A generalization of Opsut's lower bounds for the competition number of a graph

TL;DR

This paper generalizes Opsut's lower bounds for the competition number of a graph, a key measure in understanding the structure of competition graphs, which are derived from digraphs.

Contribution

The paper extends two existing lower bounds for the competition number, providing a broader theoretical framework for analyzing competition graphs.

Findings

Generalized Opsut's lower bounds for competition number

Enhanced understanding of competition graph structure

Provides tools for better estimation of competition numbers

Abstract

The notion of a competition graph was introduced by J. E. Cohen in 1968. The competition graph C(D) of a digraph $D$ is a (simple undirected) graph which has the same vertex set as D and has an edge between two distinct vertices 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. In 1978, F. S. Roberts defined the competition number k(G) of a graph G as the minimum number of such isolated vertices. In general, it is hard to compute the competition number k(G) for a graph G and it has been one of the important research problems in the study of competition graphs to characterize a graph by its competition number. In 1982, R. J. Opsut gave two lower bounds for the competition number of a graph. In this paper, we give a generalization of these two lower bounds for the competition number of a graph.