Ore-type conditions for bipartite graphs containing hexagons

Let G=(V₁,V₂;E) be a bipartite graph with |V₁|=|V₂|=3k, where k>0. In this paper it is proved that if d(x)+d(y)≥4k−1 for every pair of nonadjacent vertices x∈V₁, y∈V₂, then G contains k−1 independent cycles of order 6 and a path of order 6 such that all of them are independent. Furthermore, if d(x)+d(y)≥4k for every pair of nonadjacent vertices x∈V₁, y∈V₂ and k>2, then G contains k−2 independent cycles of order 6 and a cycle of order 12 such that all of them are independent.     

Ore-type degree condition for heavy paths in weighted graphs

A weighted graph is one in which every edge e is assigned a nonnegative number w(e), called the weight of e. For a vertex v of a weighted graph, d^w(v) is the sum of the weights of the edges incident to v. And the weight of a path is the sum of the weights of the edges belonging to it. In this paper, we give a sufficient condition for a weighted graph to have a heavy path which joins two specified vertices. Let G be a 2-connected weighted graph and let x and y be distinct vertices of G. Suppose that d^w(u)+d^w(v)2d for every pair of non-adjacent vertices u and vV(G) x,y . Then x and y are joined by a path of weight at least d, or they are joined by a Hamilton path. Also, we consider the case when G has some vertices whose weighted degree are not assumed.     

An Ore-type condition for large k-factor and disjoint perfect matchings

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 $n\ge k+2$. In this paper, we prove that Win's conjecture is true for $k\ge n∕2$, 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 include Tutte's $k$-factor theorem, the Karush-Kuhn-Tucker theorem on convex optimization and the solution to the long-standing 1-factor decomposition conjecture.     

Some Ore-type Results for Matching and Perfect Matching in <Emphasis Type="Italic">k</Emphasis>-uniform Hypergraphs

Let S₁ and S₂ be two (k-1)-subsets in a k-uniform hypergraph H. We call S₁ and S₂ strongly or middle or weakly independent if H does not contain an edge e ∈ E(H) such that S₁ ∩ e ≠∅ and S₂ ∩ e ≠∅ or e ⊆ S₁ ∪ S₂ or e ⊇ S₁ ∪ S₂, respectively. In this paper, we obtain the following results concerning these three independence. (1) For any n ≥ 2k²-k and k ≥ 3, there exists an n-vertex k-uniform hypergraph, which has degree sum of any two strongly independent (k-1)-sets equal to 2n-4(k-1), contains no perfect matching; (2) Let d ≥ 1 be an integer and H be a k-uniform hypergraph of order n ≥ kd+(k-2)k. If the degree sum of any two middle independent (k-1)-subsets is larger than 2(d-1), then H contains a d-matching; (3) For all k ≥ 3 and sufficiently large n divisible by k, we completely determine the minimum degree sum of two weakly independent (k-1)-subsets that ensures a perfect matching in a k-uniform hypergraph H of order n.     

Ore-type conditions implying 2-factors consisting of short cycles

For every graph G, let . The main result of the paper says that every n-vertex graph G with contains each spanning subgraph H all whose components are isomorphic to graphs in . This generalizes the earlier results of Justesen, Enomoto, and Wang, and is a step towards an Ore-type analogue of the Bollobás-Eldridge-Catlin Conjecture.     

关于图的最大亏格的一个定理改进

一个图G的最大亏格γM（G）主要由其参数Betti亏数ξ（G）确定．本文改进Nebesky文［5］中关于ξ（G）的一个表示定理，从而得到关于ξ（G）的一个新结果；由此，给出几个已有结果的简单证明，且其中推广文［8］中的一个结果．     

Brooks’ Theorem via the Alon-Tarsi Theorem

We give a proof of Brooks’ Theorem and its choosability extension based on the Alon-Tarsi Theorem; this also shows that Brooks’ Theorem remains valid in a more general game coloring setting.     

Packing Trees with Constraints on the Leaf Degree

If m is a positive integer then we call a tree on at least 2 vertices an m-tree if no vertex is adjacent to more than m leaves. Kaneko proved that a connected, undirected graph G = (V, E) has a spanning m-tree if and only if for every the number of isolated vertices of G − X is at most —unless we have the exceptional case of and m = 1. As an attempt to integrate this result into the theory of graph packings, in this paper we consider the problem of packing a graph with m-trees. We use an approach different from that of Kaneko, and we deduce Gallai–Edmonds and Berge–Tutte type theorems and a matroidal result for the m-tree packing problem. Jácint Szabó: Research is supported by OTKA grants K60802, TS049788 and by European MCRTN Adonet, Contract Grant No.504438.     

