Two sufficient conditions for a planar graph to be list vertex-2-arborable

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 the following results: (i) If G is a planar graph without 7-cycles, then a(l)(G) <=& nbsp;2; (ii) If G is a planar graph without 8-cycles and adjacent 3-cycles, then a(l)(G) <=& nbsp;2. The result (i) extends a result in Huang et al. (2012) [12], which says that every planar graph G without 7-cycles has a(G) <=& nbsp;2.(C) 2022 Elsevier B.V. All rights reserved.