Optimal k-fold colorings of webs and antiwebs

Details Optimal k-fold colorings of webs and antiwebs Optimal k-fold colorings of webs and antiwebs

2011-08-29

arxiv.org/abs/1108.5757v1

Computer Science

Discrete Mathematics

A short version of this paper was presented at the Simp\'osio Brasileiro de Pesquisa Operacional, Brazil, 2011

Scientific paper

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.

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