Planar graphs without intersecting 5-cycles are signed-4-choosable

Abstract

A signed graph is a pair [Formula: see text], where [Formula: see text] is a graph and [Formula: see text] is a signature of [Formula: see text]. A set [Formula: see text] of integers is symmetric if [Formula: see text] implies that [Formula: see text]. Given a list assignment [Formula: see text] of [Formula: see text], an [Formula: see text]-coloring of a signed graph [Formula: see text] is a coloring [Formula: see text] of [Formula: see text] such that [Formula: see text] for each [Formula: see text] and [Formula: see text] for every edge [Formula: see text]. The signed choice number [Formula: see text] of a graph [Formula: see text] is defined to be the minimum integer [Formula: see text] such that for any [Formula: see text]-list assignment [Formula: see text] of [Formula: see text] and for any signature [Formula: see text] on [Formula: see text], there is a proper [Formula: see text]-coloring of [Formula: see text]. List signed coloring is a generalization of list coloring. However, the difference between signed choice number and choice number can be arbitrarily large. Hu and Wu [Planar graphs without intersecting [Formula: see text]-cycles are [Formula: see text]-choosable, Discrete Math. 340 (2017) 1788–1792] showed that every planar graph without intersecting 5-cycles is 4-choosable. In this paper, we prove that [Formula: see text] if [Formula: see text] is a planar graph without intersecting 5-cycles, which extends the main result of [D. Hu and J. Wu, Planar graphs without intersecting [Formula: see text]-cycles are [Formula: see text]-choosable, Discrete Math. 340 (2017) 1788–1792].