 
Summary: Two Maps on One Surface
Dan Archdeacon
Department of Mathematics and Statistics
University of Vermont
Burlington, VT, USA 05405
dan.archdeacon@uvm.edu
C. Paul Bonnington
Department of Mathematics
University of Auckland
Auckland, New Zealand
p.bonnington@auckland.ac.nz
Abstract
Two embeddings of a graph in a surface S are said to be \equiv
alent" if they are identical under an homeomorphism of S that is
orientationpreserving for orientable S. Two graphs cellularly embed
ded simultaneously in S are said to be \jointly embedded" if the only
points of intersection involve an edge of one graph transversally cross
ing an edge of the other. The problem is to nd equivalent embeddings
of the two graphs that minimize the number of these edgecrossings;
this minimum we call the \joint crossing number" of the two graphs.
