Optimal k-fold colorings of webs and antiwebs

TL;DR

This paper determines the exact k-th chromatic number for webs and antiwebs, generalizing known results for odd cycles and exploring criticality properties in graph coloring.

Contribution

It provides exact formulas for the k-th chromatic number of webs and antiwebs, extending previous results and introducing hi_k-critical graphs.

Findings

Exact hi_k(G) values for webs and antiwebs

Necessary and sufficient conditions for bounds attainment

Identification of hi_k-critical webs and antiwebs

Abstract

A k-fold x-coloring of a graph is an assignment of (at least) k distinct colors from the set {1, 2, ..., x} to each vertex such that any two adjacent vertices are assigned disjoint sets of colors. The smallest number x such that G admits a k-fold x-coloring is the k-th chromatic number of G, denoted by \chi_k(G). We determine the exact value of this parameter when G is a web or an antiweb. Our results generalize the known corresponding results for odd cycles and imply necessary and sufficient conditions under which \chi_k(G) attains its lower and upper bounds based on the clique, the fractional chromatic and the chromatic numbers. Additionally, we extend the concept of \chi-critical graphs to \chi_k-critical graphs. We identify the webs and antiwebs having this property, for every integer k <= 1.