# Non-Hamiltonian 3-Regular Graphs with Arbitrary Girth

## TL;DR

The paper presents constructions of non-Hamiltonian 3-regular graphs with arbitrarily large girth, expanding understanding of their existence and properties.

## Contribution

It introduces three methods to construct non-Hamiltonian 3-regular graphs with large girth without reducing girth, and provides bounds and examples for their sizes.

## Key findings

- Existence of non-Hamiltonian 3-regular graphs with arbitrary girth
- Construction methods for such graphs with different connectivity levels
- Bounds on the size of smallest such graphs for various girth values

## Abstract

It is well known that 3--regular graphs with arbitrarily large girth exist. Three constructions are given that use the former to produce non-Hamiltonian 3--regular graphs without reducing the girth, thereby proving that such graphs with arbitrarily large girth also exist. The resulting graphs can be 1--, 2-- or 3--edge-connected depending on the construction chosen. From the constructions arise (naive) upper bounds on the size of the smallest non-Hamiltonian 3--regular graphs with particular girth. Several examples are given of the smallest such graphs for various choices of girth and connectedness.