Shannon Capacity and the Categorical Product

Gábor Simonyi

doi:10.37236/9113

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 therefore $C_{\rm OR}(F\times G)\leqslant\min\{C_{\rm OR}(F),C_{\rm OR}(G)\}$ holds 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 OR-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.

Highlights

The real content of the conjecture was that the right hand side is a lower bound on χ(F × G)
The Shannon capacity of a graph involves a graph product which is different of the categorical product that appears in Hedetniemi’s conjecture
If the above mentioned lower bound on the chromatic number (that we will denote by top(G) ) was multiplicative under the OR-product, it would imply by the McEliece-Posner theorem, that χf (G) = limt→∞ t χ(Gt) limt→∞ t top(Gt) = limt→∞ t top(G)t = top(G) holds for any graph G

Summary

Introduction

For two graphs F and G, their categorical ( called tensor or weak direct) product F × G is defined by. (For the definition and basic properties the electronic journal of combinatorics 28(1) (2021), #P1.51 of this parameter see Lovasz’s original paper [33] or his recent book [35].) In [19] the authors investigated the analogous question for a closely related parameter called vector chromatic number (which differs from the strict vector chromatic number) that was introduced in [29] They conjectured that the Hedetniemi-type equality holds for this parameter and proved it in special cases. In a follow-up paper by Godsil, Roberson, Roomey, Samal, and Varvitsiotis [20] the latter conjecture is proved in general as well Both the fractional chromatic number and the Lovasz theta number of the complementary graph are well-known upper bounds on the Shannon OR-capacity of the graph which is the usual Shannon capacity, or Shannon AND-capacity of the complementary graph. We will present some graph pairs with this property in Subsection 4.3

Shannon OR-capacity

On the possibilities of equality

A lower bound and identifying test cases

Lower bound

Trivial pairs

Nontrivial pairs