feng; Fuliang Lu

    Let G be a matching covered graph and let X ⊆ E ( G ) . Then X is called feasible if there exist two perfect matchings M 1 and M 2 in G such that ∣ M 1 ∩ X ∣ ≢ ∣ M 2 ∩ X ∣ ( mod 2 ) ; otherwise, it is nonfeasible. For any vertex v ∈ V ( G ) , the switching operation of X in v is defined as the symmetric difference of X and ∂ ( v ) , where ∂ ( v ) denotes the set of all edges in G incident with v .

    更新日期:2020-04-16
  • Topologically 4‐chromatic graphs and signatures of odd cycles
    J. Graph Theory (IF 0.922) Pub Date : 2020-04-14
    Gord Simons; Claude Tardif; David Wehlau

    We investigate group‐theoretic “signatures” of odd cycles of a graph, and their connections to topological obstructions to 3‐colourability. In the case of signatures derived from free groups, we prove that the existence of an odd cycle with trivial signature is equivalent to having the coindex of the hom‐complex at least 2 (which implies that the chromatic number is at least 4). In the case of signatures

    更新日期:2020-04-14
  • Chromatic roots at 2 and the Beraha number B10
    J. Graph Theory (IF 0.922) Pub Date : 2020-04-13
    Daniel J. Harvey; Gordon F. Royle

    By the construction of suitable graphs and the determination of their chromatic polynomials, we resolve two open questions concerning real chromatic roots. First we exhibit graphs for which the Beraha number B 10 = ( 5 + 5 ) ∕ 2 is a chromatic root. As it was previously known that no other noninteger Beraha number is a chromatic root, this completes the determination of precisely which Beraha numbers

    更新日期:2020-04-13
  • Arc‐disjoint strong spanning subdigraphs of semicomplete compositions
    J. Graph Theory (IF 0.922) Pub Date : 2020-04-13
    Jørgen Bang‐Jensen; Gregory Gutin; Anders Yeo

    A strong arc decomposition of a digraph D = ( V , A ) is a decomposition of its arc set A into two disjoint subsets A 1 and A 2 such that both of the spanning subdigraphs D 1 = ( V , A 1 ) and D 2 = ( V , A 2 ) are strong. Let T be a digraph with t vertices u 1 , … , u t and let H 1 , … , H t be digraphs such that H i has vertices u i , j i , 1 ≤ j i ≤ n i . Then the composition Q = T [ H 1 , … , H

    更新日期:2020-04-13
  • Flexibility of planar graphs of girth at least six
    J. Graph Theory (IF 0.922) Pub Date : 2020-04-01
    Zdeněk Dvořák; Tomáš Masařík; Jan Musílek; Ondřej Pangrác

    Let G be a planar graph with a list assignment L . Suppose a preferred color is given for some of the vertices. We prove that if G has girth at least six and all lists have size at least three, then there exists an L ‐coloring respecting at least a constant fraction of the preferences.

    更新日期:2020-04-01
  • Edge precoloring extension of hypercubes
    J. Graph Theory (IF 0.922) Pub Date : 2020-03-20
    Carl Johan Casselgren; Klas Markström; Lan Anh Pham

    We consider the problem of extending partial edge colorings of hypercubes. In particular, we obtain an analogue of the positive solution to the famous Evans' conjecture on completing partial Latin squares by proving that every proper partial edge coloring of at most d − 1 edges of the d ‐dimensional hypercube Q d can be extended to a proper d ‐edge coloring of Q d . Additionally, we characterize which

    更新日期:2020-03-20
  • Distance restricted matching extension missing vertices and edges in 5‐connected triangulations of the plane
    J. Graph Theory (IF 0.922) Pub Date : 2020-03-17
    R. E. L. Aldred; Michael D. Plummer; Watcharintorn Ruksasakchai

    In [4] it was shown that in a 5‐connected even planar triangulation G, every matching M of size | M | < | V ( G ) | / 2 can be extended to a perfect matching of G, as long as the edges of M lie at distance at least 5 from each other. Somewhat later in [7], the following result was proved. Let G be a 5‐connected triangulation of a surface Σ different from the sphere. Let χ = χ ( Σ ) be the Euler characteristic

    更新日期:2020-03-17
  • Edge proximity and matching extension in projective planar graphs
    J. Graph Theory (IF 0.922) Pub Date : 2020-03-12
    Jun Fujisawa; Hiroki Seno

    A graph G with at least 2 m + 2 vertices is said to be distance d m ‐extendable if, for any matching M of G with m edges in which the edges lie at distance at least d pairwise, there exists a perfect matching of G containing M . In this paper we prove that every 5‐connected triangulation on the projective plane of even order is distance 3 7‐extendable and distance 4 m ‐extendable for any m .

