Choosability and paintability of the lexicographic product of graphs

This paper studies the choice number and paint number of the lexicographic product of graphs. We prove that if G has maximum degree Δ, then for any graph H on n vertices ch(G[H])≤(4Δ+2)(ch(H)+log2n) and χP(G[H])≤(4Δ+2)(χP(H)+log2n).