几类图的pebbling数

金芳蓉定义了图 G上的一个 pebbling 移动是从一个顶点处移走两个pebble 而把其中的一个移到与其相邻的一个顶点上. 图G的pebbling数f(G)是最小的整数n, 使得不管n 个pebble 如何放置在G的顶点上, 总可以通过一系列的 pebbling 移动把一个pebble 移到 G的任一个顶点上. Graham 猜测对于任意的连通图G和H有f(G×H)≤f(G)f(H). 计算了两个扇图的积和两个轮图的积的pebbling数, 作为推论, 当G和H同时是扇图或轮图时, Graham 猜想成立.     

完全二部图乘积上的 Graham pebbling猜想

图G的pebbling数f(G)是最小的整数n,使得不论n个pebble如何放置在G的顶点上,总可以通过一系列的pebbling移动把1个pebble移到任意一个顶点上,其中的pebbling移动是从一个顶点处移走两个pebble而把其中的一个移到与其相邻的一个顶点上. Graham猜测对于任意的连通图G和H有f(G×H)≤f(G)f(H).证明了对于一个完全二部图和一个具有2-pebbling性质的图来说,Graham猜想是成立的,作为一个推论,当G和H都是完全二部图时,Graham猜想成立.     

几类二部图的pebbling数

Chung定义了图G上的一个pebbling移动是从一个顶点移走两个pebble而把其中的一个移到与其相邻的一个顶点上.连通图G的pebbling数f(G)是最小的正整数n,使得不管n个pebble如何放置在G的顶点上,总可以通过一系列的pebbling移动把一个pebble移到G的任意一个顶点上.Graham猜测对于任意的连通图G和H有f(G×H)≤f(G)f(H).作者们验证了三类二部图的2-pebbling性质以及当H为此类二部图,G为一个2-pebbling性质的图时,Graham猜想成立.     

圈的中间图pebbling数和Graham猜想

图G的一个pebbling移动是从一个顶点移走2个pebble, 而把其中的1个pebble移到与其相邻的一个顶点上. 图G 的pebbling数f(G)是最小的正整数n, 使得不论n个pebble 如何放置在G的顶点上, 总可以通过一系列的pebbling移动, 把1个pebble移到图G的任意一个顶点上. 图G 的中间图M(G) 就是在G 的每一条边上插入一个新点, 再把G 上相邻边上的新点用一条边连接起来的图. 对于任意两个连通图G和H, Graham猜测f(G\times H)\leq f(G)f(H). 首先研究了圈的中间图的pebbling 数, 然后讨论了一些圈的中间图满足Graham猜想.     

多扇图的Pebbling数和Graham猜想

图G的pebbling数f(G)是最小的整数n,使得不论n个pebble如何放置在G的顶点上,总可以通过一系列的pebbling移动把1个pebble移到任意一个顶点上,其中一个pebbling移动是从一个顶点处移走两个pebble而把其中的一个移到与其相邻的一个顶点上。Graham猜想对于任意的连通图G和H有f(G×H)f(G)f(H)。多扇图F_n1,n2,…,nm是指阶为n₁+n₂+…+n_m+1的联图P₁∨(P_n1∪P_n2∪…∪P_nm)。本文首先给出了多扇图的pebbling数,然后证明了多扇图F_n1,n2,…,nm具有2-pebbling性质,最后论述了对于一个多扇图和一个具有2-pebbling性质的图的乘积来说,Graham猜想是成立的。作为一个推论,当G和H都是多扇图时,Graham猜想成立。     

图乘积的星色数

星色数的概念最早是由Vince作为图的色数的推广而引入的.本文研究了两类图乘积G×H,G[H]的星色数.     

广义友谊图乘积上的Graham pebbling猜想

连通图G的pebbling数f(G)是最小的正整数n,使得不管n个pebble如何放置在G的顶点上,总可以通过一系列的pebbling移动把一个pebble移到图G的任意一个顶点上.Graham猜测对于任意的连通图G和H有f(G×H)≤f(G)f(H).文中证明了当H为友谊图或广义友谊图,G是一个具有2-pebbling性质的图时,Graham猜想成立.作为一个推论,文中也证明了当G和H是友谊图或广义友谊图时,Graham猜想成立.     

乘积图的全色数

本文得到了有关乘积图的全色数的一些结果,并利用这些结果证明了Mesh图和Tours-图均满足全色数猜想．特别,几乎所有的Mesh-图都是第一类图．     

关于图的星形因子覆盖

如果图 G 的支撑子图 M 的每个分支都同构于{K_(1,1)K_(1,2,)…,K_(1,k}(k≥2)中的某个 K_(1,i),则 M(?)叫做 G 的星形因子。进一步,如果对于图 G 的每一条边都存在一个星形因子包含这条边,则称图 G 是星形因子覆盖的。本文给出了图是{P_2,P_3}一因子覆盖的充要条件,并证明了任意正则图均存在星形因子覆盖。     

六阶图Q与星图S_n的积图交叉数

目前关于积图的交叉数的研究已经推广到六阶图与星图的积图.研究得到了一个特殊六阶图Q与n个孤立点nK_1的联图交叉数,然后通过收缩的方法,得到了Q与星图S_n的积图交叉数.     

