On a Planarity Criterion Coming from Knot Theory
Abstract
A graph G is called "minimalizable" if a diagram with minimal crossing number can be obtained from an arbitrary diagram of G by crossing changes. If, furthermore, the minimal diagram is unique up to crossing changes then G is called "strongly minimalizable". In this article, it is explained how minimalizability of a graph is related to its automorphism group and it is shown that a graph is strongly minimalizable if the automorphism group is trivial or isomorphic to a product of symmetric groups. Then, the treatment of crossing number problems in graph theory by knot theoretical means is discussed and, as an example, a planarity criterion for minimalizable graphs is given.
Cite
@article{arxiv.math/0001140,
title = {On a Planarity Criterion Coming from Knot Theory},
author = {J. Sawollek},
journal= {arXiv preprint arXiv:math/0001140},
year = {2007}
}
Comments
14 pages, 9 figures, latex2e, metafont; Theorems 1 and 7 modified, example and lemma 11 added; definition clarified and minor corrections