Minimal normal graph covers

TL;DR

This paper studies the properties of normal graph covers, establishing bounds on clique and stable set sizes relative to the number of vertices, and constructs extremal examples demonstrating these bounds.

Contribution

It introduces extremal bounds for normal covers with bounded clique and stable set sizes and constructs graphs achieving near these bounds.

Findings

For graphs with normal covers, c+s ≥ log₂(n).

Constructed graphs with clique and stable set sizes less than 0.87 log₂(n).

Existence of normal graphs with Θ(log₂(n)) clique and independence numbers.

Abstract

A graph is normal if it admits a clique cover $\mathcal C$ and a stable set cover $\mathcal S$ such that each clique in $\mathcal C$ and each stable set in $\mathcal S$ have a vertex in common. The pair $(\mathcal{C,S})$ is a normal cover of the graph. We present the following extremal property of normal covers. For positive integers $c,s$, if a graph with $n$ vertices admits a normal cover with cliques of sizes at most $c$ and stable sets of sizes at most $s$, then $c+s\geq\log_2(n)$. For infinitely many $n$, we also give a construction of a graph with $n$ vertices that admits a normal cover with cliques and stable sets of sizes less than $0.87\log_2(n)$. Furthermore, we show that for all $n$, there exists a normal graph with $n$ vertices, clique number $\Theta(\log_2(n))$ and independence number $\Theta(\log_2(n))$. When $c$ or $s$ are very small, we can describe all normal graphs with the largest possible number of vertices that allow a normal cover with cliques of sizes at most $c$ and stable sets of sizes at most $s$. However, such extremal graphs remain elusive even for moderately small values of $c$ and $s$.