Competition Numbers, Quasi-Line Graphs and Holes

TL;DR

This paper proves two conjectures in graph theory: that quasi-line graphs have a competition number at most 2, and that the competition number of any graph is closely related to the number of holes and the cycle space dimension.

Contribution

It provides new characterizations of quasi-line graphs and establishes bounds on competition numbers related to holes and cycle space dimensions.

Findings

The competition number of every quasi-line graph is at most 2.

The competition number of any graph is at most one greater than the number of holes.

The competition number is also at most one greater than the dimension of the cycle space spanned by holes.

Abstract

The competition graph of a directed acyclic graph D is the undirected graph on the same vertex set as D in which two distinct vertices are adjacent if they have a common out-neighbor in D. The competition number of an undirected graph G is the least number of isolated vertices that have to be added to G to make it the competition graph of a directed acyclic graph. We resolve two conjectures concerning competition graphs. First we prove a conjecture of Opsut by showing that the competition number of every quasi-line graph is at most 2. Recall that a quasi-line graph, also called a locally co-bipartite graph, is a graph for which the neighborhood of every vertex can partitioned into at most two cliques. To prove this conjecture we devise an alternative characterization of quasi-line graphs to the one by Chudnovsky and Seymour. Second, we prove a conjecture of Kim by showing that the competition number of any graph is at most one greater than the number of holes in the graph. Our methods also allow us to prove a strengthened form of this conjecture recently proposed by Kim, Lee, Park and Sano, showing that the competition number of any graph is at most one greater than the dimension of the subspace of the cycle space spanned by the holes.