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Decomposition of the completer-graph into completer-partiter-graphs

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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.

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References

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Authors and Affiliations

  1. Department of Mathematics, Tel Aviv University, Tel Aviv, Israel

    Noga Alon

  2. Department of Mathematics, Massachusetts Institute of Technology, 02139, Cambridge, MA, USA

    Noga Alon

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  1. Noga Alon
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Research supported in part by Air Force Contract OSR 82-0326 and by Allon Fellowship.

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Alon, N. Decomposition of the completer-graph into completer-partiter-graphs. Graphs and Combinatorics 2, 95–100 (1986). https://doi.org/10.1007/BF01788083

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  • DOI: https://doi.org/10.1007/BF01788083

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