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  • Proof of a conjecture on the nullity of a graph
    J. Graph Theory (IF 0.883) 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 0.883) 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

    更新日期:2020-05-20
  • Partitions of hypergraphs under variable degeneracy constraints
    J. Graph Theory (IF 0.883) 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 0.883) 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 0.883) 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 0.883) 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 0.883) 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
  • Isometric subgraphs for Steiner distance
    J. Graph Theory (IF 0.883) 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
  • Multithreshold graphs
    J. Graph Theory (IF 0.883) 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 0.883) 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
  • Constructions of infinite graphs with Ramsey property
    J. Graph Theory (IF 0.883) 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
  • An improved upper bound for the grid Ramsey problem
    J. Graph Theory (IF 0.883) Pub Date : 2020-02-04
    Luka Milićević

    For a positive integer r , let G (r ) be the smallest N such that, whenever the edges of the Cartesian product K N  × K N are r ‐colored, then there is a rectangle in which both pairs of opposite edges receive the same color. In this paper, we improve the upper bounds on G (r ) by proving G ( r ) ≤ ( 1 − 1 128 r − 2 ( 1 − o ( 1 ) ) ) r ( r + 1 2 ) , for r large enough. Unlike the previous improvements

    更新日期:2020-02-04
  • Bounds on the localization number
    J. Graph Theory (IF 0.883) Pub Date : 2020-01-30
    Anthony Bonato; William B. Kinnersley

    We consider the localization game played on graphs, wherein a set of cops attempt to determine the exact location of an invisible robber by exploiting distance probes. The corresponding optimization parameter for a graph G is called the localization number and is written as ζ (G ). We settle a conjecture of Bosek et al by providing an upper bound on the chromatic number as a function of the localization

    更新日期:2020-01-30
  • On clique‐inverse graphs of graphs with bounded clique number
    J. Graph Theory (IF 0.883) Pub Date : 2020-01-28
    Liliana Alcón; Sylvain Gravier; Claudia L. Sales; Fabio Protti; Gabriela Ravenna

    The clique graph K (G ) of G is the intersection graph of the family of maximal cliques of G . For a family F of graphs, the family of clique‐inverse graphs of F , denoted by K − 1 ( F ) , is defined as K − 1 ( F ) = { H | K ( H ) ∈ F } . Let F p be the family of K p ‐free graphs, that is, graphs with clique number at most p  − 1, for an integer constant p  ≥ 2. Deciding whether a graph H is a clique‐inverse

    更新日期:2020-01-28
  • Partitioning a graph into cycles with a specified number of chords
    J. Graph Theory (IF 0.883) Pub Date : 2019-12-26
    Shuya Chiba; Suyun Jiang; Jin Yan

    For a graph G , let σ 2 ( G ) be the minimum degree sum of two nonadjacent vertices in G . A chord of a cycle in a graph G is an edge of G joining two nonconsecutive vertices of the cycle. In this paper, we prove the following result, which is an extension of a result of Brandt et al for large graphs: For positive integers k and c , there exists an integer f ( k , c ) such that if G is a graph of order

    更新日期:2019-12-26
  • Rainbow saturation of graphs
    J. Graph Theory (IF 0.883) Pub Date : 2019-12-20
    António Girão; David Lewis; Kamil Popielarz

    In this paper, we study the following problem proposed by Barrus, Ferrara, Vandenbussche, and Wenger. Given a graph H and an integer t , what is sat t ( n , R ( H ) ) , the minimum number of edges in a t ‐edge‐colored graph G on n vertices such that G does not contain a rainbow copy of H , but adding to G a new edge in any color from { 1 , 2 , … , t } creates a rainbow copy of H ? Here, we completely

    更新日期:2019-12-20
  • A generalization of Tuza's conjecture
    J. Graph Theory (IF 0.883) Pub Date : 2019-12-17
    Ron Aharoni; Shira Zerbib

    A famous conjecture of Tuza is that the minimal number of edges needed to cover all triangles in a graph is at most twice the maximal size of a set of edge‐disjoint triangles. We propose a wider setting for this conjecture. For a hypergraph H let ν ( m ) ( H ) be the maximal size of a collection of edges, no two of which share m or more vertices, and let τ ( m ) ( H ) be the minimal size of a collection

    更新日期:2019-12-17
  • Vertex‐disjoint properly edge‐colored cycles in edge‐colored complete graphs
    J. Graph Theory (IF 0.883) Pub Date : 2019-12-17
    Ruonan Li; Hajo Broersma; Shenggui Zhang

    It is conjectured that every edge‐colored complete graph G on n vertices satisfying Δ m o n ( G ) ≤ n − 3 k + 1 contains k vertex‐disjoint properly edge‐colored cycles. We confirm this conjecture for k = 2 , prove several additional weaker results for general k , and we establish structural properties of possible minimum counterexamples to the conjecture. We also reveal a close relationship between

