Chromatic Numbers of Quadrangulations on Closed Surfaces

Abstract

It has been shown that every quadrangulation on any non-spherical orientable closed surface with a sufficiently large representativity has chromatic number at most 3. In this paper, we show that a quadrangulation G on a nonorientable closed surface N k has chromatic number at least 4 if G has a cycle of odd length which cuts open N k into an orientable surface. Moreover, we characterize the quadrangulations on the torus and the Klein bottle with chromatic number exactly 3. By our characterization, we prove that every quadrangulation on the torus with representativity at least 9 has chromatic number at most 3, and that a quadrangulation on the Klein bottle with representativity at least 7 has chromatic number at most 3 if a cycle cutting open the Klein bottle into an annulus has even length. As an application of our theory, we prove that every nonorientable closed surface N k admits an eulerian triangulation with chromatic number at least 5 which has arbitrarily large representativity