Independent Set Enumeration in King Graphs by Tensor Network Contractions

TL;DR

This paper introduces a tensor network contraction method to enumerate independent sets in king graphs, extending known results, approximating larger cases, and analyzing related combinatorial and physical properties.

Contribution

It transforms the enumeration problem into Wang tiling, computes exact and approximate counts for larger graphs, and adds new data to OEIS sequences, advancing combinatorial and physical modeling.

Findings

Enumerated independent sets for all m+n ≤ 79

Added thousands of new entries to OEIS sequences

Estimated entropy constants with less than 10^{-9} error

Abstract

This paper discusses the enumeration of independent sets in king graphs of size $m \times n$, based on the tensor network contractions algorithm given in reference~\cite{tilEnum}. We transform the problem into Wang tiling enumeration within an $(m+1) \times (n+1)$ rectangle and compute the results for all cases where $m + n \leq 79$ using tensor network contraction algorithm, and provided an approximation for larger $m, n$. Using the same algorithm, we also enumerated independent sets with vertex number restrictions. Based on the results, we analyzed the vertex number that maximize the enumeration for each pair $(m, n)$. Additionally, we compute the corresponding weighted enumeration, where each independent set is weighted by the number of its vertices (i.e., the total sum of vertices over all independent sets). The approximations for larger $m, n$ are given as well. Our results have added thousands of new items to the OEIS sequences A089980 and A193580. In addition, the combinatorial problems above are closely related to the hard-core model in physics. We estimate some important constants based on the existing results, and the relative error between our estimation of the entropy constant and the existing results is less than $10^{-9}$.