Betti numbers of subgraphs

TL;DR

This paper establishes a novel algebraic criterion using Betti numbers from algebraic topology to determine subgraph containment in simple graphs, focusing on complete and bipartite subgraphs.

Contribution

It introduces Betti numbers as invariants that characterize subgraph inclusion, providing a new algebraic perspective on graph substructure detection.

Findings

Betti numbers can detect the presence of complete and bipartite subgraphs.

A simplified criterion involving only the first syzygy module suffices.

Characterization of subgraphs includes specific multipartite graphs.

Abstract

Let $G$ be a simple graph on $n$ vertices. Let $H$ be either the complete graph $K_m$ or the complete bipartite graph $K_{r,s}$ on a subset of the vertices in $G$. We show that $G$ contains $H$ as a subgraph if and only if $\beta_{i,\alpha}(H) \le \beta_{i,\alpha}(G)$ for all $i \ge 0$ and $\alpha \in \mathbb{Z}^n$. In fact, it suffices to consider only the first syzygy module. In particular, we prove that $\beta_{1,\alpha}(H) \le \beta_{1,\alpha}(G)$ for all $\alpha \in \mathbb{Z}^n$ if and only if $G$ contains a subgraph that is isomorphic to either $H$ or a multipartite graph $K_{2,\dots,2,a,b}$.