New Lower Bounds for the Shannon Capacity of Odd Cycles

TL;DR

This paper improves lower bounds on the Shannon capacity of odd cycles like C7 and C15 by using stochastic search methods to find larger independent sets in their graph powers, advancing understanding of a long-standing open problem.

Contribution

The authors introduce a novel approach with stabilizers and stochastic search to establish new lower bounds for the Shannon capacity of odd cycles.

Findings

Improved lower bounds for c(C7) and c(C15)

Largest known independent sets in specific graph powers

Advancement in methods for estimating Shannon capacity

Abstract

The Shannon capacity of a graph $G$ is defined as $c(G)=\sup_{d\geq 1}(\alpha(G^d))^{\frac{1}{d}},$ where $\alpha(G)$ is the independence number of $G$. The Shannon capacity of the cycle $C_5$ on $5$ vertices was determined by Lov\'{a}sz in 1979, but the Shannon capacity of a cycle $C_p$ for general odd $p$ remains one of the most notorious open problems in information theory. By prescribing stabilizers for the independent sets in $C_p^d$ and using stochastic search methods, we show that $\alpha(C_7^5)\geq 350$, $\alpha(C_{11}^4)\geq 748$, $\alpha(C_{13}^4)\geq 1534$ and $\alpha(C_{15}^3)\geq 381$. This leads to improved lower bounds on the Shannon capacity of $C_7$ and $C_{15}$: $c(C_7)\geq 350^{\frac{1}{5}}> 3.2271$ and $c(C_{15})\geq 381^{\frac{1}{3}}> 7.2495$.