The competition number of a graph with exactly two holes

TL;DR

This paper proves Kim's conjecture that the competition number of a graph with exactly two holes is at most three, advancing understanding of the relationship between holes and competition numbers.

Contribution

It establishes the validity of Kim's conjecture specifically for graphs with exactly two holes, a case previously unresolved.

Findings

Proves the conjecture for graphs with two holes

Shows the competition number is at most h+1 for h=2

Advances the theory of competition graphs with holes

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 x and y if and only if there exists a vertex v in D such that (x,v) and (y,v) 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 a cycle of length at least 4 as an induced subgraph. In 2005, Kim [5] conjectured that the competition number of a graph with h holes is at most h+1. Though Li and Chang [8] and Kim et al. [7] showed that her conjecture is true when the holes do not overlap much, it still remains open for the case where the holes share edges in an arbitrary way. In order to share an edge, a graph must have at least two holes and so it is natural to start with a graph with exactly two holes. In this paper, the conjecture is proved true for such a graph.