The Shannon capacity of a graph and the independence numbers of its powers

TL;DR

This paper investigates the Shannon capacity of graphs through the lens of independence numbers of their strong powers, revealing complex behaviors that challenge approximation methods.

Contribution

It demonstrates that the series of independence numbers can be highly complex, making the Shannon capacity difficult to approximate with any fixed initial segment.

Findings

Independence numbers in strong powers can exhibit complex, non-monotonic behavior.

The Shannon capacity cannot be approximated by any fixed prefix of the independence number series.

Even significant increases in independence numbers may be followed by stabilization, complicating analysis.

Abstract

The independence numbers of powers of graphs have been long studied, under several definitions of graph products, and in particular, under the strong graph product. We show that the series of independence numbers in strong powers of a fixed graph can exhibit a complex structure, implying that the Shannon Capacity of a graph cannot be approximated (up to a sub-polynomial factor of the number of vertices) by any arbitrarily large, yet fixed, prefix of the series. This is true even if this prefix shows a significant increase of the independence number at a given power, after which it stabilizes for a while.