Bipartite Version of the Erd?s-Sós Conjecture

The Erd?s-Sós Conjecture states that every graph on n vertices and more than n(k-2)/2 edges contains every tree of order k as a subgraph. In this note, we study a weak(bipartite)version of Erd?s-Sós Conjecture. Based on a basic lemma, we show that every bipartite graph on n vertices and more than n(k-2)/2 edges contains the following families of trees of order k:(1) trees of diameter at most five;(2) trees with maximum degree at least [k-1/2];(3) almost balanced trees, these results are better than the corresponding known results for the general version of the Erd?s-Sós Conjecture.     

5连通图中的可去边

An edge e of a k-connected graph G is said to be a removable edge if G O e is still k-connected, where G e denotes the graph obtained from G by deleting e to get G - e, and for any end vertex of e with degree k - 1 in G- e, say x, delete x, and then add edges between any pair of non-adjacent vertices in NG-e （x）. The existence of removable edges of k-connected graphs and some properties of 3-connected and 4-connected graphs have been investigated [1, 11, 14, 15]. In the present paper, we investigate some properties of 5-connected graphs and study the distribution of removable edges on a cycle and a spanning tree in a 5- connected graph. Based on the properties, we proved that for a 5-connected graph G of order at least 10, if the edge-vertex-atom of G contains at least three vertices, then G has at least （3│G│ ＋ 2）/2 removable edges.     

Removable edges in a spanning tree of a k-connected graph

An edge e of a k-connected graph G is said to be a removable edge if G O e is still k-connected, where G O e denotes the graph obtained from G by the following way： deleting e to get G - e, and for any end vertex of e with degree k - 1 in G - e, say x, deleting x, and then adding edges between any pair of non-adjacent vertices in NG-e （x）. The existence of removable edges of k-connected graphs and some properties of k-connected graphs have been investigated. In the present paper, we investigate the distribution of removable edges on a spanning tree of a k-connected graph （k ≥ 4）.     

Domination number in graphs with minimum degree two

A set D of vertices of a graph G = （V, E） is called a dominating set if every vertex of V not in D is adjacent to a vertex of D. In 1996, Reed proved that every graph of order n with minimum degree at least 3 has a dominating set of cardinality at most 3n/8. In this paper we generalize Reed＇s result. We show that every graph G of order n with minimum degree at least 2 has a dominating set of cardinality at most （3n ＋IV21）/8, where V2 denotes the set of vertices of degree 2 in G. As an application of the above result, we show that for k ≥ 1, the k-restricted domination number rk （G, γ） ≤ （3n＋5k）/8 for all graphs of order n with minimum degree at least 3.     

Some results on spanning trees

Some structures of spanning trees with many or less leaves in a connected graph are determined.We show（1） a connected graph G has a spanning tree T with minimum leaves such that T contains a longest path,and（2） a connected graph G on n vertices contains a spanning tree T with the maximum leaves such that Δ（G） =Δ（T） and the number of leaves of T is not greater than n D（G）＋1,where D（G） is the diameter of G.     

Path Factors and Neighborhoods of Independent Sets in Graphs

A path-factor is a spanning subgraph F of G such that every component of F is a path with at least two vertices.Let k≥2 be an integer.A P_≥k-factor of G means a path factor in which each component is a path with at least k vertices.A graph G is a P_≥k-factor covered graph if for any e∈E(G),G has a P_≥k-factor including e.Let β be a real number with 1/3≤β≤1 and k be a positive integer.We verify that(ⅰ) a k-connected graph G of order n with n≥5k+2 has a P_≥3...     

Panconnectivity for interconnection networks with faulty elements

Let Go and G1 be two graphs with the same vertices. The new graph G（G0, G1; M） is a graph with the vertex set V（0o） ∪）V（G1） and the edge set E（Go） UE（G1） UM, where M is an arbitrary perfect matching between the vertices of Go and G1, i.e., a set of cross edges with one endvertex in Go and the other endvertex in G1. In this paper, we will show that if Go and G1 are f-fault q-panconnected, then for any f 〉 2, G（G0, G1; M） is （f ＋ 1）-fault （q ＋ 2）-panconnected.     

An iterative rounding 2-approximation algorithm for the k-partial vertex cover problem

We study a generalization of the vertex cover problem. For a given graph with weights on the vertices and an integer k, we aim to find a subset of the vertices with minimum total weight, so that at least k edges in the graph are covered. The problem is called the k-partial vertex cover problem. There are some 2-approximation algorithms for the problem. In the paper we do not improve on the approximation ratios of the previous algorithms, but we derive an iterative rounding algorithm. We present our technique in two algorithms. The first is an iterative rounding algorithm and gives a （2 ＋ Q/OPT ）-approximation for the k-partial vertex cover problem where Q is the largest finite weight in the problem definition and OPT is the optimal value for the instance. The second algorithm uses the first as a subroutine and achieves an approximation ratio of 2.     

