Improved Bounds for the Graham-Pollak Problem for Hypergraphs

For a fixed $r$, let $f_r(n)$ denote the minimum number of complete $r$-partite $r$-graphs needed to partition the complete $r$-graph on $n$ vertices. The Graham-Pollak theorem asserts that $f_2(n)=n-1$. An easy construction shows that $f_r(n) \leq (1+o(1))\binom{n}{\lfloor r/2 \rfloor}$, and we write $c_r$ for the least number such that $f_r(n) \leq c_r (1+o(1))\binom{n}{\lfloor r/2 \rfloor}$. It was known that $c_r<1$ for each even $r \geq 4$, but this was not known for any odd value of $r$. In this short note, we prove that $c_{295}<1$. Our method also shows that $c_r \rightarrow 0$, answering another open problem.


Introduction
The edge set of K n , the complete graph on n vertices, can be partitioned into n − 1 complete bipartite subgraphs: this may be done in many ways, for example by taking n − 1 stars centred at different vertices. Graham and Pollak [4,5] proved that the number n − 1 cannot be decreased. Several other proofs of this result have been found, by Tverberg [8], Peck [7], and Vishwanathan [9,10], among others.
Generalising this to hypergraphs, for n ≥ r ≥ 1, let f r (n) be the minimum number of complete r-partite r-graphs needed to partition the edge set of K (r) n , the complete r-uniform hypergraph on n vertices (i.e., the collection of all r-sets from an n-set). Thus the Graham-Pollak theorem asserts that f 2 (n) = n − 1. For r ≥ 3, an easy upper bound of n−⌈r/2⌉ ⌊r/2⌋ may be obtained by generalising the star example above. Indeed, for r even, having ordered the vertices, consider the collection of r-sets whose 2nd, 4th, . . . , rth vertices are fixed. This forms a complete r-partite r-graph, and the collection of all n−r/2 r/2 such is a partition of K (r) n . For r odd, we instead fix the 2nd, 4th, . . . , (r − 1)th vertices, yielding a partition into n−(r+1)/2 (r−1)/2 parts.
Alon [1] showed that f 3 (n) = n − 2. More generally, for each fixed r ≥ 1, he showed that where the upper bound follows from the construction above. Writing c r for the least c such that f r (n) ≤ c(1 + o(1)) n ⌊r/2⌋ , the above results assert that c 2 = 1, c 3 = 1, and How do the c r behave?
Cioabǎ, Kündgen and Verstraëte [2] gave an improvement (in a lower-order term) to Alon's lower bound, and Cioabǎ and Tait [3] showed that the construction above is not sharp in general, but Alon's asymptotic bounds (i.e., the above bounds on c r ) remained unchanged. Recently, Leader, Milićević and Tan [6] showed that c r ≤ 14 15 for each even r ≥ 4. However, they could not improve the bound of c r ≤ 1 for any odd r -the point being that the construction above is better for r odd than for r even (the exponent of n is (r − 1)/2 for r odd versus r/2 for r even), and so is harder to improve.
In this note, we give a simple argument to show that c 295 < 1. Our method also shows that c r → 0, answering another question from [6].
It would be interesting to know what happens for smaller odd values of r: for example, is c 5 < 1? Determining the precise value of c 4 (i.e., the asymptotic behaviour of f 4 (n)) would also be of great interest, as would determining the decay rate of the c r . See [6] for several related questions and conjectures.

