Circular mixed hypergraphs II: the upper chromatic number

Authors:
Vitaly Voloshin;Heinz-Jürgen Voss
Affiliations:
Department of Mathematics and Physics, Troy University, Troy, Alabama;Institut für Algebra, Technische Universität Dresden, Dresden, Germany
Venue:
Discrete Applied Mathematics
Year:
2006

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Abstract

A mixed hypergraph is a triple H = (X, C, D), where X is the vertex set and each of C, D is a family of subsets of X, the C-edges and D-edges, respectively. A proper k-coloring of H is a mapping c: X → |k| such that each C-edge has two vertices with a common color and each D-edge has two vertices with distinct colors. A mixed hypergraph H is called circular if there exists a host cycle on the vertex set X such that every edge (C-or D-) induces a connected subgraph of this cycle.We suggest a general procedure for coloring circular mixed hypergraphs and prove that if H is a reduced colorable circular mixed hypergraph with n vertices, upper chromatic number χ- and sieve number s, then n-s - 2 ≤ χ- ≤ n-s+2.