任意两个洞至多共享一条边的图的竞争数
The competition number of a graph in which any two holes share at most one edge
摘要：
有向图D的竞争图是具有与D相同的顶点集且在x和y之间有一条边的（简单无向）图当且仅当存在D中的顶点v使得（x，v）和（y，v）是D的弧。对任意图G，G加上足够多的孤立点是某个无圈有向图的竞争图。图G的竞争数k（G）是图G中孤立点的最小个数。一般来说，图G的竞争数k（G）是很难计算的，用图G的竞争数来刻画图G是竞争图研究中的重要问题之一。一个图的洞是长度至少为4的圈作为诱导子图。在一定条件下，图G的竞争数不能超过1与洞数之和。本文证明了具有恰好h个洞且其中任意两个洞至多共享一条边的图的竞争数至多为h+1，推广了已有的结果.
英文摘要：
The competition graph of a digraph D is a (simple undirected) graph which has the same vertex set as D and has an edge between x and y if and only if there exists a vertex v in D such that (x,v) and (y,v) are arcs of D. For any graph G, G together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number k(G) of G is the smallest number of such isolated vertices. In general, it is hard to compute the competition number k(G) for a graph G and it has been one of important research problems in the study of competition graphs to characterize a graph by its competition number. A hole of a graph is a cycle of length at least 4 as an induced subgraph. It holds that the competition number of a graph cannot exceed one plus the number of its holes if G satisfies a certain condition. In this paper, we show that the competition number of a graph with exactly h holes any two of which share at most one edge is at most h+1, which generalizes the existing results on this subject.
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