Conjecture on Self Hamiltonian Path Graph

The Hamiltonian path graph H(G) of a graph G isa that graph having the samevertex set as G and in which two vertices u and v are adjacent if and only if Gcontains a Hamiltonian u - v path. A graph G is a self-Hamiltonian path graph ifG≌H(G).G. Chartrand conjecture: A graph G of order p is a self-Hamiltonian path:graph if and only if G is chord additive or G is isomorphic to one of the graphsK_p, (?)_p, C_p(p≥3),K((1/2)p, (1/2)p), and K_(p/2) (?)_(p/2),the last two for even P.     

The Criticality for Circular Coloring of Graph G_d~k

For two positive integers k and d such that k ≥ 2d, Gkd is the graph with vertex set {0,1, ...,k-1} in which ij is an edge if and only if d ≤ |i-j| ≤ k-d. Clearly, Gk1 is a complete graph of k vertices and we always assume d ≥ 2 in the following. It is easy to see (also [1]) that a graph G is (k, d)-colorable if and only if there exists a homomorphism from G to Gkd.     

Induced Matching Number of the Plane Grid Graph

An induced matching M in a graph G is a matching such that V(M) induces a 1-regular subgraph of G. The induced matching number of a graph G, denoted by I M(G), is the maximum number r such that G has an induced matching of r edges. Induced matching number of Pm×Pn is investigated in this paper. The main results are as follows:(1) If at least one of m and n is even, then IM(Pm×Pn=[(mn)/4].(2) If m is odd, then     

The Binding Number of a Graph

The binding number of a graph is an important characteristic quantity of agraph.In1973 Woodall first introduced the concept of the binding number ofa graph and then gave the binding number of some special graphs in[1].In1981 Kane,Mohanty and Hales gave the binding number of some product graphsin[2].Wang Jianfang,Tian Songlin,Liu Ziuqiang(1983-1987)gave the bin-ding number of some multi-product graphs,and it is the limit character.Up to     

A Note on the Graph of Class One

1 Introduction Let G be a simple graph. An edge colouring ∏ of G is a map ∏: E(G)→{1,2,…} such that no two adjacent edges have the same image. The chromatic index X' (G) of G is the minimum cardinality of all possible images of colouring of G. By Vizing's theorem if △ is the maximum degree of G.     

On Primitive Idempotents of the Strong Endomorphism Monoid of a Graph

51.IntroductionandPreIiminariesThemonoidofendomorphismsofagraph,inparticular,thatofstrongendomorphismsofagraph,hasbeentheobjectofresearchesinthetheoryofsemigroupsforquitesometime(cf.Llj-Lloj).LlJandL2jcanserveasasurvey.Theaimoftheseresearchesistocon-tributetothealgebraicanalysisofgraphs.Thesubjectyieldssomeinterestsincetheresultsoftheseresearchesmayopenvastpossibilitiesforapplicationsofsemigrouptheorytographtheo-ry.InL1]andL3j,ithasbeenprovedthatforfinitegraphG,sEnd(G),themonoidofstrongen…     

Bounds for Two Classes of Graph Spectrum and Proof of Their Conjectures

Let λ_k be the kth greatest eigenvalue of forest F (or tree T) on n vertices, then λ_k=-λ_(n-k 1). Hong Yuan proposed the following conjecture: Conjecture 1. Suppose T is a tree with n vertices and edge independence number q. For k≤q, λ_k(T)≥λ_k(S_(n-2k 2)~(2h-2) with equality iff T≌S_(n-2k 2)~(2k-2), where S_(n-2k 2)~(2k-2) is formed from a K_(1,n-2k 1) and a path P_(2k-2) by joining with an edge a vertex of degree one of P_(2k-2) to the vertex of degree n-2k 1 of K_(1,n-2k 1).     

The Criticality for Circular Coloring of Graph

For two positive integers k and d such that k ≥ 2d, Gkd is the graph with vertex set{0,1,…,k-1} in which ij is an edge if and only ifd ≤ |i-j|≤ k-d. Clearly, Gk1 is acomplete graph of k vertices and we always assume d ≥ 2 in the following.     

