Proof of the squashed cube conjecture


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.

This is a preview of subscription content, access via your institution.


  1. [1]

    R. L. Graham andH. O. Pollak, On the addressing problem for loop switching,Bell System Tech. J. 50 (1971), 2495–2519.

    MathSciNet  Google Scholar 

  2. [2]

    R. L. Graham andH. O. Pollak, On embedding graphs in squashed cubes,Graph Theory and Applications, Lecture Notes in Mathematics303, Springer-Verlag (Proc. of a conference held at Western Michigan University, May 10–13, 1972).

  3. [3]

    J. R. Pierce, Network for block switching of data,Bell System Tech. J. 51 (1972), 1133–1145.

    Google Scholar 

  4. [4]

    P. M. Winkler, Every connected graph is a query graph,J. Combinatorics, Information and System Sciences, to appear.

  5. [5]

    A. C.-C. Yao, On the loop switching addressing problem,SIAM J. on Computing, Vol. 7 No. 4 (1978), 515–523.

    MATH  Article  Google Scholar 

Download references

Author information



Rights and permissions

Reprints and Permissions

About this article

Cite this article

Winkler, P.M. Proof of the squashed cube conjecture. Combinatorica 3, 135–139 (1983).

Download citation

AMS subject classification (1980)

  • 05 C 99
  • 55 M 15