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Graph theoretic analysis demonstrates upper bounds on the 2-distance chromatic number in triangle-free planar graphs, refining theoretical limits for vertex coloring algorithms.
A [Formula: see text]-distance [Formula: see text]-coloring of a graph is a coloring of the vertices with [Formula: see text] colors in which any two vertices at distance at most [Formula: see text] receive distinct colors. The [Formula: see text]-distance chromatic number of [Formula: see text], denoted by [Formula: see text], is the minimum integer [Formula: see text] such that [Formula: see text] admits a [Formula: see text]-distance [Formula: see text]-coloring. The girth of a graph [Formula: see text], denoted by [Formula: see text], is the length of its shortest cycle. We show that if [Formula: see text] is a planar graph with [Formula: see text] and [Formula: see text], then [Formula: see text]. Moreover, we prove that if [Formula: see text] is a planar graph with [Formula: see text] and [Formula: see text], then [Formula: see text].
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Sara Al Hajjar (2026) studied this question.
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