On partitions of K2,3-free graphs under degree constraints

Suppose that $s$, $t$ are two positive integers, and $?$ is a set of graphs. Let $g(s,t;?)$ be the least integer $g$ such that any $?$-free graph with minimum degree at least $g$ can be partitioned into two sets which induced subgraphs have minimum degree at least $s$ and $t$, respectively. For a given graph $H$, we simply write $g(s,t;H)$ for $g(s,t;?)$ when $?=\left\{H\right\}$. In this paper, we show that if $s,t\ge 2$, then $g(s,t;K_{2,3})\le s+t$ and $g(s,t;\{K_3,C_8,K_{2,3}\})\le s+t?1$. Moreover, if $?$ is the set of graphs obtained by connecting a single vertex to exactly two vertices of $K_4?e$, then $g(s,t;?)\le s+t$ on $?$-free graphs with at least five vertices, which generalize a result of Liu and Xu (2017).     

Decomposing graphs with girth at least five under degree constraints

We prove that the vertex set of a simple graph with minimum degree at least s + t − 1 and girth at least 5 can be decomposed into two parts, which induce subgraphs with minimum degree at least s and t, respectively, where s, t are positive integers ≥ 2. © 2000 John Wiley & Sons, Inc: J Graph Theory 33: 237–239, 2000     

Efficient algorithms for decomposing graphs under degree constraints

Stiebitz [Decomposing graphs under degree constraints, J. Graph Theory 23 (1996) 321-324] proved that if every vertex v in a graph G has degree d(v)?a(v)+b(v)+1 (where a and b are arbitrarily given nonnegative integer-valued functions) then G has a nontrivial vertex partition (A,B) such that d_A(v)?a(v) for every v∈A and d_B(v)?b(v) for every v∈B. Kaneko [On decomposition of triangle-free graphs under degree constraints, J. Graph Theory 27 (1998) 7-9] and Diwan [Decomposing graphs with girth at least five under degree constraints, J. Graph Theory 33 (2000) 237-239] strengthened this result, proving that it suffices to assume d(v)?a+b (a,b?1) or just d(v)?a+b-1 (a,b?2) if G contains no cycles shorter than 4 or 5, respectively.The original proofs contain nonconstructive steps. In this paper we give polynomial-time algorithms that find such partitions. Constructive generalizations for k-partitions are also presented.     

Proximity and remoteness in triangle-free and C4-free graphs in terms of order and minimum degree

Let G be a finite, connected graph. The average distance of a vertex v of G is the arithmetic mean of the distances from v to all other vertices of G. The remoteness $\rho (G)$ and the proximity $\pi (G)$ of G are the maximum and the minimum of the average distances of the vertices of G. In this paper, we present a sharp upper bound on the remoteness of a triangle-free graph of given order and minimum degree, and a corresponding bound on the proximity, which is sharp apart from an additive constant. We also present upper bounds on the remoteness and proximity of $C_4$-free graphs of given order and minimum degree, and we demonstrate that these are close to being best possible.     

Lower bounds on size and independence in K4-free graphs

We investigate lower bounds on the size of K₄-free graphs. For several ranges of independence relative to order and for graphs with maximum degree 3 and 4, we find sharp lower bounds. We also evaluate Ramsey-type numbers over the classes of graphs with maximum degree 3 and with maximum degree 4. © 1997 John Wiley & Sons, Inc. J Graph Theory 26: 61–71, 1997     

Polychromatic colorings of bounded degree plane graphs

A polychromatic k‐coloring of a plane graph G is an assignment of k colors to the vertices of G such that every face of G has all k colors on its boundary. For a given plane graph G, one seeks the maximum number k such that G admits a polychromatic k ‐coloring. In this paper, it is proven that every connected plane graph of order at least three, and maximum degree three, other than K₄ or a subdivision of K₄ on five vertices, admits a 3‐coloring in the regular sense (i.e., no monochromatic edges) that is also a polychromatic 3‐coloring. Our proof is constructive and implies a polynomial‐time algorithm. © 2009 Wiley Periodicals, Inc. J Graph Theory 60: 269‐283, 2009     

Extremal graphs with bounded densities of small subgraphs

Let Ex(n, k, μ) denote the maximum number of edges of an n-vertex graph in which every subgraph of k vertices has at most μ edges. Here we summarize some known results of the problem of determining Ex(n, k, μ), give simple proofs, and find some new estimates and extremal graphs. Besides proving new results, one of our main aims is to show how the classical Turáan theory can be applied to such problems. The case μ = is the famous result of Turáan. © 1998 John Wiley & Sons, Inc. J. Graph Theory 29: 185–207, 1998     

On decomposition of triangle-free graphs under degree constraints

We prove that if s and t are positive integers and if G is a triangle-free graph with minimum degree s + t, then the vertex set of G has a decomposition into two sets which induce subgraphs of minimum degree at least s and t, respectively. © 1998 John Wiley & Sons, Inc. J Graph Theory 27: 7–9, 1998     

