Bounds for the Graham-Pollak Theorem for Hypergraphs

TL;DR

This paper investigates bounds on the minimum number of complete r-partite r-graphs needed to partition the edges of complete r-uniform hypergraphs, improving known bounds and reducing the smallest r for which certain bounds hold.

Contribution

The authors improve the known bounds for hypergraph partitions and reduce the minimal r for which the bounds are less than one, advancing understanding of hypergraph decompositions.

Findings

Improved upper bounds for f_r(n) for small even r.

Reduced the smallest r with c_r<1 from 295 to 113.

Provided tighter bounds for hypergraph edge partitions.

Abstract

Let $f_r(n)$ represent the minimum number of complete $r$-partite $r$-graphs required to partition the edge set of the complete $r$-uniform hypergraph on $n$ vertices. The Graham-Pollak theorem states that $f_2(n)=n-1$. An upper bound of $(1+o(1)){n \choose \lfloor{\frac{r}{2}}\rfloor}$ was known. Recently this was improved to $\frac{14}{15}(1+o(1)){n \choose \lfloor{\frac{r}{2}}\rfloor}$ for even $r \geq 4$. A bound of $\bigg[\frac{r}{2}(\frac{14}{15})^{\frac{r}{4}}+o(1)\bigg](1+o(1)){n \choose \lfloor{\frac{r}{2}}\rfloor}$ was also proved recently. The smallest odd $r$ for which $c_r < 1$ that was known was for $r=295$. In this note we improve this to $c_{113}<1$ and also give better upper bounds for $f_r(n)$, for small values of even $r$.