Improved bounds of r-hued colorings of planar graphs

For integers k , r > 0 , a ( k , r ) -coloring of a graph G is a proper k -coloring such that for any vertex v , there are at least min { r , d G ( v ) } different colors in its neighbors. Such a coloring is also called an r -hued coloring. The r -hued chromatic number χ r ( G ) of graph G is the least k such that there exists a ( k , r ) -coloring of G . In this paper, we show that χ r ( G ) ≤ 2 r + 7 for any integer r ≥ 9 and any planar graph G . This improves a result of Hu, Kong, Wang and Yang (a note on the r -hued coloring of planar graphs, Discrete Math. 349 (2026) 114829) saying that χ r ( G ) ≤ 2 r + 8 if G is a planar graph with r ≥ 8 . This extends a result of Bousquet, Deschamps, de Meyer and Pierron (Improved square coloring of planar graphs, Discrete Math. 346 (2023) 113288.) saying χ ( G 2 ) ≤ 2 Δ + 7 for every planar graph G with Δ ≥ 9 .