任意长圈覆盖指定的独立的点

An invariant σ2（G） of a graph is defined as follows： σ2（G） ：= min{d（u） ＋ d（v）｜u, v ∈V（G）,uv ∈ E（G）,u ≠ v} is the minimum degree sum of nonadjacent vertices （when G is a complete graph, we define σ2（G） = ∞）. Let k, s be integers with k ≥ 2 and s ≥ 4, G be a graph of order n sufficiently large compared with s and k. We show that if σ2（G） ≥ n ＋ k- 1, then for any set of k independent vertices v1,..., vk, G has k vertex-disjoint cycles C1,..., Ck such that ｜Ci｜ ≤ s and vi ∈ V（Ci） for all 1 ≤ i ≤ k.
The condition of degree sum σs（G） ≥ n ＋ k - 1 is sharp.     

A New Lower Bound on the Potential-Ramsey Number of Two Graphs

A nonincreasing sequenceπ=(d1,…,dn)of nonnegative integers is a graphic sequence if it is realizable by a simple graph G on n vertices.In this case,G is referred to as a realization ofπ.Given a graph H,a graphic sequenceπis potentially H-graphic ifπhas a realization containing H as a subgraph.For graphs G1 and G2,the potential-Ramsey number rpot(G1,G2)is the smallest integer k such that for every k-term graphic sequenceπ,eitherπis potentially G1-graphic or the complementary sequenceπ=(k-1-dk,…,k-1-d1)is potentially G2-graphic.For 0≤k≤[t/2],denote Kt-k to be the graph obtained from Kt by deleting k independent edges.If k=0,Busch et al.(Graphs Combin.,30(2014)847-859)present a lower bound on rpot(G,Kt)by using the 1-dependence number of G.In this paper,we utilize i-dependence number of G for i≥1 to give a new lower bound on rpot(G,Kt-k)for any k with 0≤k≤[T/2].Moreover,we also determine the exact values of rpot(Kn,Kt-k)for 1≤k≤2.     

Hamiltonicity of 4-connected graphs

Let σk（G） denote the minimum degree sum of k independent vertices in G and α（G） denote the number of the vertices of a maximum independent set of G. In this paper we prove that if G is a 4-connected graph of order n and σ5（G） 〉 n ＋ 3σ（G） ＋ 11, then G is Hamiltonian.     

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.     

A Rao-type characterization for a sequence to have a realization containing an arbitrary subgraph H

Let G be an arbitrary spanning subgraph of the complete graph Kr+1 on r+1 vertices and Kr+1-E(G) be the graph obtained from Kr+1 by deleting all edges of G.A non-increasing sequence π=(d1,d2,...,dn) of nonnegative integers is said to be potentially Kr+1-E(G)-graphic if there is a graph on n vertices that has π as its degree sequence and contains Kr+1-E(G) as a subgraph.In this paper,a characterization of π that is potentially Kr+1-E(G)-graphic is given,which is analogous to the Erdo s–Gallai characterization of graphic sequences using a system of inequalities.This is a solution to an open problem due to Lai and Hu.As a corollary,a characterization of π that is potentially Ks,tgraphic can also be obtained,where Ks,t is the complete bipartite graph with partite sets of size s and t.This is a solution to an open problem due to Li and Yin.     

VERTEX-FAULT-TOLERANT CYCLES EMBEDDING ON ENHANCED HYPERCUBE NETWORKS

In this paper, we study the enhanced hypercube, an attractive variant of the hypercube and obtained by adding some complementary edges from a hypercube, and focus on cycles embedding on the enhanced hypercube with faulty vertices. Let Fu be the set of faulty vertices in the n-dimensional enhanced hypercube Qn,k （n ≥ 3, 1 ≤ k 〈≤n - 1）. When IFvl = 2, we showed that Qn,k - Fv contains a fault-free cycle of every even length from 4 to 2n - 4 where n （n ≥ 3） and k have the same parity; and contains a fault-free cycle of every even length from 4 to 2n - 4, simultaneously, contains a cycle of every odd length from n-k ＋ 2 to 2^n-3 where n （≥ 3） and k have the different parity. Furthermore, when ｜Fv｜ = fv ≤ n - 2, we prove that there exists the longest fault-free cycle, which is of even length 2^n - 2fv whether n （n ≥ 3） and k have the same parity or not; and there exists the longest fault-free cycle, which is of odd length 2^n - 2fv ＋ 1 in Qn,k - Fv where n （≥ 3） and k have the different parity.     

