This research demonstrates that the r-hued chromatic number is accurately defined for maximal planar graphs with diameter two, confirming a prior conjecture.
Let [Formula: see text] be an integer. The [Formula: see text]-hued chromatic number [Formula: see text] of a graph [Formula: see text] is the minimum [Formula: see text] such that [Formula: see text] admits a proper [Formula: see text]-coloring where each vertex [Formula: see text] has at least [Formula: see text] distinct colors in its neighborhood. In this paper, we prove that if [Formula: see text] is a maximal planar graphs with diameter two, then [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text] if [Formula: see text]. These upper bounds are sharp. The result confirms the r-hued coloring conjecture proposed by Song at al. in [Discrete Math. 315 (2014) 47-52] for maximal planar graphs with diameter two.
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Duan et al. (2026) studied this question.
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