An upper bound for the competition numbers of graphs

A hole of a graph is an induced cycle of length at least 4. Kim (2005)  [2] conjectured that the competition number k(G)k(G) is bounded by h(G)+1h(G)+1 for any graph GG, where h(G)h(G) is the number of holes of GG. In Lee et al. [3], it is proved that the conjecture is true for a graph whose holes are mutually edge-disjoint. In Li et al. (2009) [4], it is proved that the conjecture is true for a graph, all of whose holes are independent. In this paper, we prove that Kim’s conjecture is true for a graph GG satisfying the following condition: for each hole CC of GG, there exists an edge which is contained only in CC among all induced cycles of GG.