On the Long-Repetition-Free 2-Colorability of Trees

TL;DR

This paper investigates the conditions under which trees can be colored with two colors to avoid long repeated patterns in paths, establishing that trees with radius up to seven are colorable, while some larger trees are not.

Contribution

It proves that all rooted trees with radius ≤7 are long-repetition-free two-colorable, and identifies classes of trees that are not, advancing understanding of pattern avoidance in graph colorings.

Findings

Trees with radius ≤7 are long-repetition-free two-colorable.

Some trees cannot be colored to avoid long repetitions.

The study links word pattern avoidance to graph coloring constraints.

Abstract

A word $\bar{w} = \bar{u}\bar{u}$ is a $long$ $square$ if $\bar{u}$ is of length at least 3; a word $\bar{w}$ is $long$-$square$-$free$ if $\bar{w}$ contains no sub-word that is a long square. We can use words to generate graph colorings; a graph coloring is called $long$-$repetition$-$free$ if the word formed by the coloring of each path in the graph is long-square-free. Our results show that every rooted tree of radius less than or equal to seven is long-repetition-free two-colorable. We also prove there exists a class of trees which are not long-repetition-free two-colorable.