Faithful subgraphs and Hamiltonian circles of infinite graphs

TL;DR

This paper explores methods to identify Hamiltonian circles in infinite graphs, extending finite graph results to infinite cases, with a focus on locally finite graphs and their properties.

Contribution

It introduces a method for finding Hamiltonian circles in infinite graphs and extends finite graph results, such as the Hamiltonicity of prisms of 3-connected cubic graphs, to infinite graphs.

Findings

Prism of every 3-connected cubic graph has a Hamiltonian circle

Extended finite graph Hamiltonian results to infinite graphs

Developed a method for identifying Hamiltonian circles in infinite graphs

Abstract

A circle of an infinite locally finite graph $G$ is the imagine of a homeomorphic mapping of the unit circle $S^1$ in $|G|$, the Freudenthal compactification of $G$. A circle of $G$ is Hamiltonian if it meets every vertex (and then every end) of $G$. In this paper, we study a method for finding Hamiltonian circles of graphs. We illustrate this by extending several results on finite graphs to Hamiltonian circles in infinite graphs. For example, we prove that the prism of every 3-connected cubic graph has a Hamiltonian circle, extending the result of the finite case by Paulraja.