 
Summary: CHROMATIC NUMBERS OF QUADRANGULATIONS
ON CLOSED SURFACES
Dan Archdeacon , Joan Hutchinson y , Atsuhiro Nakamoto z ,
Seiya Negami x and Katsuhiro Ota {
Abstract
It has been shown that every quadrangulation on any nonspherical orientable
closed surface with a suÆciently 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 quadrangu
lations on the torus and the Klein bottle with chromatic number exactly 3. By our
characterization, we prove that every quadrangulation on the torus with represen
tativity 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 appli
cation 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.
1 Introduction
A quadrangulation G on a closed surface F 2 is a xed open2cell embedding of a simple
