Shannon Capacity and the Categorical Product

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