Fractional Thue chromatic number of graphs

Abstract

This paper introduces the concept of the fractional Thue chromatic number of graphs and studies the relation between the fractional Thue chromatic number and the Thue chromatic number. We determine the fractional Thue chromatic number of all paths, all trees with no vertices of degree two, and all cycles, except $C_{10}$, $C_{14}$, $C_{17}$. As a consequence, we prove that if $G$ is a path or a tree with no degree two vertices, then its fractional Thue chromatic number equals its Thue chromatic number. On the other hand, we show that there are trees and cycles whose fractional Thue chromatic numbers are strictly less than their Thue chromatic numbers.