    更新日期:2020-03-12
  • On induced Ramsey numbers for multiple copies of graphs
    J. Graph Theory (IF 0.922) Pub Date : 2020-03-12
    Maria Axenovich; Izolda Gorgol

    We say that a graph F strongly arrows a pair of graphs ( G , H ) and write F → ind ( G , H ) if any coloring of its edges with red and blue leads to either a red G or a blue H appearing as induced subgraphs of F . The induced Ramsey number, IR ( G , H ) , is defined as min { | V ( F ) | : F → ind ( G , H ) } . We consider the connection between the induced Ramsey number for a pair of two connected

    更新日期:2020-03-12
  • Pure pairs. III. Sparse graphs with no polynomial‐sized anticomplete pairs
    J. Graph Theory (IF 0.922) Pub Date : 2020-03-07
    Maria Chudnovsky; Jacob Fox; Alex Scott; Paul Seymour; Sophie Spirkl

    A graph is H ‐free if it has no induced subgraph isomorphic to H , and |G| denotes the number of vertices of G . A conjecture of Conlon, Sudakov and the second author asserts that: For every graph H , there exists ε > 0 such that in every H ‐free graph G with |G| there are two disjoint sets of vertices, of sizes at least ε | G | ε and ε | G | , complete or anticomplete to each other. This is equivalent

    更新日期:2020-03-07
  • On nonfeasible edge sets in matching‐covered graphs
    J. Graph Theory (IF 0.922) Pub Date : 2020-03-07
    Xiao Zhao; Fengming Dong; Sheng Chen

    Let G = ( V , E ) be a matching‐covered graph and X be an edge set of G . X is said to be feasible if there exist two perfect matchings M 1 and M 2 in G such that | M 1 ∩ X | ≢ | M 2 ∩ X | ( mod 2 ) . For any V 0 ⊆ V , X is said to be switching‐equivalent to X ⊕ ∇ G ( V 0 ) , where ∇ G ( V 0 ) is the set of edges in G each of which has exactly one end in V 0 and A ⊕ B is the symmetric difference of

    更新日期:2020-03-07
  • Graph‐like compacta: Characterizations and Eulerian loops
    J. Graph Theory (IF 0.922) Pub Date : 2020-03-06
    Benjamin Espinoza; Paul Gartside; Max Pitz

    A compact graph‐like space is a triple ( X , V , E ) , where X is a compact, metrizable space, V ⊆ X is a closed zero‐dimensional subset, and E is an index set such that X ⧹ V ≅ E × ( 0 , 1 ) . New characterizations of compact graph‐like spaces are given, connecting them to certain classes of continua, and to standard subspaces of Freudenthal compactifications of locally finite graphs. These are applied

    更新日期:2020-03-06
  • Low independence number and Hamiltonicity implies pancyclicity
    J. Graph Theory (IF 0.922) Pub Date : 2020-02-26
    Attila Dankovics

    A graph on n vertices is called pancyclic if it contains a cycle of every length 3 ≤ ℓ ≤ n . Given a Hamiltonian graph G with independence number at most k , we are looking for the minimum number of vertices f ( k ) that guarantees that G is pancyclic. The problem of finding f ( k ) was raised by Erdős in 1972, who showed that f ( k ) ≤ 4 k 4 and conjectured that f ( k ) = Θ ( k 2 ) . Improving on

    更新日期:2020-02-26
  • Isometric subgraphs for Steiner distance
    J. Graph Theory (IF 0.922) Pub Date : 2020-02-25
    Daniel Weißauer

    Let G be a connected graph and ℓ   : E ( G ) → R + a length‐function on the edges of G . The Steiner distance sdG (A ) of A  ⊆ V (G ) within G is the minimum length of a connected subgraph of G containing A , where the length of a subgraph is the sum of the lengths of its edges. It is clear that every subgraph H  ⊆ G , endowed with the induced length‐function ℓ ∣E (H ), satisfies sdH (A ) ≥ sdG (A

    更新日期:2020-02-25
  • t‐Cores for ( Δ + t )‐edge‐coloring
    J. Graph Theory (IF 0.922) Pub Date : 2020-02-18
    Jessica McDonald; Gregory J. Puleo

