On the (3,1)-choosability of planar graphs without adjacent cycles of length 5, 6, 7

A (k, d)-list assignment L of a graph G is a mapping that assigns to each vertex v a list L(v) of at least k colors satisfying vertical bar L(x) boolean AND L(y)vertical bar <= d for each edge xy. A graph G is (k, d)-choosable if there exists an L-coloring of G for every (k, d)-list assignment L. This concept is also known as choosability with separation. In this paper, we prove that any planar graph G is (3, 1)-choosable if any i-cycle is not adjacent to a j-cycle, where 5 <= i <= 6 and 5 <= j <= 7. (C) 2019 Elsevier B.V. All rights reserved.