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  • Edge clique covers in graphs with independence number two
    J. Graph Theory (IF 0.922) Pub Date : 2021-01-21
    Pierre Charbit; Geňa Hahn; Marcin Kamiński; Manuel Lafond; Nicolas Lichiardopol; Reza Naserasr; Ben Seamone; Rezvan Sherkati

    The edge clique cover number ecc ( G ) of a graph G is the size of the smallest collection of complete subgraphs whose union covers all edges of G . Chen, Jacobson, Kézdy, Lehel, Scheinerman, and Wang conjectured in 2000 that if G is claw‐free, then ecc ( G ) is bounded above by its order (denoted n ). Recently, Javadi and Hajebi verified this conjecture for claw‐free graphs with an independence number

    更新日期:2021-01-21
  • On graphs with no induced five‐vertex path or paraglider
    J. Graph Theory (IF 0.922) Pub Date : 2021-01-19
    Shenwei Huang; T. Karthick

    Given two graphs H 1 and H 2 , a graph is ( H 1 , H 2 ) ‐free if it contains no induced subgraph isomorphic to H 1 or H 2 . For a positive integer t , P t is the chordless path on t vertices. A paraglider is the graph that consists of a chorless cycle C 4 plus a vertex adjacent to three vertices of the C 4 . In this paper, we study the structure of ( P 5 , paraglider)‐free graphs, and show that every

    更新日期:2021-01-19
  • Seymour's second‐neighborhood conjecture from a different perspective
    J. Graph Theory (IF 0.922) Pub Date : 2021-01-13
    Farid Bouya; Bogdan Oporowski

    Seymour's Second‐Neighborhood Conjecture states that every directed graph whose underlying graph is simple has at least one vertex v such that the number of vertices of out‐distance two from v is at least as large as the number of vertices of out‐distance one from it. We present alternative statements of the conjecture in the language of linear algebra.

    更新日期:2021-01-13
  • At least half of the leapfrog fullerene graphs have exponentially many Hamilton cycles
    J. Graph Theory (IF 0.922) Pub Date : 2021-01-11
    František Kardoš; Martina Mockovčiaková

    A fullerene graph is a 3‐connected cubic planar graph with pentagonal and hexagonal faces. The leapfrog transformation of a planar graph produces the dual of the truncation of the given graph. A fullerene graph is a leapfrog if it can be obtained from another fullerene graph by the leapfrog transformation. We prove that leapfrog fullerene graphs on n = 12 k − 6 vertices have at least 2 k Hamilton cycles

    更新日期:2021-01-11
  • Strongly perfect claw‐free graphs—A short proof
    J. Graph Theory (IF 0.922) Pub Date : 2021-01-11
    Maria Chudnovsky; Cemil Dibek

    A graph is strongly perfect if every induced subgraph H of it has a stable set that meets every maximal clique of H . A graph is claw‐free if no vertex has three pairwise nonadjacent neighbors. The characterization of claw‐free graphs that are strongly perfect by a set of forbidden induced subgraphs was conjectured by Ravindra in 1990 and was proved by Wang in 2006. Here we give a shorter proof of

    更新日期:2021-01-11
  • Perfect matchings and derangements on graphs
    J. Graph Theory (IF 0.922) Pub Date : 2021-01-06
    Matija Bucic; Pat Devlin; Mo Hendon; Dru Horne; Ben Lund

    We show that each perfect matching in a bipartite graph G intersects at least half of the perfect matchings in G . This result has equivalent formulations in terms of the permanent of the adjacency matrix of a graph, and in terms of derangements and permutations on graphs. We give several related results and open questions.