    更新日期:2019-12-17
  • Graph homomorphism reconfiguration and frozen H‐colorings
    J. Graph Theory (IF 0.883) Pub Date : 2019-12-09
    Richard C. Brewster; Jae‐Baek Lee; Benjamin Moore; Jonathan A. Noel; Mark Siggers

    For a fixed graph H, the reconfiguration problem for H‐colorings (ie, homomorphisms to H) asks: given a graph G and two H‐colorings φ and ψ of G, does there exist a sequence f 0 , … , f m of H‐colorings such that f 0 = φ , f m = ψ , and f i ( u ) f i + 1 ( v ) ∈ E ( H ) for every 0 ≤ i < m and u v ∈ E ( G ) ? If the graph G is loop‐free, then this is the equivalent to asking whether it possible to

    更新日期:2019-12-09
  • Characterizing and decomposing classes of threshold, split, and bipartite graphs via 1‐Sperner hypergraphs
    J. Graph Theory (IF 0.883) Pub Date : 2019-12-04
    Endre Boros; Vladimir Gurvich; Martin Milanič

    A hypergraph is said to be 1‐Sperner if for every two hyperedges the smallest of their two set differences is of size one. We present several applications of 1 ‐Sperner hypergraphs to graphs. First, we consider several ways of associating hypergraphs to graphs, namely, vertex cover, clique, independent set, dominating set, and closed neighborhood hypergraphs. For each of them, we characterize graphs

    更新日期:2019-12-04
  • Hamiltonian cycles in 3‐tough 2K2‐free graphs
    J. Graph Theory (IF 0.883) Pub Date : 2019-11-27
    Songling Shan

    A graph is called 2 K 2 ‐free if it does not contain two independent edges as an induced subgraph. Broersma, Patel, and Pyatkin showed that every 25‐tough 2 K 2 ‐free graph with at least three vertices is Hamiltonian. In this paper, we improve the required toughness in this result from 25 to 3.

    更新日期:2019-11-27
  • An Ore‐type condition for large k‐factor and disjoint perfect matchings
    J. Graph Theory (IF 0.883) Pub Date : 2019-11-25
    Hongliang Lu; Bo Ning

    Win conjectured that a graph G on n vertices contains k disjoint perfect matchings, if the degree sum of any two nonadjacent vertices is at least n + k − 2 , where n is even and n ≥ k + 2 . In this paper, we prove that Win's conjecture is true for k ≥ n ∕ 2 , where n is sufficiently large. To show this result, we prove a theorem on k ‐factor in a graph under some Ore‐type condition. Our main tools

    更新日期:2019-11-25
  • Few H copies in F‐saturated graphs
    J. Graph Theory (IF 0.883) Pub Date : 2019-11-25
    Jürgen Kritschgau; Abhishek Methuku; Michael Tait; Craig Timmons

    A graph is F ‐saturated if it is F ‐free but the addition of any edge creates a copy of F . In this paper we study the function sat ( n , H , F ) which is the minimum number of copies of H that an F ‐saturated graph on n vertices may contain. This function is a natural saturation analogue of Alon and Shikhelman's generalized Turán problem, and letting H = K 2 recovers the well‐studied saturation function

    更新日期:2019-11-25
  • Improved bounds for minimal feedback vertex sets in tournaments.
    J. Graph Theory Pub Date : 2018-06-05
    M Mnich,E Teutrine

    We study feedback vertex sets (FVS) in tournaments, which are orientations of complete graphs. As our main result, we show that any tournament on n nodes has at most 1.5949 n minimal FVS. This significantly improves the previously best upper bound of 1.6667 n by Fomin et al. [STOC 2016] and 1.6740 n by Gaspers and Mnich [J. Graph Theory72(1):72-89, 2013]. Our new upper bound almost matches the best-known

    更新日期:2019-11-01
  • Monochromatic Clique Decompositions of Graphs.
    J. Graph Theory Pub Date : 2015-12-01
    Henry Liu,Oleg Pikhurko,Teresa Sousa

    Let G be a graph whose edges are colored with k colors, and H=(H1,⋯,Hk) be a k-tuple of graphs. A monochromatic H-decomposition of G is a partition of the edge set of G such that each part is either a single edge or forms a monochromatic copy of Hi in color i, for some 1≤i≤k. Let φk(n,H) be the smallest number ϕ, such that, for every order-n graph and every k-edge-coloring, there is a monochromatic

    更新日期:2019-11-01
  • On dihedral flows in embedded graphs.
    J. Graph Theory Pub Date : null
    Bart Litjens

    Let Γ be a multigraph with for each vertex a cyclic order of the edges incident with it. For n ≥ 3 , let D 2 n be the dihedral group of order 2 n . Define D ≔ { ( ± 1 a 0 1 ) ∣ a ∈ Z } . Goodall et al in 2016 asked whether Γ admits a nowhere-identity D 2 n -flow if and only if it admits a nowhere-identity D -flow with ∣ a ∣ < n (a "nowhere-identity dihedral n -flow"). We give counterexamples to this

    更新日期:2019-11-01
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