About a new class of matroid-inducing packing families

Let T be a family of graphs. A T-packing of a graph G is a subgraph of G, components of which are isomorphic to members of T. We are concerned with families T, such that in every graph G, the subsets of vertices that can be saturated by some T-packing form a collection of independent sets of a matroid. The main purpose of this paper is to introduce a new class of families with this property.     

Packing cycles in complete graphs

We introduce a new technique for packing pairwise edge-disjoint cycles of specified lengths in complete graphs and use it to prove several results. Firstly, we prove the existence of dense packings of the complete graph with pairwise edge-disjoint cycles of arbitrary specified lengths. We then use this result to prove the existence of decompositions of the complete graph of odd order into pairwise edge-disjoint cycles for a large family of lists of specified cycle lengths. Finally, we construct new maximum packings of the complete graph with pairwise edge-disjoint cycles of uniform length.     

An analogue of Hajós' Theorem for the circular chromatic number

This paper designs a set of graph operations and proves that starting from , by repeatedly applying these operations, one can construct all graphs with (for ). This can be viewed as an analogue of Hajós' Theorem for the circular chromatic number.

     

An Ore-type condition for arbitrarily vertex decomposable graphs

Let G be a graph of order n and r, 1≤r≤n, a fixed integer. G is said to be r-vertex decomposable if for each sequence (n₁,…,n_r) of positive integers such that n₁+?+n_r=n there exists a partition (V₁,…,V_r) of the vertex set of G such that for each i∈{1,…,r}, V_i induces a connected subgraph of G on n_i vertices. G is called arbitrarily vertex decomposable if it is r-vertex decomposable for each r∈{1,…,n}.In this paper we show that if G is a connected graph on n vertices with the independence number at most ⌈n/2⌉ and such that the degree sum of any pair of non-adjacent vertices is at least n−3, then G is arbitrarily vertex decomposable or isomorphic to one of two exceptional graphs. We also exhibit the integers r for which the graphs verifying the above degree-sum condition are not r-vertex decomposable.     

On the ratios between packing and domination parameters of a graph

The relationship ρ_L(G)≤ρ(G)≤γ(G) between the lower packing number ρ_L(G), the packing number ρ(G) and the domination number γ(G) of a graph G is well known. In this paper we establish best possible bounds on the ratios of the packing numbers of any (connected) graph to its six domination-related parameters (the lower and upper irredundance numbers ir and IR, the lower and upper independence numbers i and β, and the lower and upper domination numbers γ and Γ). In particular, best possible constants a_θ, b_θ, c_θ and d_θ are found for which the inequalities and hold for any connected graph G and all θ∈{ir,γ,i,β,Γ,IR}. From our work it follows, for example, that and for any connected graph G, and that these inequalities are best possible.     

The Erdős-Sós Conjecture for Graphs Whose Complements Contain No <Emphasis Type="Italic">C</Emphasis><Subscript>4</Subscript>

Abstract Erdős and Sós conjectured in 1963 (see [1], Problem 12 in 247) that every graph G on n vertices with size contains every tree T of size k. In this paper, we prove the conjecture for graphs whose complements contain no cycles of length 4. Supported by the National Natural Science Foundation of China (No.19971086) and the Doctoral Foundation of Hainan University.     

The Erds-Sós Conjecture for Graphs Whose Complements Contain No C_4

Erds and Sós conjectured in 1963 (see [1],Problem 12 in 247) that every graph G on n verticeswith size e(G)>1/2n(k-1) contains every tree T of size k.In this paper,we prove the conjecture for graphswhose complements contain no cycles of length 4.     

关于图论中棱形式的Menger定理的注记

本注记给出图论中棱形式Menger定理的一个直接而又简单的证明.     

An algorithm for the number of path homomorphisms

A homomorphism of a graph G₁=(V₁,E₁) to a graph G₂=(V₂,E₂) is a mapping from the vertex set V₁ of G₁ to the vertex set V₂ of G₂ which preserves edges. In this paper we provide an algorithm to determine the number of homomorphisms from an arbitrary finite undirected path to another arbitrary finite undirected path.     

Quadripartite version of the Hajnal-Szemerédi theorem

Let G be a quadripartite graph with N vertices in each vertex class and each vertex is adjacent to at least vertices in each of the other classes. There exists an N₀ such that, if N?N₀, then G contains a subgraph that consists of N vertex-disjoint copies of K₄.     

An eigenvalue bound for the Laplacian of a graph

We present a lower bound for the smallest non-zero eigenvalue of the Laplacian of an undirected graph. The bound is primarily useful for graphs with small diameter.