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

Abstract

A sequence $$a_1a_2\ldots a_p$$a1a2?ap is an r-repetition (for a real number $$r >1 $$r>1) if $$p=\lceil rq \rceil $$p=?rq? for some positive integer q, and $$a_j=a_{j+q}$$aj=aj+q for $$j=1,2,\ldots , p-q$$j=1,2,?,p-q. In other words, the sequence can be divided into $$\lceil r \rceil $$?r? blocks where all the blocks are the same, say, all the blocks equal to $$a_1a_2\ldots a_q$$a1a2?aq for some $$q \ge 1$$q?1, except that when r is not an integer, the last block is the prefix of $$a_1...a_q$$a1...aq of length $$ \lceil (r - \lfloor r \rfloor )q \rceil $$?(r-?r?)q?. 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 $$\pi _r(G)$$?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 $$\pi _{rch}(G)$$?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 $$c(v)\in L(v)$$c(v)?L(v) for every vertex v of G. A classical result of Thue asserts that $$\pi _2(P_n)\le 3$$?2(Pn)≤3 for all n. It is known that $$ \pi _{2ch}(P_n) \le 4$$?2ch(Pn)≤4 for all n. However, it remains an open problem whether $$\pi _{2ch}(P_n) \le 3$$?2ch(Pn)≤3 for all n. This paper proves that for any $$\epsilon > 0$$∈>0, $$\pi _{(2+\epsilon )ch}(P_n) \le 3$$?(2+∈)ch(Pn)≤3 for all n.