    更新日期:2021-01-07
  • Induced path factors of regular graphs
    J. Graph Theory (IF 0.922) Pub Date : 2020-12-18
    Saieed Akbari; Daniel Horsley; Ian M. Wanless

    An induced path factor of a graph G is a set of induced paths in G with the property that every vertex of G is in exactly one of the paths. The induced path number ρ ( G ) of G is the minimum number of paths in an induced path factor of G . We show that if G is a connected cubic graph on n > 6 vertices, then ρ ( G ) ⩽ ( n − 1 ) / 3 . Fix an integer k ⩾ 3 . For each n , define ℳ n to be the maximum

    更新日期:2020-12-18
  • The Hamiltonicity of essentially 9‐connected line graphs
    J. Graph Theory (IF 0.922) Pub Date : 2020-12-07
    Tomáš Kaiser; Petr Vrána

    Yang et al proved that every 3‐connected, essentially 11‐connected line graph is Hamilton‐connected. This was extended by Li and Yang to 3‐connected, essentially 10‐connected line graphs. Strengthening their result further, we prove that 3‐connected, essentially 9‐connected line graphs are Hamilton‐connected. We use a method based on quasigraphs in combination with the discharging technique. The result

    更新日期:2020-12-07
  • On supersaturation and stability for generalized Turán problems
    J. Graph Theory (IF 0.922) Pub Date : 2020-12-02
    Anastasia Halfpap; Cory Palmer

    Fix a graph F . We say that a graph is F ‐free if it contains no copy of F as a subgraph. Let ex ( n , H , F ) denote the maximum number of copies of a graph H in an n ‐vertex F ‐free graph. In this note, we will give a new general supersaturation result for ex ( n , H , F ) in the case when χ ( H ) < χ ( F ) as well as a new proof of a stability theorem for ex ( n , K r , F ) .

    更新日期:2020-12-02
  • Extremal digraphs on Woodall‐type condition for Hamiltonian cycles in balanced bipartite digraphs
    J. Graph Theory (IF 0.922) Pub Date : 2020-11-25
    Ruixia Wang

    Adamus et al proved that: a balanced bipartite digraph D of order 2 a is Hamiltonian if d + ( u ) + d − ( v ) ≥ a + 2 whenever u and v belong to different partite sets and u v ∉ A ( D ) . They also showed that the bound is sharp. In this paper, we shall show that the extremal digraph of this condition is a digraph of six vertices.

    更新日期:2020-11-25
  • Towards Gallai's path decomposition conjecture
    J. Graph Theory (IF 0.922) Pub Date : 2020-11-24
    Fábio Botler; Maycon Sambinelli

    A path decomposition of a graph G is a collection of edge‐disjoint paths of G that covers the edge set of G . Gallai conjectured that every connected graph on n vertices admits a path decomposition of cardinality at most ⌈ n / 2 ⌉ . Seminal results toward its verification consider the graph obtained from G by removing its vertices of odd degree, which is called the E‐subgraph of G . Lovász verified

    更新日期:2020-11-25
  • On induced subgraphs of the Hamming graph
    J. Graph Theory (IF 0.922) Pub Date : 2020-10-27
    Dingding Dong

    In connection with his solution of the Sensitivity Conjecture, Hao Huang (arXiv: 1907.00847, 2019) asked the following question: Given a graph G with high symmetry, what can we say about the smallest maximum degree of induced subgraphs of G with α ( G ) + 1 vertices, where α ( G ) denotes the size of the largest independent set in G ? We study this question for H ( n , k ) , the n ‐dimensional Hamming

    更新日期:2020-11-25
  • Perfect matchings in random subgraphs of regular bipartite graphs
    J. Graph Theory (IF 0.922) Pub Date : 2020-11-23
    Roman Glebov; Zur Luria; Michael Simkin

    Consider the random process in which the edges of a graph G are added one by one in a random order. A classical result states that if G is the complete graph K 2 n or the complete bipartite graph K n , n , then typically a perfect matching appears at the moment at which the last isolated vertex disappears. We extend this result to arbitrary k ‐regular bipartite graphs G on 2 n vertices for all k =

    更新日期:2020-11-23
  • Unavoidable chromatic patterns in 2‐colorings of the complete graph
    J. Graph Theory (IF 0.922) Pub Date : 2020-11-10
    Yair Caro; Adriana Hansberg; Amanda Montejano

