$\ell ^{2}$-homology and planar graphs
Volume 131 / 2013
Abstract
In his 1930 paper, Kuratowski proves that a finite graph $\varGamma $ is planar if and only if it does not contain a subgraph that is homeomorphic to $K_5$, the complete graph on five vertices, or $K_{3,3}$, the complete bipartite graph on six vertices. This result is also attributed to Pontryagin. In this paper we present an $\ell ^2$-homological method for detecting non-planar graphs. More specifically, we view a graph $\varGamma $ as the nerve of a related Coxeter system and construct the associated Davis complex, $\varSigma _\varGamma $. We then use a result of the author regarding the (reduced) $\ell ^2$-homology of Coxeter groups to prove that if $\varGamma $ is planar, then the orbihedral Euler characteristic of $\varSigma _\varGamma /W_\varGamma $ is non-positive. This method not only implies as subcases the classical inequalities relating the number of vertices $V$ and edges $E$ of a planar graph (that is, $E\leq 3V-6$ or $E\leq 2V-4$ for triangle-free graphs), but it is stronger in that it detects non-planar graphs in instances the classical inequalities do not.
