Topologically $4$-chromatic graphs and signatures of odd cycles
Combinatorics
2016-02-25 v2
Abstract
We investigate group-theoretic "signatures" of odd cycles of a graph, and their connections to topological obstructions to 3-colourability. In the case of signatures derived from free groups, we prove that the existence of an odd cycle with trivial signature is equivalent to having the coindex of the hom-complex at least 2 (which implies that the chromatic number is at least 4). In the case of signatures derived from elementary abelian 2-groups we prove that the existence of an odd cycle with trivial signature is a sufficient condition for having the index of the hom-complex at least 2 (which again implies that the chromatic number is at least 4).
Keywords
Cite
@article{arxiv.1601.07856,
title = {Topologically $4$-chromatic graphs and signatures of odd cycles},
author = {Gord Simons and Claude Tardif and David Wehlau},
journal= {arXiv preprint arXiv:1601.07856},
year = {2016}
}
Comments
Fixed an incorrect assertion in the abstract of the previous version