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.
Sara Al Hajjar (Fri,) studied this question.