    Given a graph G on k edges, we consider the following two extremal problems: provided n is large enough, what is the minimum integer bal ( n , G ) , if it exists, such that any 2‐coloring of the edges of a complete graph on n vertices having more than bal ( n , G ) edges in each color class, contains a balanced copy of G , that is, a copy of G with exactly ⌊ k ∕ 2 ⌋ edges in one of the colors? Graphs

    更新日期:2020-11-12
  • On extremal nonsolid bricks
    J. Graph Theory (IF 0.922) Pub Date : 2020-11-08
    Xing Feng; Fuliang Lu

    A 3‐connected graph is a brick if, after the removal of any two distinct vertices, the resulting graph has a perfect matching. Lovász proved that the dimension of the matching lattice of a brick G is equal to | E ( G ) | − | V ( G ) | + 1 . We say a brick G is extremal if the number of perfect matchings in G is exactly d i m ( G ) . de Carvalho et al. characterized extremal bricks and conjectured that

    更新日期:2020-11-09
  • Single‐conflict colouring
    J. Graph Theory (IF 0.922) Pub Date : 2020-11-04
    Zdeněk Dvořák; Louis Esperet; Ross J. Kang; Kenta Ozeki

    Given a multigraph, suppose that each vertex is given a local assignment of k colours to its incident edges. We are interested in whether there is a choice of one local colour per vertex such that no edge has both of its local colours chosen. The least k for which this is always possible given any set of local assignments we call the single‐conflict chromatic number of the graph. This parameter is

    更新日期:2020-11-06
  • Extremal total distance of graphs of given radius I
    J. Graph Theory (IF 0.922) Pub Date : 2020-11-03
    Stijn Cambie

    In 1984, Plesník determined the minimum total distance for given order and diameter and characterized the extremal graphs and digraphs. We prove the analog for given order and radius, when the order is sufficiently large compared to the radius. This confirms asymptotically a conjecture of Chen et al. We also state an analog of the conjecture of Chen et al for digraphs and prove it for sufficiently

    更新日期:2020-11-04
  • Canonical double covers of generalized Petersen graphs, and double generalized Petersen graphs
    J. Graph Theory (IF 0.922) Pub Date : 2020-11-03
    Yan‐Li Qin; Binzhou Xia; Sanming Zhou

    The canonical double cover D ( Γ ) of a graph Γ is the direct product of Γ and K 2 . If Aut ( D ( Γ ) ) ≅ Aut ( Γ ) × Z 2 then Γ is called stable; otherwise Γ is called unstable. An unstable graph is said to be nontrivially unstable if it is connected, non‐bipartite and no two vertices have the same neighborhood. In 2008 Wilson conjectured that, if the generalized Petersen graph GP ( n , k ) is nontrivially

    更新日期:2020-11-03
  • The average size of a connected vertex set of a graph—Explicit formulas and open problems
    J. Graph Theory (IF 0.922) Pub Date : 2020-11-03
    Andrew Vince

    Although connectivity is a basic concept in graph theory, the enumeration of connected subgraphs of a graph has only recently received attention. The topic of this paper is the average order of a connected induced subgraph of a graph. This generalizes, to graphs in general, the average order of a subtree of a tree. For various infinite families of graphs, we investigate the asymptotic behavior of the

    更新日期:2020-11-03
  • Edge‐decomposing graphs into coprime forests
    J. Graph Theory (IF 0.922) Pub Date : 2020-10-31
    Tereza Klimošová; Stéphan Thomassé

    The Barát‐Thomassen conjecture, recently proved in Bensmail et al. (2017), asserts that for every tree T , there is a constant c T such that every c T ‐edge‐connected graph G with number of edges (size) divisible by the size of T admits an edge partition into copies of T (a T ‐decomposition). In this paper, we investigate in which case the connectivity requirement can be dropped to a minimum degree

