# A sufficient local degree condition for Hamiltonicity in locally finite claw-free graphs

## TL;DR

This paper proves a conjecture by Diestel that a known degree condition for Hamiltonicity in finite graphs also guarantees a Hamiltonian circle in the Freudenthal compactification of locally finite claw-free graphs.

## Contribution

It extends a weak degree condition for Hamiltonicity from finite to infinite claw-free graphs, confirming Diestel's conjecture.

## Key findings

- The degree condition guarantees a Hamiltonian circle in the compactification.
- The result applies specifically to claw-free graphs.
- It confirms the conjecture for an important class of infinite graphs.

## Abstract

Among the well-known sufficient degree conditions for the Hamiltonicity of a finite graph, the condition of Asratian and Khachatrian is the weakest and thus gives the strongest result. Diestel conjectured that it should extend to locally finite infinite graphs~$G$, in that the same condition implies that the Freudenthal compactification of $G$ contains a circle through all its vertices and ends. We prove Diestel's conjecture for claw-free graphs.