奇数圈香农容量的改进下界
Improved lower bounds for the Shannon capacity of odd cycles
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中文总结 AI 辅助
研究通过与大语言模型迭代交互，在奇数圈\(C_7^{10}\)、\(C_{11}^{6}\)、\(C_{13}^{6}\)中构造独立集，改进了这些图香农容量的已知下界，还改进了几个奇数圈单个强幂独立数的已知下界。
中文摘要 AI 辅助
图\(G\)的香农容量\(\Theta(G)\)量化了在有噪声信道上无差错传输信息的最大速率，它以\(\alpha(G^d)^{1/d}\)为下界，其中\(\alpha(G^d)\)是\(G\)的\(d\)次强幂的独立数。本文在\(C_7^{10}\)中构造了大小为\(134753\)的独立集，在\(C_{11}^{6}\)中构造了\(21909\)的独立集，在\(C_{13}^{6}\)中构造了\(62530\)的独立集，改进了这些图香农容量的已知下界。还改进了几个奇数圈单个强幂独立数的已知下界。构造通过与大语言模型迭代交互发现。
英文摘要
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 product of $G$. We construct independent sets of size $134753$ in $C_7^{10}$, $21909$ in $C_{11}^{6}$, $62530$ in $C_{13}^{6}$, and $8076974$ in $C_{15}^{8}$, improving the best known lower bounds for the Shannon capacity of these graphs to $Θ(C_7)\geq 134753^{1/10}>3.258020$, $Θ(C_{11})\geq 21909^{1/6}>5.289773$, $Θ(C_{13})\geq 62530^{1/6}>6.300109$, and $Θ(C_{15})\geq 8076974^{1/8}>7.301399$. We also improve the best known lower bounds on the independence numbers of several individual strong products 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.
发表机构
- Worcester Polytechnic Institute（沃斯特理工学院）
机构由 AI 辅助整理，请以论文原文为准。
