Local Clique Covering of Graphs

TL;DR

This paper investigates the local clique cover number of graphs, especially claw-free and linear interval graphs, establishing bounds related to maximum degree and introducing a new combinatorial inequality.

Contribution

It provides new bounds for the local clique cover number in specific graph classes and introduces a Bollobas-type inequality for intersecting set systems.

Findings

Local clique cover number of claw-free graphs is at most cΔ / logΔ.

Bound on local clique number for linear interval graphs is logΔ + O(1).

A new Bollobas-type inequality for intersecting set systems is derived.

Abstract

A k-clique covering of a simple graph G, is an edge covering of G by its cliques such that each vertex is contained in at most k cliques. The smallest k for which G admits a k-clique covering is called local clique cover number of G and is denoted by $lcc(G)$. Local clique cover number can be viewed as the local counterpart of the clique cover number which is equal to the minimum total number of cliques covering all edges. In this paper, several aspects of the problem are studied and its relationships to other well-known problems are discussed. Moreover, the local clique cover number of claw-free graphs and its subclasses are notably investigated. In particular, it is proved that local clique cover number of every claw-free graph is at most $c\Delta / \log\Delta$, where $\Delta$ is the maximum degree of the graph and $c$ is a universal constant. It is also shown that the bound is tight, up to a constant factor. Furthermore, it is established that local clique number of the linear interval graphs is bounded by $\log\Delta + 1/2 \log \log\Delta + O(1)$. Finally, as a by-product, a new Bollobas-type inequality is obtained for the intersecting pairs of set systems.