Vertex 2-arboricity of planar graphs without 4-cycles adjacent to 6-cycles

The vertex-arboricity of a graph G, denoted by symbolscript is the minimum number of subsets in which V symbolscript can be partitioned so that each subset induces a forest. It is known that symbolscript <= 3 for every planar graph G and symbolscript <= 2 if G is a planar graph without k-cycles where k E {3, 4, 5, 6, 7}. Recently, Cui et al. proved that symbolscript <= 2 if G is a planar graph without 4-cycles intersecting with 6-cycles. In this paper, we extend the result of Cui et al. by showing that symbolscript <= 2 if G is a planar graph without 4-cycles adjacent to 6-cycles.(c) 2022 Elsevier B.V. All rights reserved.