A sufficient condition for Kim’s conjecture on the competition numbers of graphs

A hole   of a graph GG is an induced cycle of length at least 4. Kim (2005) [3] 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. Li and Chang (2009) [5] proved that the conjecture is true for a graph whose holes all satisfy a property called ‘independence’. In this paper, by using similar proof techniques in Li and Chang (2009) [5], we prove the conjecture for graphs satisfying two conditions that allow the holes to overlap a lot.