On the Shannon capacity of sums and products of graphs

TL;DR

This paper provides an elementary proof that the Shannon capacity of graph sums and products exceeds the sum and product of individual capacities, clarifying fundamental inequalities in graph theory.

Contribution

It offers a simple, elementary proof of known inequalities relating Shannon capacities of graph sums and products, previously shown using advanced duality and choice axioms.

Findings

Established equivalence of inequalities for graph sums and products

Provided elementary proof avoiding advanced duality methods

Clarified fundamental properties of Shannon capacity in graph theory

Abstract

Let $\Theta(G)$ denote the Shannon capacity of a graph $G$. We give an elementary proof of the equivalence, for any graphs $G$ and $H$, of the inequalities $\Theta(G\sqcup H)>\Theta(G)+\Theta(H)$ and $\Theta(G\boxtimes H)>\Theta(G)\Theta(H)$. This was shown independently by Wigderson and Zuiddam [2022] using Kadison-Dubois duality and the Axiom of choice.