Maximum pebbling number of graphs of diameter three

Given a configuration of pebbles on the vertices of a graph G, a pebbling move consists of taking two pebbles off some vertex v and putting one of them back on a vertex adjacent to v. A graph is called pebbleable if for each vertex v there is a sequence of pebbling moves that would place at least one pebble on v. The pebbling number of a graph G is the smallest integer m such that G is pebbleable for every configuration of m pebbles on G. We prove that the pebbling number of a graph of diameter 3 on n vertices is no more than (3/2)n + O(1), and, by explicit construction, that the bound is sharp. © 2006 Wiley Periodicals, Inc. J Graph Theory     

The signed star domination numbers of the Cartesian product graphs

Let G be a graph with vertex set V(G) and edge set E(G). A function f:E(G)→{-1,1} is said to be a signed star dominating function of G if for every v∈V(G), where E_G(v)={uv∈E(G)|u∈V(G)}. The minimum of the values of , taken over all signed star dominating functions f on G, is called the signed star domination number of G and is denoted by γ_SS(G). In this paper, a sharp upper bound of γ_SS(G×H) is presented.     

Graham’s pebbling conjecture on product of thorn graphs of complete graphs

The pebbling number of a graph G, f(G), is the least n such that, no matter how n pebbles are placed on the vertices of G, we can move a pebble to any vertex by a sequence of pebbling moves, each move taking two pebbles off one vertex and placing one on an adjacent vertex. Let p₁,p₂,…,p_n be positive integers and G be such a graph, V(G)=n. The thorn graph of the graph G, with parameters p₁,p₂,…,p_n, is obtained by attaching p_i new vertices of degree 1 to the vertex u_i of the graph G, i=1,2,…,n. Graham conjectured that for any connected graphs G and H, f(G×H)≤f(G)f(H). We show that Graham’s conjecture holds true for a thorn graph of the complete graph with every by a graph with the two-pebbling property. As a corollary, Graham’s conjecture holds when G and H are the thorn graphs of the complete graphs with every .     

Independence number of products of Kneser graphs

We study the independence number of a product of Kneser graph $K(n,k)$ with itself, where we consider all four standard graph products. The cases of the direct, the lexicographic and the strong product of Kneser graphs are not difficult (the formula for $\alpha (K(n,k)⊠K(n,k))$ is presented in this paper), while the case of the Cartesian product of Kneser graphs is much more involved. We establish a lower bound and an upper bound for the independence number of $K(n,2)\square K(n,2)$, which are asymptotically tending to $n^3∕3$ and $3n^3∕8$, respectively. The former is obtained by a construction, which differs from the standard diagonalization procedure, while for the upper bound the $\ell$-independence number of Kneser graphs can be applied. We also establish some constructions in odd graphs $K(2k+1,k)$, which give a lower bound for the 2-independence number of these graphs, and prove that two such constructions give the same lower bound as a previously known one. Finally, we consider the $s$-stable Kneser graphs $K{(ks+1,k)}_{s-stab}$, derive a formula for their $\ell$-independence number, and give the exact value of the independence number of the Cartesian square of $K{(ks+1,k)}_{s-stab}$.     

Game coloring the Cartesian product of graphs

This article proves the following result: Let G and G′ be graphs of orders n and n′, respectively. Let G^* be obtained from G by adding to each vertex a set of n′ degree 1 neighbors. If G^* has game coloring number m and G′ has acyclic chromatic number k, then the Cartesian product G□G′ has game chromatic number at most k(k + m ? 1). As a consequence, the Cartesian product of two forests has game chromatic number at most 10, and the Cartesian product of two planar graphs has game chromatic number at most 105. © 2008 Wiley Periodicals, Inc. J Graph Theory 59: 261–278, 2008     

Game chromatic number of strong product graphs

The graph coloring game is a two-player game in which the two players properly color an uncolored vertex of G alternately. The first player wins the game if all vertices of G are colored, and the second wins otherwise. The game chromatic number of a graph G is the minimum integer k such that the first player has a winning strategy for the graph coloring game on G with k colors. There is a lot of literature on the game chromatic number of graph products, e.g., the Cartesian product and the lexicographic product. In this paper, we investigate the game chromatic number of the strong product of graphs, which is one of major graph products. In particular, we completely determine the game chromatic number of the strong product of a double star and a complete graph. Moreover, we estimate the game chromatic number of some King's graphs, which are the strong products of two paths.     

On the geodetic number and related metric sets in Cartesian product graphs

A set S of vertices of a graph G is a geodetic set if every vertex of G lies in at least one interval between the vertices of S. The size of a minimum geodetic set in G is the geodetic number of G. Upper bounds for the geodetic number of Cartesian product graphs are proved and for several classes exact values are obtained. It is proved that many metrically defined sets in Cartesian products have product structure and that the contour set of a Cartesian product is geodetic if and only if their projections are geodetic sets in factors.