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

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. (C) 2010 Elsevier B.V. All rights reserved.