A Homology Theory of Graphs: First Homology Group of Hamiltonian Graphs

TL;DR

This paper applies a geometric homology approach to Hamiltonian graphs, demonstrating that their first homology group is torsion-free, extending previous results on cycle graphs.

Contribution

It introduces a geometric method to analyze the first homology group of Hamiltonian graphs, showing they are torsion-free, which was not previously established.

Findings

First homology group of Hamiltonian graphs is torsion-free.

Method extends previous homology analysis from cycle graphs to Hamiltonian graphs.

Provides new insights into the topological structure of Hamiltonian graphs.

Abstract

An integral homology theory on the category of undirected reflexive graphs was constructed in [2]. A geometrical method to understand behaviors of $1$- and $2$-simplices under differential maps of the theory was developed in [3] and led us to an independent proof that the first homology group of any cycle graphs is $\mathbb{Z}$, as it was proved before by a version of Hurewicz theorem harshly defined and shown in [1] and [2]. In this work, we use the old method in [3] to study behaviors of the first homology group of Hamiltonian graphs. We discovered that $H_1(G)$ is torsion-free, for any Hamiltonian graphs $G$.