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Snakes, coils, and single-track circuit codes with spread \(k\)

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Abstract

The snake-in-the-box problem is concerned with finding a longest induced path in a hypercube \(Q_n\). Similarly, the coil-in-the-box problem is concerned with finding a longest induced cycle in \(Q_n\). We consider a generalization of these problems that considers paths and cycles where each pair of vertices at distance at least \(k\) in the path or cycle are also at distance at least \(k\) in \(Q_n\). We call these paths \(k\)-snakes and the cycles \(k\)-coils. The \(k\)-coils have also been called circuit codes. By optimizing an exhaustive search algorithm, we find 13 new longest \(k\)-coils, 21 new longest \(k\)-snakes and verify that some of them are optimal. By optimizing an algorithm by Paterson and Tuliani to find single-track circuit codes, we additionally find another 8 new longest \(k\)-coils. Using these \(k\)-coils with some basic backtracking, we find 18 new longest \(k\)-snakes.

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Notes

  1. This work was made possible by the facilities of the Shared Hierarchical Academic Research Computing Network (SHARCNET: www.sharcnet.ca) and Compute/Calcul Canada.

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Authors and Affiliations

  1. School of Computer Science, University of Guelph, Guelph, ON, Canada

    Simon Hood, Daniel Recoskie, Joe Sawada & Dennis Wong

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  1. Simon Hood
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  2. Daniel Recoskie
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  3. Joe Sawada
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  4. Dennis Wong
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Correspondence to Joe Sawada.

Appendices

Appendices

1.1 Instances of longest \(k\)-coils and \(k\)-snakes

Each \(k\)-coil and \(k\)-snake can be represented compactly by its transition (or coordinate) sequence starting from some arbitrary starting vertex, generally assumed to be \(0^n\). Assuming that the first bit position is 0 and that the symbols \(a\) to \(g\) represent the positions 10–16, we provide the transition sequences for each new longest \(k\)-coil and \(k\)-snake discovered. Each sequence is prefaced by \((n,k,t)\) where \(n\) is the length of the bitstrings, \(k\) is the spread, and \(t\) is the length of the \(k\)-coil or \(k\)-snake. Observe that for \(k\)-snakes this means the transition sequence will have \(t\) symbols, while for the \(k\)-coils the transition sequence will have \(t{-}1\) symbols.

The \(k\)-coils (13,3,494), (14,3,812), (15,3,1380), (16,3,2240), (17,3,3910), (15,4, 480), (16,4,768), and (17,4,1224) are also single-track circuit codes. However, to have the single-track property, the starting vertex is important. For these coils, we add in this starting string as a 4th parameter.

1.1.1 \(k\)-coils

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1.1.2 \(k\)-snakes

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Hood, S., Recoskie, D., Sawada, J. et al. Snakes, coils, and single-track circuit codes with spread \(k\) . J Comb Optim 30, 42–62 (2015). https://doi.org/10.1007/s10878-013-9630-z

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