完全$r$部图乘积上的Graham猜想

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移到任意一点上,其中Pebbling移动是从一个顶点处移走两个Pebble而把其中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).本文证明对于一个完全γ部图和一个具有2-Pebbleing性质的图来说,Graham猜想成立.作为一个推论,当G和H均为完全γ部图时,Graham猜想成立.     

补图是$r$-部的图的顶点可分解性质

Let G be a non-complete graph such that its complement G is r-partite.In this paper,properties of the graph G are studied,including the Cohen-Macaulay property and the sequential Cohen-Macaulay property.For r=2,3,some constructions are established for G to be vertex decomposable and some sufficient conditions are provided for r≥4.