Abstract
Forn ≥ r ≥ 1, letf r (n) denote the minimum numberq, such that it is possible to partition all edges of the completer-graph onn vertices intoq completer-partiter-graphs. Graham and Pollak showed thatf 2(n) =n − 1. Here we observe thatf 3(n) =n − 2 and show that for every fixedr ≥ 2, there are positive constantsc 1(r) andc 2(r) such thatc 1(r) ≤f r (n)⋅n −[r/2] ≤n 2(r) for alln ≥ r. This solves a problem of Aharoni and Linial. The proof uses some simple ideas of linear algebra.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aharoni, R., Linial, N.: Private communication
Graham, R.L., Lovász, L.: Distance matrix polynomials of trees. Advances in Math.29, 60–88 (1978)
Graham, R.L., Pollak, H.O.: On the addressing problem for loop switching. Bell Syst. Tech. J.50, 2495–2519 (1971)
Graham, R.L., Pollak, H.O.: On embedding graphs in squashed cubes. In: Lecture Notes in Mathematics 303, pp 99–110. New York-Berlin-Heidelberg: Springer-Verlag 1973
Lovász, L.: Problem 11.22. In: Combinatorial Problems and Exercises, p 73. Amsterdam: North Holland 1979
Peck, G.W.: A new proof of a theorem of Graham and Pollak. Discrete Math.49, 327–328 (1984)
Tverberg, H.: On the decomposition ofK n into complete bipartite graphs. J. Graph Theory6, 493–494 (1982)
Author information
Authors and Affiliations
Additional information
Research supported in part by Air Force Contract OSR 82-0326 and by Allon Fellowship.
Rights and permissions
About this article
Cite this article
Alon, N. Decomposition of the completer-graph into completer-partiter-graphs. Graphs and Combinatorics 2, 95–100 (1986). https://doi.org/10.1007/BF01788083
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01788083