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CHROMATIC NUMBERS OF QUADRANGULATIONS ON CLOSED SURFACES
 

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 non-spherical orientable
closed surface with a suĂciently 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 open-2-cell embedding of a simple

  

Source: Archdeacon, Dan - Department of Mathematics and Statistics, University of Vermont

 

Collections: Mathematics