A new upper bound for the clique cover number with applications

TL;DR

This paper establishes a new upper bound for the clique cover number of a graph based on its relationship with an interval graph, providing insights and improvements for intersection graph analysis.

Contribution

The paper introduces a novel upper bound for the clique cover number involving interval graphs and a parameter (G,H), unifying and enhancing previous results.

Findings

Proves (G)  (H)  (G,H) ((G) is the clique cover number)

Provides a generalized bound applicable to various intersection graphs

Improves upon existing bounds for specific classes of graphs

Abstract

Let $\alpha(G)$ and $\beta(G)$, denote the size of a largest independent set and the clique cover number of an undirected graph $G$. Let $H$ be an interval graph with $V(G)=V(H)$ and $E(G)\subseteq E(H)$, and let $\phi(G,H)$ denote the maximum of ${\beta(G[W])\over \alpha(G[W])}$ overall induced subgraphs $G[W]$ of $G$ that are cliques in $H$. The main result of this paper is to prove that for any graph $G$ $${\beta(G)}\le 2 \alpha(H)\phi(G,H)(\log \alpha(H)+1),$$ where, $\alpha(H)$ is the size of a largest independent set in $H$. We further provide a generalization that significantly unifies or improves some past algorithmic and structural results concerning the clique cover number for some well known intersection graphs.