# Hamilton cycles in infinite cubic graphs

## TL;DR

This paper proves that one-ended Hamiltonian cubic graphs with end degree 3 always have a second Hamilton cycle, but does not extend to three cycles or two-ended graphs, highlighting specific structural properties.

## Contribution

It establishes the existence of a second Hamilton cycle in one-ended Hamiltonian cubic graphs with end degree 3, addressing a problem posed by B. Mohar.

## Key findings

- Every one-ended Hamiltonian cubic graph with end degree 3 contains a second Hamilton cycle.
- The result does not hold for the existence of a third Hamilton cycle.
- The result does not extend to two-ended graphs.

## Abstract

Investigating a problem of B. Mohar, we show that every one-ended Hamiltonian cubic graph with end degree 3 contains a second Hamilton cycle. We also construct two examples showing that this result does not extend to give a third Hamilton cycle, nor that it extends to the two-ended case.