Cyclic coloring of plane graphs

Let G be a plane graph, and let χ_k(G) be the minimum number of colors to color the vertices of G so that every two of them which lie in the boundary of the same face of the size at most k, receive different colors. In 1966, Ore and Plummer proved that χ_k(G)2k for any k3. It is also known that χ₃(G)4 (Appel and Haken, 1976) and χ₄(G)6 (Borodin, 1984). The result in the present paper is: χ₅(G)9, χ₆(G)11, χ₇(G)12, and χ_k(G)2k − 3 if k8.