On the chromatic number of the lexicographic product and the Cartesian sum of graphs Original Research Article

Let G[H] be the lexicographic product of graphs G and H and let G ⊕ H be their Cartesian sum. It is proved that if G is a nonbipartite graph, then for any graph H, χ(G(H])⩾2χ(H)+⌜ χ(H)/k ⌝, where 2k+1 is the length of a shortest odd cycle of G. Chromatic numbers of the Cartesian sum of graphs are also considered. It is shown in particular that for χ-critical and not complete graphs G and H, χ(G ⊕ H)⩽ χ(G)χ(H) − 1. These bounds are used to calculate chromatic numbers of the Cartesian sum of two odd cycles. Finally, a connection of some colorings with hypergraphs is given.