Upper bounds of r-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 χr(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 χr(G)≤⌈3r2⌉+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 χr(G)≤2r+16.