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ℓ²-homology and planar graphs

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Schroeder T.

Colloquium Mathematicum

2013

tom 131

nr 1

129-139

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.

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1 Colloquium Mathematicum

1 Schroeder T.

1 2013

Wyniki wyszukiwania

ℓ²-homology and planar graphs