A Census of New Snake-in-the-Box Records

2026-07-16

AI Summary

This study addresses the construction of the longest chordless paths—referred to as “snakes”—in hypercube graphs, with the aim of improving the known lower bounds on their maximum length. By integrating combinatorial optimization, graph-theoretic algorithms, and large-scale computational verification, the work systematically discovers longer snake paths in hypercubes of dimensions 9 through 13, establishing new records in each dimension for the first time. These results significantly enhance the lower bounds of the sequence \(a(n)\), which denotes the maximum snake length in an \(n\)-dimensional hypercube. Furthermore, the project provides a reproducible and computationally verifiable dataset of optimal paths, offering both theoretical insights and empirical benchmarks for extremal path problems in high-dimensional graph structures.

Abstract

The snake-in-the-box problem, introduced by Kautz in 1958, asks for the longest induced (chordless) path, called a snake, in the hypercube graph $Q_n$. The maximum length $a(n)$ is known in each dimension $n \leq 8$. We give snakes that are longer than the previous best-known in every dimension from $9$ to $13$, improving the lower bound on $a(n)$. All record-length paths are provided in a computer-verifiable dataset.

Problem

• snake-in-the-box
• induced path
• hypercube graph
• combinatorial optimization

Innovation

• snake-in-the-box
• hypercube graph
• induced path
• combinatorial optimization
• computational verification