# Halin graphs are 3-vertex-colorable except even wheels

## TL;DR

This paper proves that all Halin graphs are 3-vertex-colorable except for even wheels and provides algorithms for coloring and chordal completion, with implementations and random graph generators.

## Contribution

It establishes the 3-colorability of Halin graphs excluding even wheels and offers algorithms for coloring and chordal completion with open-source code.

## Key findings

- Halin graphs are 3-colorable except even wheels.
- Algorithms for coloring and chordal completion are provided.
- Open-source Python implementations and random graph generators are available.

## Abstract

A Halin graph is a graph obtained by embedding a tree having no nodes of degree two in the plane, and then adding a cycle to join the leaves of the tree in such a way that the resulting graph is planar. According to the four color theorem, Halin graphs are 4-vertex-colorable. On the other hand, they are not 2-vertex-colorable because they have triangles. We show that all Halin graphs are 3-vertex-colorable except even wheels. We also show how to find the perfect elimination ordering of a chordal completion for a given Halin graph. The algorithms are implemented in Python using the graphtheory package. Generators of random Halin graphs (general or cubic) are included. The source code is available from the public GitHub repository.