On optimal k-fold colorings of webs and antiwebs

Authors:
Manoel CampêLo;Ricardo C. CorrêA;Phablo F. S. Moura;Marcio C. Santos
Affiliations:
Universidade Federal do Ceará, Departamento de Estatística e Matemática Aplicada, Campus do Pici, Bloco 910, 60440-554 Fortaleza - CE, Brazil;Universidade Federal do Ceará, Departamento de Computação, Campus do Pici, Bloco 910, 60440-554 Fortaleza - CE, Brazil;Universidade de São Paulo, Instituto de Matemática e Estatística, Rua do Matão 1010, 05508-090 São Paulo - SP, Brazil;Universidade Federal do Ceará, Departamento de Computação, Campus do Pici, Bloco 910, 60440-554 Fortaleza - CE, Brazil
Venue:
Discrete Applied Mathematics
Year:
2013

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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 @g"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 @g"k(G) attains its lower and upper bounds based on clique and integer and fractional chromatic numbers. Additionally, we extend the concept of @g-critical graphs to @g"k-critical graphs. We identify the webs and antiwebs having this property, for every integer k=1.