Shannon capacity and the categorical product

TL;DR

This paper investigates when the Shannon OR-capacity of the categorical product of two graphs equals the minimum of their individual capacities, revealing connections to other graph invariants and providing bounds and examples using Paley graphs.

Contribution

It initiates the study of equality conditions in Shannon OR-capacity for graph products and links this to other graph invariants with nicer algebraic properties.

Findings

Equality in Shannon OR-capacity for graph products is characterized via other invariants.

A natural lower bound on the OR-capacity of graph products is provided.

Examples of graph pairs with strict inequality are constructed using Paley graphs.

Abstract

Shannon OR-capacity $C_{\rm OR}(G)$ 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 $C_{\rm OR}(F\times G)\le\min\{C_{\rm OR}(F),C_{\rm OR}(G)\}$ for every pair of graphs, where $F\times G$ 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 $C_{\rm OR}(F\times G)$ and elaborate on the question of how to find graph pairs for which it is known to be strictly less, than the upper bound $\min\{C_{\rm OR}(F),C_{\rm OR}(G)\}$. We present such graph pairs using the properties of Paley graphs.