Generation of isospectral graphs

We discuss a discrete version of Sunada's Theorem on isospectral manifolds, which allows the generation of isospectral simple graphs, i.e., nonisomorphic simple graphs that have the same Laplace spectrum. We also consider additional boundary conditions and Buser's transplantation technique applied to a discrete situation. © John Wiley & Sons, Inc. J Graph Theory 31: 255–265, 1999     

On second order degree of graphs

Given a vertex v of a graph G the second order degree of v denoted as d 2(v) is defined as the number of vertices at distance 2 from v.In this paper we address the following question:What are the sufficient conditions for a graph to have a vertex v such that d2(v) ≥ d(v),where d(v) denotes the degree of v? Among other results,every graph of minimum degree exactly 2,except four graphs,is shown to have a vertex of second order degree as large as its own degree.Moreover,every K-4-free graph or every maximal planar graph is shown to have a vertex v such that d2(v) ≥ d(v).Other sufficient conditions on graphs for guaranteeing this property are also proved.     

Vertex partitions of (C3,C4,C6)-free planar graphs

A graph is $(k_1,k_2)$-colorable if it admits a vertex partition into a graph with maximum degree at most $k_1$ and a graph with maximum degree at most $k_2$. We show that every $(C_3,C_4,C_6)$-free planar graph is $(0,6)$-colorable. We also show that deciding whether a $(C_3,C_4,C_6)$-free planar graph is $(0,3)$-colorable is NP-complete.     

Color degree and monochromatic degree conditions for short properly colored cycles in edge‐colored graphs

For an edge‐colored graph, its minimum color degree is defined as the minimum number of colors appearing on the edges incident to a vertex and its maximum monochromatic degree is defined as the maximum number of edges incident to a vertex with a same color. A cycle is called properly colored if every two of its adjacent edges have distinct colors. In this article, we first give a minimum color degree condition for the existence of properly colored cycles, then obtain the minimum color degree condition for an edge‐colored complete graph to contain properly colored triangles. Afterwards, we characterize the structure of an edge‐colored complete bipartite graph without containing properly colored cycles of length 4 and give the minimum color degree and maximum monochromatic degree conditions for an edge‐colored complete bipartite graph to contain properly colored cycles of length 4, and those passing through a given vertex or edge, respectively.     

Sparse halves in K 4-free graphs

A conjecture of Chung and Graham states that every $K_4$-free graph on $n$ vertices contains a vertex set of size $?n∕2?$ that spans at most $n^2∕18$ edges. We make the first step toward this conjecture by showing that it holds for all regular graphs.     

Polynomial algorithms for kernels in comparability, permutation and P 4-free graphs

In this paper, we give three polynomial algorithms which detect a kernel in comparability graphs relatively to an M-orientation, in permutation graphs and in P₄-free graphs with a normal orientation. MSC classification: 05C69, 05C85 Correspondence to: Saoula Youcef     

无C_4图的邻域同调分类

本文证明了两个连通的无C4图是邻域同调的充要条件为它们的二分性相同，并且圈秩相等．     

Geometry of the vacant set left by random walk on random graphs,Wright's constants,and critical random graphs with prescribed degrees

We provide an explicit algorithm for sampling a uniform simple connected random graph with a given degree sequence. By products of this central result include: (1) continuum scaling limits of uniform simple connected graphs with given degree sequence and asymptotics for the number of simple connected graphs with given degree sequence under some regularity conditions, and (2) scaling limits for the metric space structure of the maximal components in the critical regime of both the configuration model and the uniform simple random graph model with prescribed degree sequence under finite third moment assumption on the degree sequence. As a substantive application we answer a question raised by ?erný and Teixeira study by obtaining the metric space scaling limit of maximal components in the vacant set left by random walks on random regular graphs.     

Minimum degree conditions for -linked graphs

For a fixed multigraph H with vertices w₁,…,w_m, a graph G is H-linked if for every choice of vertices v₁,…,v_m in G, there exists a subdivision of H in G such that v_i is the branch vertex representing w_i (for all i). This generalizes the notions of k-linked, k-connected, and k-ordered graphs.Given a connected multigraph H with k edges and minimum degree at least two and n7.5k, we determine the least integer d such that every n-vertex simple graph with minimum degree at least d is H-linked. This value D(H,n) appears to equal the least integer d^′ such that every n-vertex graph with minimum degree at least d^′ is b(H)-connected, where b(H) is the maximum number of edges in a bipartite subgraph of H.     

Partial characterizations of clique-perfect and coordinated graphs: Superclasses of triangle-free graphs

A graph G is clique-perfect if the cardinality of a maximum clique-independent set of H equals the cardinality of a minimum clique-transversal of H, for every induced subgraph H of G. A graph G is coordinated if the minimum number of colors that can be assigned to the cliques of H in such a way that no two cliques with non-empty intersection receive the same color equals the maximum number of cliques of H with a common vertex, for every induced subgraph H of G. Coordinated graphs are a subclass of perfect graphs. The complete lists of minimal forbidden induced subgraphs for the classes of clique-perfect and coordinated graphs are not known, but some partial characterizations have been obtained. In this paper, we characterize clique-perfect and coordinated graphs by minimal forbidden induced subgraphs when the graph is either paw-free or {gem, W₄, bull}-free, both superclasses of triangle-free graphs.