$(2+\epsilon )$-Nonrepetitive List Colouring of Paths

Abstract

A sequence $a_1a_2\ldots a_p$ is an r-repetition (for a real number $r >1$) if $p=\lceil rq \rceil$ for some positive integer q, and $a_j=a_{j+q}$ for $j=1,2,\ldots , p-q$. In other words, the sequence can be divided into $\lceil r \rceil$ blocks where all the blocks are the same, say, all the blocks equal to $a_1a_2\ldots a_q$ for some $q \ge 1$, except that when r is not an integer, the last block is the prefix of $a_1...a_q$ of length $\lceil (r - \lfloor r \rfloor )q \rceil$. A colouring of the vertices of a graph G is r-nonrepetitive if there is no path in G for which the colour sequence of its vertices forms an r-repetition. The r-nonrepetitive chromatic number $\pi _r(G)$ of G is the minimum number of colours needed in an r-nonrepetitive colouring of G. A k-list assignment of a graph G is a mapping L which assigns a set L(v) of k permissible colours to each vertex v of G. The r-nonrepetitive choice number $\pi _{rch}(G)$ of G is the least integer k such that for every k-list assignment L, there is an r-nonrepetitive colouring c of G satisfying $c(v)\in L(v)$ for every vertex v of G. A classical result of Thue asserts that $\pi _2(P_n)\le 3$ for all n. It is known that $\pi _{2ch}(P_n) \le 4$ for all n. However, it remains an open problem whether $\pi _{2ch}(P_n) \le 3$ for all n. This paper proves that for any $\epsilon > 0$, $\pi _{(2+\epsilon )ch}(P_n) \le 3$ for all n.