关于0类图的一个注记

A note on Class 0 graphs

在图G的顶点上放置一些Pebble,图G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).若f(G)=|V(G)|,称G是0类的(Class 0).证明了有关0类图的一个结果.作为推论,得到了P×C5和P×P都是0类图,其中P是Petersen图.

Given a distribution of Pebbles on the vertices of a graph, a Pebbling move on a graph G is de- fined to be the removal of two Pebbles from one vertex and the addition of one Pebble to an adjacent vertex. The Pebbling number of a connected graph G is the smallest number f（G） so that any distribution off（G） Pebbles on G allows one Pebble to be moved to any specified but arbitrary vertex by a sequence of Pebbling moves. Graham conjectured that for any connected graphs G and H, f（ G x H） ≤f（G）f（H）. A graph is of Class 0 if f（G） = IV （G） I. Class 0 graphs were proved, and P x C5 and P x P are of Class 0, where P is the Petersen graph.