Fractional coloring problem of 1-planar graphs without short cycles

Abstract

We consider a new coloring as follows such that a subset of [Formula: see text] needs very few colors. Let [Formula: see text] be a graph with vertex set [Formula: see text] and [Formula: see text] be three integers with [Formula: see text]. [Formula: see text] is a vertex subset of [Formula: see text], graph [Formula: see text] is said to be [Formula: see text]-weak [Formula: see text]-choosable about set [Formula: see text], if the following can be satisfied: For any list assignment [Formula: see text], there is a way that each vertex [Formula: see text] of [Formula: see text] is assigned [Formula: see text] colors of [Formula: see text] and each vertex [Formula: see text] of [Formula: see text] is assigned [Formula: see text] colors of [Formula: see text] such that the color sets corresponding to adjacent vertices do not intersect. In this paper, we consider the [Formula: see text]-weak [Formula: see text]-choosable problem of [Formula: see text]-planar graphs without short cycles and prove our results by the discharging method. In this paper, we prove that every [Formula: see text]-planar graph without [Formula: see text]-cycles, [Formula: see text]-cycles and adjacent [Formula: see text]-cycles is [Formula: see text]-weak [Formula: see text]-choosable about set [Formula: see text].