Upper bounds of [formula omitted]-hued colorings of planar graphs.

For positive integers k and r , a ( k , r )-coloring of a graph G is a proper k -coloring of the vertices such that every vertex of degree d is adjacent to vertices with at least min { d , r } different colors. The r -hued chromatic number of a graph G , denoted by χ r ( G ) , is the smallest integer k such that G has a ( k , r ) -coloring. In Song et al. (2014), it is conjectured that if r ≥ 8 , then every planar graph G satisfies χ r ( G ) ≤ ⌈ 3 r 2 ⌉ + 1 . Wegner in 1977 conjectured that the above-mentioned conjecture holds when r = Δ ( G ) . This conjecture, if valid, would be best possible in some sense. In this paper, we prove that, if G is a planar graph and r ≥ 8 , then χ r ( G ) ≤ 2 r + 16 . [ABSTRACT FROM AUTHOR]