Author:
Campion Loth Jesse,Mohar Bojan
Abstract
Abstract
A random two-cell embedding of a given graph
$G$
is obtained by choosing a random local rotation around every vertex. We analyse the expected number of faces of such an embedding, which is equivalent to studying its average genus. In 1991, Stahl [5] proved that the expected number of faces in a random embedding of an arbitrary graph of order
$n$
is at most
$n\log (n)$
. While there are many families of graphs whose expected number of faces is
$\Theta (n)$
, none are known where the expected number would be super-linear. This led the authors of [1] to conjecture that there is a linear upper bound. In this note we confirm their conjecture by proving that for any
$n$
-vertex multigraph, the expected number of faces in a random two-cell embedding is at most
$2n\log (2\mu )$
, where
$\mu$
is the maximum edge-multiplicity. This bound is best possible up to a constant factor.
Publisher
Cambridge University Press (CUP)
Subject
Applied Mathematics,Computational Theory and Mathematics,Statistics and Probability,Theoretical Computer Science