On Planar Mixed Hypergraphs

A mixed hypergraph H is a triple (V, C,D) where V is its vertex set and C and D are families of subsets of V , C–edges and D–edges. A mixed hypergraph is a bihypergraph iff C = D. A hypergraph is planar if its bipartite incidence graph is planar. A vertex coloring of H is proper if each C–edge contains two vertices with the same color and each D–edge contains two vertices with different colors. The set of all k’s for which there exists a proper coloring using exactly k colors is the feasible set of H; the feasible set is called gap-free if it is an interval. The minimum (maximum) number of the feasible set is called a lower (upper) chromatic number. We prove that the feasible set of any planar mixed hypergraph without edges of size two and with an edge of size at least four is gap-free. We further prove that a planar mixed hypergraph with at most two D–edges of size two is two-colorable. We describe a polynomial-time algorithm to decide whether the lower chromatic number of a planar mixed hypergraph equals two. We prove that it is NP-complete to find the upper chromatic number of a mixed hypergraph even for 3-uniform planar bihypergraphs. In order to prove the latter statement, we prove that it is NP-complete to determine whether a planar 3-regular bridgeless graph contains a 2-factor with at least a given number of components. ∗The author acknowledges partial support by GAČR 201/1999/0242 and GAUK 158/1999. †The author acknowledges partial support by GAČR 201/1999/0242, GAUK 158/1999 and KONTAKT 338/99. ‡Institute for Theoretical Computer Science is supported by Ministry of Education of Czech Republic as project LN00A056. the electronic journal of combinatorics 8 (2001), #R35 1