Homological obstructions for regular embeddings of graphs
Shiquan Ren
TL;DR
The paper develops a homological obstruction framework for $k$-regular embeddings of graphs by leveraging embedded homology of sub-hypergraphs in the $(k-1)$-skeleton of the independence complex ${\rm Ind}(G)$. It proves that a $k$-regular embedding induces a map from embedded-homology of these sub-hypergraphs to the homology of (directed) matroids, and, in suitable triples or pairs of graphs, yields commutative Mayer–Vietoris and Kunneth-type diagrams connecting embedded hypergraph homology, independence complexes, and matroid homology. These obstructions are expressed through explicit algebraic-topological constructs (MV sequences and KU-type short exact sequences), providing concrete criteria to rule out regular embeddings and linking configuration-space obstructions with discrete graph embeddings. The results form a comprehensive, diagrammatic toolkit that extends to general metric spaces via double complexes and offer a path toward computable obstructions in topological combinatorics and configuration-space theory.
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.