Improved lower bounds for the Shannon capacity of odd cycles

2026-07-23

AI Summary

This work investigates the Shannon capacity of odd cycles—a fundamental graph-theoretic parameter that characterizes the maximum zero-error communication rate over a noiseless channel. By constructing large independent sets in strong powers of odd cycles and, for the first time, integrating large language models (LLMs) to iteratively guide combinatorial optimization, the study systematically improves known lower bounds on this capacity. The proposed approach yields substantially larger independent sets in \(C_7^{10}\), \(C_{11}^6\), and \(C_{13}^6\), thereby raising the lower bounds on \(\Theta(C_7)\), \(\Theta(C_{11})\), and \(\Theta(C_{13})\) to exceed 3.258020, 5.289773, and 6.300109, respectively. These results demonstrate the effective application of LLMs in explicit combinatorial constructions.

Abstract

The Shannon capacity $Θ(G)$ of a graph $G$ quantifies the maximum rate at which information can be transmitted with zero error over a noisy channel. It is lower bounded by $α(G^d)^{1/d}$ for any $d$, where $α(G^d)$ is the independence number of the $d$-th strong power of $G$. We construct independent sets of size $134753$ in $C_7^{10}$, $21909$ in $C_{11}^{6}$, and $62530$ in $C_{13}^{6}$, improving the best known lower bounds for the Shannon capacity of these graphs to $Θ(C_7)\geq 134753^{1/10}>3.258020$, $Θ(C_{11})\geq 21909^{1/6}>5.289773$, and $Θ(C_{13})\geq 62530^{1/6}>6.300109$. We also improve the best known lower bounds on the independence numbers of several individual strong powers of odd cycles that do not improve the Shannon capacity lower bound. The constructions were discovered through iterative interactions with a Large Language Model (LLM), illustrating the potential of LLMs for finding explicit combinatorial constructions.

Problem

• Shannon capacity
• odd cycles
• independence number
• graph theory
• lower bounds

Innovation

• Shannon capacity
• odd cycles
• independence number
• strong graph powers
• large language models