    更新日期:2020-11-02
  • The chromatic number of { ISK 4, diamond, bowtie}‐free graphs
    J. Graph Theory (IF 0.922) Pub Date : 2020-10-31
    Guantao Chen; Yuan Chen; Qing Cui; Xing Feng; Qinghai Liu

    A graph is said to be ISK 4 ‐free if it does not contain any subdivision of K 4 as an induced subgraph. Lévêque, Maffray and Trotignon conjectured that every ISK 4 ‐free graph is 4‐colorable. In this paper, we show that this conjecture is true for the class of { ISK 4 , diamond, bowtie}‐free graphs, where a diamond is the graph obtained from K 4 by removing one edge and a bowtie is the graph consisting

    更新日期:2020-11-02
  • Toughness and spanning trees in K4‐minor‐free graphs
    J. Graph Theory (IF 0.922) Pub Date : 2020-10-31
    M. N. Ellingham; Songling Shan; Dong Ye; Xiaoya Zha

    For an integer k , a k ‐tree is a tree with maximum degree at most k . More generally, if f is an integer‐valued function on vertices, an f ‐tree is a tree in which each vertex v has degree at most f ( v ) . Let c ( G ) denote the number of components of a graph G . We show that if G is a connected K 4 ‐minor‐free graph and c ( G − S ) ≤ ∑ v ∈ S ( f ( v ) − 1 ) for all S ⊆ V ( G ) with S ≠ ∅ , then

    更新日期:2020-10-31
  • Large homogeneous subgraphs in bipartite graphs with forbidden induced subgraphs
    J. Graph Theory (IF 0.922) Pub Date : 2020-10-28
    Maria Axenovich; Casey Tompkins; Lea Weber

    For a bipartite graph G , let h ˜ ( G ) be the largest t such that either G contains K t , t , a complete bipartite subgraph with parts of size t , or the bipartite complement of G contains K t , t as a subgraph. For a class of graphs F , let h ˜ ( F ) = min { h ˜ ( G ) : G ∈ F } . We say that a bipartite graph H is strongly acyclic if neither H nor its bipartite complement contains a cycle. By Forb

    更新日期:2020-10-30
  • A unified approach to construct snarks with circular flow number 5
    J. Graph Theory (IF 0.922) Pub Date : 2020-10-28
    Jan Goedgebeur; Davide Mattiolo; Giuseppe Mazzuoccolo

    The well‐known 5‐flow Conjecture of Tutte, stated originally for integer flows, claims that every bridgeless graph has circular flow number at most 5. It is a classical result that the study of the 5‐flow Conjecture can be reduced to cubic graphs, in particular to snarks. However, very few procedures to construct snarks with circular flow number 5 are known. In the first part of this paper, we summarise

    更新日期:2020-10-30
  • Coloring rings
    J. Graph Theory (IF 0.922) Pub Date : 2020-10-28
    Frédéric Maffray; Irena Penev; Kristina Vušković

    A ring is a graph whose vertex set can be partitioned into k ≥ 4 nonempty sets, X 1 , … , X k , such that for all i ∈ { 1 , … , k } , the set X i can be ordered as X i = { u i 1 , … , u i | X i | } so that X i ⊆ N R [ u i ∣ X i ∣ ] ⊆ ⋯ ⊆ N R [ u i 1 ] = X i − 1 ∪ X i ∪ X i + 1 . A hyperhole is a ring R such that for all i ∈ { 1 , … , k } , X i is complete to X i − 1 ∪ X i + 1 . In this paper, we prove

    更新日期:2020-10-30
  • 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:2020-09-23
  • 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:2020-09-23
  • 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:2020-09-14
  • 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:2020-08-05
  • 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:2020-07-14
  • 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:2020-07-14
  • 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
  • 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
  • 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:2020-06-30
  • 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:2020-06-27
  • 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:2020-06-27
  • 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:2020-06-27
  • 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
  • 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
  • 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
  • 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
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