List vertex arboricity of planar graphs without 5-cycles intersecting with 6-cycles

Abstract

<abstract> The vertex arboricity $ a(G) $ of a graph $ G $ is the minimum number of colors required to color the vertices of $ G $ such that no cycle is monochromatic. The list vertex arboricity $ a_l(G) $ is the list version of this concept. In this paper, we prove that if $ G $ is a planar graph without 5-cycles intersecting with 6-cycles, then $ a_l(G)\le 2 $. </abstract>