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

  • Authors:

  • Suh-Ryung Kim;Jung Yeun Lee;Yoshio Sano

  • Affiliations:

  • Department of Mathematics Education, Seoul National University, Seoul 151-742, Republic of Korea;Department of Mathematics Education, Seoul National University, Seoul 151-742, Republic of Korea;Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2010

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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.