Shannon capacity, Chess, DNA and Umbrellas

TL;DR

This paper explores the challenging problem of determining Shannon capacity for certain graphs using chess puzzles and quantum entanglement concepts, highlighting the role of pure states and posing open questions.

Contribution

It introduces a novel approach linking chess puzzles and quantum states to bounds on Shannon capacity of cyclic graphs, and discusses the nature of optimal quantum states.

Findings

Lower bounds derived from King chess puzzles

Upper bounds obtained via Lovasz umbrellas and quantum states

Optimal states are always pure states

Abstract

A vexing open problem in information theory is to find the Shannon capacity of odd cyclic graphs larger than the pentagon and especially for the heptagon. Lower bounds for the capacity are obtained by solving King chess puzzles. Upper bounds are obtained by solving entanglement problems, that is to find good Lovasz umbrellas, quantum state realizations of the graph. We observe that optimal states are always pure states. The rest is expository. One general interesting question is whether the Shannon capacity is always some n-th root of the independence number of the n'th power of the graph.