Forcing Hamiltonicity in locally finite graphs via forbidden induced subgraphs I: nets and bulls

TL;DR

This paper investigates conditions that guarantee Hamiltonian cycles in locally finite infinite graphs by analyzing forbidden induced subgraphs like claws, nets, and bulls, extending finite graph results to infinite cases.

Contribution

It extends Hamiltonicity conditions involving claws, nets, and bulls from finite to locally finite infinite graphs, including classifications and relaxed conditions.

Findings

Extended Hamiltonicity results to locally finite graphs.

Classified locally finite claw-free and net-free graphs.

Generalized bull-free Hamiltonicity conditions.

Abstract

In a series of papers, of which this is the first, we study sufficient conditions for Hamiltonicity in terms of forbidden induced subgraphs and extend such results to locally finite infinite graphs. For this we use topological circles within the Freudenthal compactification of a locally finite graph as infinite cycles. In this paper we focus on conditions involving claws, nets and bulls as induced subgraphs. We extend Hamiltonicity results for finite claw-free and net-free graphs by Shepherd to locally finite graphs. Moreover, we generalise a classification of finite claw-free and net-free graphs by Shepherd to locally finite ones. Finally, we extend to locally finite graphs a Hamiltonicity result by Ryj\'{a}\v{c}ek involving a relaxed condition of being bull-free.