    We extend the edge‐coloring notion of core (subgraph induced by the vertices of maximum degree) to t ‐core (subgraph induced by the vertices v with d ( v ) + μ ( v ) > Δ + t ), and find a sufficient condition for ( Δ + t ) ‐edge‐coloring. In particular, we show that for any t ≥ 0 , if the t ‐core of G has multiplicity at most t + 1 , with its edges of multiplicity t + 1 inducing a multiforest, then

    更新日期:2020-02-18
  • Three‐colored asymmetric bipartite Ramsey number of connected matchings and cycles
    J. Graph Theory (IF 0.922) Pub Date : 2020-02-16
    Zhidan Luo; Yuejian Peng

    Guaranteed by Szemerédi's Regularity Lemma, a technique originated by Łuczak is to reduce the problem of showing the existence of a monochromatic cycle to show the existence of a monochromatic matching in a component. So determining the Ramsey number of connected matchings is crucial in determining the Ramsey number of cycles. Let k , l , m be integers and r ( k , l , m ) be the minimum integer N such

    更新日期:2020-02-16
  • Colorings versus list colorings of uniform hypergraphs
    J. Graph Theory (IF 0.922) Pub Date : 2020-02-15
    Wei Wang; Jianguo Qian; Zhidan Yan

    Let r be an integer with r ≥ 2 and G be a connected r ‐uniform hypergraph with m edges. By refining the broken cycle theorem for hypergraphs, we show that if k > m − 1 ln ( 1 + 2 ) ≈ 1.135 ( m − 1 ) , then the k ‐list assignment of G admitting the fewest colorings is the constant list assignment. This extends the previous results of Donner, Thomassen, and the current authors for graphs.

    更新日期:2020-02-15
  • Multithreshold graphs
    J. Graph Theory (IF 0.922) Pub Date : 2020-02-11
    Robert E. Jamison; Alan P. Sprague

    Multithreshold graphs are defined in terms of a finite sequence of real thresholds that break the real line into a set of regions, alternating between NO and YES. If real ranks can be assigned to the vertices of a graph in such a way that two vertices are adjacent iff the sum of their ranks lies in a YES region, then that graph is a multithreshold graph with respect to the given set of thresholds.

    更新日期:2020-02-11
  • Positively curved graphs
    J. Graph Theory (IF 0.922) Pub Date : 2020-02-11
    Matthew P. Yancey

    This paper consists of two halves. In the first half of the paper, we consider real‐valued functions f whose domain is the vertex set of a graph G and that are Lipschitz with respect to the graph distance. By placing a uniform distribution on the vertex set, we treat as a random variable. We investigate the link between the isoperimetric function of G and the functions f that have maximum variance

    更新日期:2020-02-11
  • Long cycles and spanning subgraphs of locally maximal 1‐planar graphs
    J. Graph Theory (IF 0.922) Pub Date : 2020-02-10
    I. Fabrici; J. Harant; T. Madaras; S. Mohr; R. Soták; C. T. Zamfirescu

    A graph is 1‐planar if it has a drawing in the plane such that each edge is crossed at most once by another edge. Moreover, if this drawing has the additional property that for each crossing of two edges the end vertices of these edges induce a complete subgraph, then the graph is locally maximal 1‐planar . For a 3‐connected locally maximal 1‐planar graph G , we show the existence of a spanning 3‐connected

    更新日期:2020-02-10
  • Bounded‐excess flows in cubic graphs
    J. Graph Theory (IF 0.922) Pub Date : 2020-02-05
    Michael Tarsi

    An (r , α )‐bounded‐excess flow ((r , α )‐flow) in an orientation of a graph G  = (V , E ) is an assignment f  : E  → [1, r −1], such that for every vertex x  ∈ V , | ∑ e ∈ E + ( x ) f ( e ) − ∑ e ∈ E − ( x ) f ( e ) | ≤ α . E +(x ), respectively E −(x ), is the set of edges directed from, respectively toward x . Bounded‐excess flows suggest a generalization of Circular nowhere‐zero flows (cnzf), which

    更新日期:2020-02-05
  • Constructions of infinite graphs with Ramsey property
    J. Graph Theory (IF 0.922) Pub Date : 2020-02-04
    Péter Komjáth

    For every infinite cardinal λ and 2 ≤ n < ω there is a directed graph D of size λ such that D does not contain directed circuits of length ≤n and if its vertices are colored with <λ colors, then there is a monochromatic directed circuit of length n  + 1. For every infinite cardinal λ and finite graph X there is a λ ‐sized graph Y such that if the vertices of Y are colored with <λ colors, then there

    更新日期:2020-02-04
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