Competition numbers of planar graphs

TL;DR

This paper establishes a new inequality relating the competition number of a graph to its edge clique cover number and applies it to derive results specifically for planar graphs.

Contribution

It introduces a tight inequality connecting competition number and edge clique cover number, and uses it to analyze competition numbers of planar graphs.

Findings

Derived a tight inequality for competition numbers of graphs.

Obtained new results on competition numbers of planar graphs.

Introduced the concept of competition-effective edge clique cover.

Abstract

In this paper, we relate the competition number of a graph to its edge clique cover number by presenting a tight inequality $k(G) \ge \theta_e(G)-|V(G)|+\widetilde{k}(G)$ where $\theta_e(G)$, $k(G)$, and $\widetilde{k}(G)$ are the edge clique cover number, the competition number, and the co-competition number of a graph $G$, respectively. By utilizing this inequality and a notion of competition-effective edge clique cover, we obtain some meaningful results on competition numbers of planar graphs.