Cyclic coloration of 3‐polytopes

Abstract

Abstract A cyclic coloration of a planar graph G is an assignment of colors to the points of G such that for any face bounding cycle the points of F have different colors. We observe that the upper bound 2ρ*( G ), due to O. Ore and M. D. Plummer, can be improved to ρ*( G ) + 9 when G is 3‐connected (ρ* denotes the size of a maximum face). The proof uses two principal tools: the theory of Euler contributions and recent results on contractible lines in 3‐connected graphs by K. Ando, H. Enomoto and A. Saito.