Bounds for the $b$-chromatic number of some families of graphs

TL;DR

This paper establishes tight upper bounds for the $b$-chromatic number in various graph families, including $K_{1,t}$-free, graphs with specified clique partitions, and bipartite graphs, based on parameters like clique number and chromatic number.

Contribution

It provides new tight upper bounds for the $b$-chromatic number for specific graph classes, linking it to clique and biclique parameters.

Findings

Bounds are tight for all considered graph families.

Upper bounds depend on clique number, chromatic number, or biclique number.

Results unify and extend previous bounds for $b$-chromatic number.

Abstract

In this paper we obtain some upper bounds for $b$-chromatic number of $K_{1,t}$ -free graphs, graphs with given minimum clique partition and bipartite graphs. These bounds are in terms of either clique number or chromatic number of graphs or biclique number for bipartite graphs. We show that all the bounds are tight.