Estimating the Minimal Number of Colors in Acyclic $k$-Strong Colorings of Maps on Surfaces

Abstract

A coloring of the vertices of a graph is called acyclic if the ends of each edge are colored in distinct colors, and there are no two-colored cycles. Suppose each face of rank $k$, $k \geqslant 4$, in a map on a surface $S^N$ is replaced by the clique having the same number of vertices. It is proved in [1] that the resulting pseudograph admits an acyclic coloring with the number of colors depending linearly on N and $k$. In the present paper we prove a sharper estimate $55( - Nk)^{4/7}$ for the number of colors provided that $k \geqslant 1$ and $- N \geqslant 5^7 k^{4/3}$.