ℓ²-homology and planar graphs

• Autorzy: Timothy A. Schroeder
• Języki publikacji: EN

Abstrakty

EN

In his 1930 paper, Kuratowski proves that a finite graph Γ is planar if and only if it does not contain a subgraph that is homeomorphic to K₅, 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 ℓ²-homological method for detecting non-planar graphs. More specifically, we view a graph Γ as the nerve of a related Coxeter system and construct the associated Davis complex, $Σ_Γ$. We then use a result of the author regarding the (reduced) ℓ²-homology of Coxeter groups to prove that if Γ is planar, then the orbihedral Euler characteristic of $Σ_Γ/W_Γ$ 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 ≤ 3V-6 or E ≤ 2V-4 for triangle-free graphs), but it is stronger in that it detects non-planar graphs in instances the classical inequalities do not.

Kategorie tematyczne

• 05C10: Planar graphs; geometric and topological aspects of graph theory
• 57M15: Relations with graph theory

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2013
• Tom: 131
• Numer: 1
• Strony: 129-139

Daty

wydano

2013

Twórcy

autor

Timothy A. Schroeder

• Department of Mathematics and Statistics, Murray State University, Murray, KY 42071, U.S.A.

Identyfikatory

DOI

10.4064/cm131-1-11