Cover numbers by certain graph families

TL;DR

This paper introduces a formula for the cover number of a graph by certain graph classes based on chromatic and clique numbers, and explores inequalities among various classes, showing their differences can be arbitrarily large.

Contribution

It provides an exact formula for the cover number by classes defined via chromatic and clique numbers and analyzes inequalities among different graph classes.

Findings

Exact formula for cover number with (\u03a9)

Chain of inequalities with five graph classes

Differences between inequalities can grow arbitrarily large

Abstract

We define the cover number of a graph $G$ by a graph class $\mathcal P$ as the minimum number of graphs of class $\mathcal P$ required to cover the edge set of $G$. Taking inspiration from a paper by Harary, Hsu and Miller, we find an exact formula for the cover number by the graph classes $\{ G \mid \chi(G) \leq f(\omega(G))\}$ for an arbitrary non-decreasing function $f$. After this, we establish a chain of inequalities with five cover numbers, the one by the class $\{ G \mid \chi(G) = \omega(G)\}$, by the class of perfect graphs, generalized split graphs, co-unipolar graphs and finally by bipartite graphs. We prove that at each inequality, the difference between the two sides can grow arbitrarily large. We also prove that the cover number by unipolar graphs cannot be expressed in terms of the chromatic or the clique number.