A Tighter Bound on the 2-Distance Chromatic Number of Planar Graphs with Specific Maximum Degree

A [Formula: see text]-distance [Formula: see text]-coloring of a graph is a coloring of the vertices of the graph using [Formula: see text] colors such that any two vertices at distance two or less receive different colors. The 2-distance chromatic number of a graph [Formula: see text], denoted as [Formula: see text], is the minimum integer [Formula: see text] such that [Formula: see text] has a 2-distance [Formula: see text]-coloring. M. Krzyzinski [7] proved that [Formula: see text] for planar graphs with maximum degree [Formula: see text] Later, N. Bousquet [4] proved that [Formula: see text] for planar graphs with [Formula: see text] and [Formula: see text] for planar graphs with [Formula: see text]. In this paper, we will prove that [Formula: see text] for planar graphs with a maximum degree [Formula: see text] hence improving the bound of [Formula: see text] for planar graphs when [Formula: see text].