Shannon capacity and the categorical product
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Shannon OR-capacity C_ OR(G) of a graph G, that is the traditionally more often used Shannon AND-capacity of the complementary graph, is a homomorphism monotone graph parameter satisfying C_ OR(F× G)<min{C_ OR(F),C_ OR(G)} for every pair of graphs, where F× G is the categorical product of graphs F and G. Here we initiate the study of the question when could we expect equality in this inequality. Using a strong recent result of Zuiddam, we show that if this "Hedetniemi-type" equality is not satisfied for some pair of graphs then the analogous equality is also not satisfied for this graph pair by some other graph invariant that has a much "nicer" behavior concerning some different graph operations. In particular, unlike Shannon capacity or the chromatic number, this other invariant is both multiplicative under the OR-product and additive under the join operation, while it is also nondecreasing along graph homomorphisms. We also present a natural lower bound on C_ OR(F× G) and elaborate on the question of how to find graph pairs for which it is known to be strictly less, than the upper bound min{C_ OR(F),C_ OR(G)}. We present such graph pairs using the properties of Paley graphs.
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