 
Summary: Graphs and Combinatorics 2, 95100 (1986)
Graphsand
Combinatorics
© SpringerVerlag1986
Decomposition of the CompleterGraphinto
CompleterPartiterGraphs*
Noga Alon
Department of Mathematics, Tel Aviv University, Tel Aviv, Israel, and
Department ofMathematics, Massachusetts Institute ofTechnology,Cambridge, MA 02139, USA
Abstract. For n > r > 1,letf,(n) denote the minimum number q, such that it is possible to partition
all edges of the complete rgraph on n vertices into q complete rpartite rgraphs. Graham and
Pollak showed that fz(n) = n  1.Here we observe that f3(n) = n  2 and show that forevery fixed
r > 2, there are positive constants cx(r) and c2(r) such that q(r) < f,(n)" nf'/2J < c2(r) for all n > r.
This solves a problem of Aharoni and Linial. The proof uses some simple ideas of linear algebra.
1. Introduction
For n _> r _> 1, let £(n) denote the minimum number q, such that it is possible to
partition all edges of the complete runiform hypergraph on n vertices into q
pairwise edgedisjoint complete rpartite runiform hypergraphs.
Obviously, f~(n) = 1. Graham and Pollak (I3, 4], see also [2, 5]) proved that
f2(n) = n  1 for all n _> 2. Simple proofs for this result were found by Tverberg [7]
