Edge Clique Covers in Graphs with Independence Number Two: a Special Case

TL;DR

This paper investigates edge clique covers in graphs with independence number two, proving bounds for specific classes and providing insights into the structure of such graphs.

Contribution

It introduces new classes of graphs satisfying the conjecture and establishes an upper bound of 1.5n for the edge clique cover number in a broad class.

Findings

A class of graphs satisfying the conjecture with difficult-to-cover edges.

A large, easily characterized class of graphs with $ecc(G) \

ecc(G) \

Abstract

The edge clique cover number $ecc(G)$ of a graph $G$ is the size of the smallest set of complete subgraphs whose union covers all edges of $G$. It has been conjectured that all the simple graphs with independence number two satisfy $ecc(G)\leq n$. First, we present a class of graphs containing edges difficult to cover but that satisfy the conjecture. Second, we describe a large class of graphs $G$ such that $ecc(G)\leq \frac{3}{2}n$. This class is easy to characterize.