Nonrepetitive colorings of lexicographic product of graphs

TL;DR

This paper investigates nonrepetitive vertex colorings of lexicographic product graphs, establishing bounds on the number of colors needed for paths and introducing fractional coloring concepts.

Contribution

It provides new bounds for nonrepetitive colorings of lexicographic product graphs and introduces fractional nonrepetitive colorings, expanding understanding of coloring complexities.

Findings

At least 3k + floor(k/2) colors are needed for nonrepetitive coloring of P[K_k].

Exactly 2k + 1 colors suffice for P[E_k] when k > 2.

Fractional nonrepetitive colorings are connected to these bounds.

Abstract

A coloring $c$ of the vertices of a graph $G$ is nonrepetitive if there exists no path $v_1v_2\ldots v_{2l}$ for which $c(v_i)=c(v_{l+i})$ for all $1\le i\le l$. Given graphs $G$ and $H$ with $|V(H)|=k$, the lexicographic product $G[H]$ is the graph obtained by substituting every vertex of $G$ by a copy of $H$, and every edge of $G$ by a copy of $K_{k,k}$. %Our main results are the following. We prove that for a sufficiently long path $P$, a nonrepetitive coloring of $P[K_k]$ needs at least $3k+\lfloor k/2\rfloor$ colors. If $k>2$ then we need exactly $2k+1$ colors to nonrepetitively color $P[E_k]$, where $E_k$ is the empty graph on $k$ vertices. If we further require that every copy of $E_k$ be rainbow-colored and the path $P$ is sufficiently long, then the smallest number of colors needed for $P[E_k]$ is at least $3k+1$ and at most $3k+\lceil k/2\rceil$. Finally, we define fractional nonrepetitive colorings of graphs and consider the connections between this notion and the above results.