Proof of the squashed cube conjecture

Abstract

We prove a conjecture of R. L. Graham and H. O. Pollak to the effect that every connected graph onn vertices has a distance-preserving embedding in “squashed cube” of dimensionn−1. This means that to each vertex of the graph a string ofn−1 symbols from the alphabet {0, 1, *} can be assigned in such a way that the length of the shortest path between two vertices is equal to the Hamming distance between the corresponding strings, with * being regarded as at distance zero from either 1 or 0. Our labelling thus provides an efficient addressing scheme for the loop-switching communications system proposed by J. R. Pierce.

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References

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Winkler, P.M. Proof of the squashed cube conjecture. Combinatorica 3, 135–139 (1983). https://doi.org/10.1007/BF02579350

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AMS subject classification (1980)

  • 05 C 99
  • 55 M 15