Main Result
The motivation for our proof is as follows. The key to the approach used in [6] in proving c r < 1 for each even r ≥ 4 was to investigate the minimum number of products of complete bipartite graphs, that is, sets of the form Writing g(n) for this minimum value, it is trivial that g(n) ≤ (n − 1) 2 , by taking the products of the complete bipartite graphs appearing in a decomposition of K n into n − 1 complete bipartite graphs. It was shown in [6] that g(n) ≤ 14 15 + o(1) n 2 . It turned out that this upper bound on g(n) was enough (via an iterative construction) to bound c r below 1 for each even r ≥ 4. Now, as remarked above, for r odd the construction in the Introduction is much better than for r even. In fact, while there are many iterative ways to redo the construction when r is even, passing from n/2 to n, these fail when r is odd: it turns out that an extra factor is introduced at each stage. However, rather unexpectedly, we will see that (at least if r is large) if we partition into many pieces, instead of just two pieces, then the gain we obtain from the 14/15 improvement in g(n) outweighs the loss arising from this extra factor -even though this extra factor grows as the number of pieces grows.
A minimal decomposition of a complete r-partite r-graph K (r) n is a partition of the edge set into f r (n) complete r-partite r-graphs. A block is a product of the edge sets of two complete bipartite graphs. Similarly, a minimal decomposition of E(K n ) × E(K n ) is a partition of E(K n ) × E(K n ) into g(n) blocks. Finally, for a set V , we may write E(V ) to denote the edge set of the complete graph on V , that is, the set of all 2-subsets of V . Theorem 1. Let r = 2d + 1 be fixed. Then for each k there exists ǫ k , with ǫ k → 0 as k → ∞, such that for all n we have (Here the o(1) term is as n → ∞, with k and d fixed.) Proof. In order to decompose the edge set of K (r) kn , we start by splitting the kn vertices into k equal parts, say V K We consider the r-edges based on their intersection sizes with the k vertex classes. For each partition of r into positive integers r 1 + r 2 + · · · + r l with r 1 ≤ r 2 ≤ · · · ≤ r l and for each collection of l vertex classes V i 1 , V i 2 , . . . , V i l , the set of r-edges e with |e ∩ V i j | = r j for all j can be decomposed into f r 1 (n)f r 2 (n) · · · f r l (n) complete r-partite r-graphs: take a complete r j -partite r j -graph from a minimal decomposition of K (r j ) n for each j, and form a complete r-partite r-graph by taking the product of them.
Note that if at least three values of the r j are odd, then f r 1 (n)f r 2 (n) · · · f r l (n) = O(n d−1 ), as f s (n) ≤ n ⌊s/2⌋ for any s. So the set of r-edges e with |e ∩ V i | is odd for at least three distinct V i can be decomposed into Cn d−1 complete r-partite r-graphs, for some constant C depending on d and k.
Let C ′ be the number of partitions of r into at most d − 1 positive integers where exactly one of them is odd. Then we observe that the set of r-edges e such that e intersects with at most d − 1 vertex classes and |e ∩ V i | is odd for exactly one V i can be decomposed into at most C ′ k d−1 n d complete r-partite r-graphs.
We are now only left with two partitions of r: r = 1 + 2 + 2 + · · · + 2 and r = 2 + 2 + · · · + 2 + 3. The first case corresponds to the set of r-edges with r 1 = 1, r 2 = · · · = r d+1 = 2. For each of the k d collections of d vertex classes V i 1 , V i 2 , . . . , V i d , we claim that the set of r-edges {e : |e ∩ V i j | = 2, j = 1, 2, . . . , d} can be decomposed into g(n) d/2 or ng(n) (d−1)/2 complete r-partite r-graphs, depending on whether d is even or odd. This is done by pairing up the V i j s (or all but one of the V i j s if d is odd), and forming complete r-partite r-graphs using products of blocks in a minimal decomposition of E(K n ) × E(K n ). [For example, for d = 4, we would take a decomposition of , and now the set of all 9-edges e with |e ∩ V i j | = 2 for all 1 ≤ j ≤ 4 may be decomposed into g(n) 2 complete 9-partite 9-graphs by taking the E Finally, the second case corresponds to the set of r-edges with r 1 = r 2 = · · · = r d−1 = 2, r d = 3. These can be decomposed in a similar fashion. Indeed, for each collection of d vertex classes V i 1 , V i 2 , . . . , V i d , the set of r-edges {e : |e ∩ V i d | = 3 and |e ∩ V i j | = 2, j = 1, 2, . . . , d − 1} can be decomposed into n 2 g(n) (d−2)/2 or ng(n) (d−1)/2 complete r-partite r-graphs, depending on whether d is even or odd. There are d k d such sets of r-edges.
Combining the above and the bound on g(n), we have Corollary 3 implies that c r → 0 as r → ∞, proving Conjecture 16 in [6].