Theoretical graph analysis demonstrates a tightened upper bound on the 2-distance chromatic number for planar graphs with maximum degree at most six, improving previous theoretical limits.
A 2-distance [Formula: see text]-coloring of a graph is a proper coloring of the vertices of the graph using [Formula: see text] colors such that any two vertices at distance two or less get distinct 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] admits a 2-distance [Formula: see text]-coloring. In [4], N. Bousquet proved that [Formula: see text] for planar graphs with maximum degree [Formula: see text] For a planar graph [Formula: see text] with a maximum degree [Formula: see text] at most 6, we prove that [Formula: see text] hence improving the bound of [Formula: see text] for planar graphs with [Formula: see text].
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Sara Al Hajjar (2026) studied this question.
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