Graphs having many holes but with small competition numbers

TL;DR

This paper investigates the relationship between the number of holes in a graph and its competition number, showing that graphs with many holes can still have small competition numbers under certain conditions.

Contribution

It establishes bounds on the competition number for connected graphs with many holes, especially when the graph has limited maximal cliques and specific hole configurations.

Findings

Graphs with many holes can have competition number as low as 2.

For graphs with h holes and limited maximal cliques, the competition number is at most h - ω + 3.

Conditions involving edge-disjoint holes and clique number influence the competition number bounds.

Abstract

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 graph which does not contain a hole is at most one and the competition number of a complete bipartite graph $K_{\lfloor \frac{n}{2} \rfloor, \lceil \frac{n}{2} \rceil}$ 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 connected graph G has many holes, the competition number of G can be as small as 2 under some assumption. In addition, we show that, for a connected graph G with exactly h holes and at most one non-edge maximal clique, if all the holes of G are pairwise edge-disjoint and the clique number $\omega = \omega (G)$ of G satisfies $2 \leq \omega \leq h+1$, then the competition number of G is at most $h - \omega + 3$.