On the chromatic number of the lexicographic product and the Cartesian sum of graphs
نویسندگان
چکیده
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)+d k e, 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.
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عنوان ژورنال:
- Discrete Mathematics
دوره 134 شماره
صفحات -
تاریخ انتشار 1994