On the Covering of the Vertices of a Graph by Cliques

In conversation I was told by Professor R.Brigham the following conjecture [1].Let G(n) be a graph of n vertices.Denote by f(G(n))=t the smallest integer for which the vertices of G(n) can be covered by t cliques. Denote further by h(G(n)) =l the largest integer for which there are l edges of our G(n) no two of Which are in the same clique.Clearly h(G(n)) can be much larger than f(G(n))e.g.if n=2m and G(n) is the complete bipartite graph of m white and m black vertices.Then l(G(n))=m and l(G(n))=m~2. It was conjectured that if G(n).has no isolated vertices then     

一个图的匹配数条件

设G是一个简单图,在图G中任意一个最大匹配的基数叫做G的匹配数,记作v(G),在这篇文章中我们获得了下面的结果,(1)设G是连通的和不完全的,则对于x,y∈v(G)和xyE(G),v(G-{x,y}=v(G)-1的充分必要条件是(a)G[A(G)]是完全的和A(G)的每一个点和C(G)的每一个点相邻,(b)c(D(G))=|A(G)| 1,和(c)y∈D(G-x)对于x,y∈C(G)。(2)设G是连通的和不完全的,则v(G-{x,y})=v(G)-2对于x,y∈V(G)和xyE(G)的充分必要条件是GK_(n,n),其中n≥2。     

3-调和的5-圈图

设v1，v2，v3，…，vn是图G的n个顶点，（d（v1），d（u2），d（u3），…，d（vn））^T是图G邻接矩阵A的特征向量，则称G是调和图，其中d（vi）表示顶点弘的度．1—4圈的调和图已经确定，本文确定了所有的3-调和的5-圈调和图．     

2p2阶3度Cayley图

Cayley图Cay(G,S)称之为正规的,如果G的右正则表示是Cay(G,S)全自同构群的正规子群。本文决定了2p~2(p为素数)阶群上3度连通Cayley图的正规性,作为该结果的一个应用,对每一个1(?)s(?)5,对2p~2阶3度s-正则Cayley图作了分类。     

Graph relations,clique divergence and surface triangulations

This work has two aims: first, we introduce a powerful technique for proving clique divergence when the graph satisfies a certain symmetry condition. Second, we prove that each closed surface admits a clique divergent triangulation. By definition, a graph is clique divergent if the orders of its iterated clique graphs tend to infinity, and the clique graph of a graph is the intersection graph of its maximal complete subgraphs. © 2005 Wiley Periodicals, Inc. J Graph Theory     

Sparsity-certifying Graph Decompositions

We describe a new algorithm, the (k, ℓ)-pebble game with colors, and use it to obtain a characterization of the family of (k, ℓ)-sparse graphs and algorithmic solutions to a family of problems concerning tree decompositions of graphs. Special instances of sparse graphs appear in rigidity theory and have received increased attention in recent years. In particular, our colored pebbles generalize and strengthen the previous results of Lee and Streinu [12] and give a new proof of the Tutte-Nash-Williams characterization of arboricity. We also present a new decomposition that certifies sparsity based on the (k, ℓ)-pebble game with colors. Our work also exposes connections between pebble game algorithms and previous sparse graph algorithms by Gabow [5], Gabow and Westermann [6] and Hendrickson [9]. Ileana Streinu; Research of both authors funded by the NSF under grants NSF CCF-0430990 and NSF-DARPA CARGO CCR-0310661 to the first author.     

Erratum to “On a Classical Theorem on the Diameter and Minimum Degree of a Graph”

The original version of the article was published in[1]. Unfortunately, the original version of this article contains a mistake:in Theorem 6.2 appears that β(n, Δ)=(n-Δ+5)/4 but the correct statement is β(n, Δ)=(n-Δ+4)/4. In this erratum we correct the theorem and give the correct proof.     

On Geometric Graph Ramsey Numbers

For any two-colouring of the segments determined by 3n − 3 points in general position in the plane, either the first colour class contains a triangle, or there is a noncrossing cycle of length n in the second colour class, and this result is tight. We also give a series of more general estimates on off-diagonal geometric graph Ramsey numbers in the same spirit. Finally we investigate the existence of large noncrossing monochromatic matchings in multicoloured geometric graphs. Research partially supported by Hungarian Scientific Research Grants OTKA T043631 and NK67867.     

太阳图与路的笛卡儿积图的任意可分性（英文）

一个阶为n的图G称为是任意可分的(简作AP),如果对于任一正整数序列τ=(n₁,n₂,…,n_k)满足n=n₁+n₂+…+n_k,总是存在顶点集V(G)的一个划分(V₁,V₂,…,V_k)满足:对于i∈[1,k],|V_i|=n_i,且子图G|V_i|是图G的V_i导出的一个连通子图.我们用S~*=S(n;m₁,m₂,…,m_n)来表示最大度△(S~*)=3的太阳图.本文讨论了图S~*P_m(m≥3)的任意可分性.     

Graph theoretic approach to parallel gene assembly

We study parallel complexity of signed graphs motivated by the highly complex genetic recombination processes in ciliates. The molecular gene assembly operations have been modeled by operations of signed graphs, i.e., graphs where the vertices have a sign + or −. In the optimization problem for signed graphs one wishes to find the parallel complexity by which the graphs can be reduced to the empty graph. We relate parallel complexity to matchings in graphs for some natural graph classes, especially bipartite graphs. It is shown, for instance, that a bipartite graph G has parallel complexity one if and only if G has a unique perfect matching. We also formulate some open problems of this research topic.