Note on 2-edge-colorings of complete graphs with small monochromatic k-connected subgraphs

Bollob′as and Gy′arf′as conjectured that for n4(k-1) every 2-edge-coloring of Kn contains a monochromatic k-connected subgraph with at least n-2k+2 verticesLiu, et alproved that the conjecture holds when n≥13k-15In this note, we characterize all the2-edge-colorings of Kn where each monochromatic k-connected subgraph has at most n-2k+2 vertices for n≥13k-15.     

The Smallest Degree Sum That Yields Potentially Kr＋1 - K3-Graphic Sequences

Let a（Kr,＋1 - K3,n） be the smallest even integer such that each n-term graphic sequence п= （d1,d2,…dn） with term sum σ（п） = d1 ＋ d2 ＋…＋ dn 〉 σ（Kr＋1 -K3,n） has a realization containing Kr＋1 - K3 as a subgraph, where Kr＋1 -K3 is a graph obtained from a complete graph Kr＋1 by deleting three edges which form a triangle. In this paper, we determine the value σ（Kr＋1 - K3,n） for r ≥ 3 and n ≥ 3r＋ 5.     

A class of antimagic join graphs

A labeling f of a graph G is a bijection from its edge set E(G) to the set {1, 2, . . . , |E(G)|}, which is antimagic if for any distinct vertices x and y, the sum of the labels on edges incident to x is different from the sum of the labels on edges incident to y. A graph G is antimagic if G has an f which is antimagic. Hartsfield and Ringel conjectured in 1990 that every connected graph other than K 2 is antimagic. In this paper, we show that if G 1 is an n-vertex graph with minimum degree at least r, and G 2 is an m-vertex graph with maximum degree at most 2r-1 (m ≥ n), then G1 ∨ G2 is antimagic.     

Acyclic improper colouring of graphs with maximum degree 4

A k-colouring(not necessarily proper) of vertices of a graph is called acyclic, if for every pair of distinct colours i and j the subgraph induced by the edges whose endpoints have colours i and j is acyclic. We consider acyclic k-colourings such that each colour class induces a graph with a given(hereditary) property. In particular, we consider acyclic k-colourings in which each colour class induces a graph with maximum degree at most t, which are referred to as acyclic t-improper k-colourings. The acyclic t-improper chromatic number of a graph G is the smallest k for which there exists an acyclic t-improper k-colouring of G. We focus on acyclic colourings of graphs with maximum degree 4. We prove that 3 is an upper bound for the acyclic 3-improper chromatic number of this class of graphs. We also provide a non-trivial family of graphs with maximum degree4 whose acyclic 3-improper chromatic number is at most 2, namely, the graphs with maximum average degree at most 3. Finally, we prove that any graph G with Δ(G) 4 can be acyclically coloured with 4 colours in such a way that each colour class induces an acyclic graph with maximum degree at most 3.     

K_(1,p)~k-FACTORIZATION OF COMPLETE BIPARTITE GRAPHS

§ 1　IntroductionLet Km,nbe a complete bipartite graph with two vertex sets having m and n vertices,respectively.A subgraph F of Km,n is called a spanning subgraph of Km,nif F contains allthe vertices of Km,n.Itis clearthata graph with no isolated vertices is uniquely determinedby the setofits edges.So in this paper,we considera graph with no isolated vertices to bea setof2 -elementsets ofits vertices.Letk be a positive integer.A K1 ,k-factor of Km,nis aspanning subgraph F of Km,nsuch th…     

The Rainbow Vertex-disconnection in Graphs

Let G be a nontrivial connected and vertex-colored graph. A subset X of the vertex set of G is called rainbow if any two vertices in X have distinct colors. The graph G is called rainbow vertex-disconnected if for any two vertices x and y of G, there exists a vertex subset S of G such that when x and y are nonadjacent, S is rainbow and x and y belong to different components of G-S; whereas when x and y are adjacent, S + x or S + y is rainbow and x and y belong to different components of(G-xy)-S. For a connected graph G, the rainbow vertex-disconnection number of G, denoted by rvd(G), is the minimum number of colors that are needed to make G rainbow vertexdisconnected. In this paper, we characterize all graphs of order n with rainbow vertex-disconnection number k for k ∈ {1, 2, n}, and determine the rainbow vertex-disconnection numbers of some special graphs. Moreover, we study the extremal problems on the number of edges of a connected graph G with order n and rvd(G) = k for given integers k and n with 1 ≤ k ≤ n.