The competition number of a graph whose holes do not overlap much

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• 作者：Suh-Ryung Kim ; Jung Yeun Lee ; Yoshio Sano
• 关键词：Competition graph ; Competition number ; Hole
• 刊名：Discrete Applied Mathematics
• 出版年：2010
• 出版时间：6 July 2010
• 年：2010
• 卷：158
• 期：13
• 页码：1456-1460
• 全文大小：235 K

文摘

Let D be an acyclic digraph. The competition graph of D is a graph which has the same vertex set as D and has an edge between u and v if and only if there exists a vertex x in D such that (u,x) and (v,x) 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.

A hole of a graph is an induced cycle of length at least four. Kim (2005) [8] conjectured that the competition number of a graph with h holes is at most h+1. Recently, Li and Chang (2009) [11] showed that the conjecture is true when the holes are independent. In this paper, we show that the conjecture is true though the holes are not independent but mutually edge-disjoint.