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

Abstract

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.