Graphs having many holes but with small competition numbers

The competition number k(G) of a graph G is the smallest number k such that G together with k isolated vertices added is the competition graph of an acyclic digraph. A chordless cycle of length at least 4 of a graph is called a hole of the graph. The number of holes of a graph is closely related to its competition number as the competition number of a chordal graph which does not contain a hole is at most one and the competition number of a complete bipartite graph K⌊n/2⌋,⌈n/2⌉ which has so many holes that no more holes can be added is the largest among those of graphs with n vertices. In this paper, we show that even if a graph G has many holes, as long as just one maximal clique of size one more than the number of holes is allowed, k(G) can be as small as 2. In addition, we show that if a graph G has h holes and just one maximal clique of size ω, and all the holes of G are edge-disjoint, then the competition number is at most h − ω + 3.