Nonrepetitive colorings of lexicographic product of graphs
Authors: Balázs Keszegh ^{1}; Balázs Patkós ^{2}; Xuding Zhu ^{3}
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Balázs Keszegh;Balázs Patkós;Xuding Zhu
1 Alfréd Rényi Institute of Mathematics
2 Geometric and Algebraic Combinatorics Research Group
3 Department of Mathematics [Jinhua]
A coloring c of the vertices of a graph G is nonrepetitive if there exists no path v1v2\textellipsisv2l 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.