Homological obstructions for regular embeddings of graphs

TL;DR

This paper develops homological obstructions for the existence of k-regular embeddings of graphs, linking graph embeddings to algebraic topology of independence complexes and matroids.

Contribution

It concretely formulates hypergraph obstructions for k-regular graph embeddings using embedded homology and homological algebra tools.

Findings

Homological conditions for k-regular embeddings of graphs.

Existence of induced homomorphisms between embedded homology and matroid homology.

Construction of commutative diagrams involving Mayer-Vietoris sequences and Kunneth sequences.

Abstract

In [36, Section 8], the present author proposed the hypergraph obstruction for the existence of k-regular embeddings. In this paper, we develop the hypergraph obstruction concretely and give some homological obstructions for the k-regular embeddings of graphs by using the embedded homology of sub-hypergraphs of the (k-1)-skeleton of the independence complexes. Regular embeddings of graphs can be regarded equivalently as geometric realizations of the independence complexes and consequently be regarded equivalently as simplicial embeddings of the independence complexes into the vectorial matroids. We prove that if there exists a k-regular embedding of a graph, then there is an induced homomorphism from the embedded homology of the sub-hyper(di)graphs of the (k-1)-skeleton of the (directed) independence complexes to the homology of (directed) matroids. Moreover, if there exists certain triple of graphs where each graph has a k-regular embedding, then there are induced commutative diagrams of certain Mayer-Vietoris sequences of the embedded homology of hyper(di)graphs, the homology of (directed) independence complexes and the homology of matroids. Furthermore, if there exists certain couple of graphs where each graph has a k-regular embedding, then there are induced commutative diagrams of certain Kunneth type short exact sequences of the embedded homology of hyper(di)graphs, the homology of (directed) independence complexes and the homology of matroids.