Threecoloring trianglefree graphs on surfaces IV. Bounding face sizes of 4critical graphs
Abstract
Let G be a 4critical graph with t triangles, embedded in a surface of genus g. Let c be the number of 4cycles in G that do not bound a 2cell face. We prove that the sum of lengths of (>=5)faces of G is at most linear in g+t+c1.
 Publication:

arXiv eprints
 Pub Date:
 April 2014
 arXiv:
 arXiv:1404.6356
 Bibcode:
 2014arXiv1404.6356D
 Keywords:

 Mathematics  Combinatorics;
 05C15 (Primary);
 05C10 (Secondary);
 G.2.2
 EPrint:
 28 pages, 1 figure v2: Several changes necessitated by further papers in the series included. Fixed an error: excluding only separating noncontractible 4cycles is not enough. v3: Added a result on number of vertices needed to remove to make a graph 3colorable v4: Updated for reviewer comments