Nonrepetitive colorings of lexicographic product of graphs

A coloring c of the vertices of a graph G is nonrepetitive if there exists no path v1v2 . . . v2l for which c(vi) = c(vl+i) for all 1 ≤ i ≤ 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 Kk,k. We prove that for a sufficiently long path P , a nonrepetitive coloring of P [Kk] needs at least 3k + ⌊k/2⌋ colors. If k > 2 then we need exactly 2k + 1 colors to nonrepetitively color P [Ek], where Ek is the empty graph on k vertices. If we further require that every copy of Ek be rainbow-colored and the path P is sufficiently long, then the smallest number of colors needed for P [Ek] is at least 3k + 1 and at most 3k + ⌈k/2⌉. Finally, we define fractional nonrepetitive colorings of graphs and consider the connections between this notion and the above results.