Vertex-arboricity of toroidal graphs without [formula omitted] and 6-cycles

Abstract

The vertex-arboricity va(G) of a graph G is defined to be the minimum number of colors needed to color the vertices of G such that no cycle is monochromatic. The list vertex-arboricity val(G) is the list-coloring version of this concept. In this paper, we prove that every toroidal graph G with neither K5− (a K5 missing at most one edge) nor 6-cycles satisfies val(G)≤2. This will be best possible in the sense that forbidding only one of the two structures cannot guarantee its (list) vertex-arboricity being at most 2.