Optimal k-fold colorings of webs and antiwebs

• Published in: arXiv.org
• Main Authors: Campêlo, Manoel, Corrêa, Ricardo C, Moura, Phablo F S, Santos, Marcio C
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 29.08.2011
• Subjects: Apexes; Coloring; Computer Science - Discrete Mathematics; Economic models; Graphs; Upper bounds; Webs
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1108.5757

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.