A 2-distance coloring of [Formula: see text] is a function [Formula: see text]: [Formula: see text], such that for every two distinct vertices [Formula: see text], [Formula: see text] in [Formula: see text], [Formula: see text] if [Formula: see text]. The 2-distance chromatic number of [Formula: see text] is the least integer [Formula: see text] such that [Formula: see text] has a [Formula: see text]-[Formula: see text]-distance coloring, denoted by [Formula: see text]. Similarly, the list 2-distance chromatic number of [Formula: see text] is denoted by [Formula: see text]. In this paper, we proved that: (1) for every planar graph with [Formula: see text] and [Formula: see text], [Formula: see text]; (2) for every planar graph with [Formula: see text] and [Formula: see text], [Formula: see text].
A proper coloring of the vertices of a graph is called a star coloring if the union of every two color classes induces a star forest. The star chromatic number s (G) is the smallest number of colors required to obtain a star coloring of G. In this paper, we study the relationship between the star chromatic number s (G) and the maximum average
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