List-coloring clique-hypergraphs of [formula omitted]-minor-free graphs strongly

Abstract

Let G be a connected simple graph with at least one edge. The hypergraph H=H(G) with the same vertex set as G whose hyper-edges are the maximal cliques of G is called the clique-hypergraph of G. A list-assignment of G is a function L which assigns to each vertex v∈V(G) a set L(v) (called the list of v). A k-list-assignment of G is a list-assignment L such that L(v) has at least k elements for every v∈V(G). For a given list assignment L, a list-coloring of H(G) is a function c:V(G)→∪vL(v) such that c(v)∈L(v) for every v∈V(G) and no hyper-edge of H(G) is monochromatic. A list-coloring of H(G) is strong if no 3-cycle of G is monochromatic. H(G) is (strongly) k-choosable if, for every k-list assignment L, there exists a (strong) list-coloring of H(G). Mohar and Sˇkrekovski proved that the clique-hypergraphs of planar graphs are strongly 4-choosable (Electr. J. Combin. 6 (1999), #R26). In this paper we give a short proof of the result and present a linear time algorithm for the strong list-4-coloring of H(G) if G is a planar graph. In addition, we prove that H(G) is strongly 4-choosable if G is a K5-minor-free graph, which is a generalization of their result.