Improved Bounds for the Graham-Pollak Problem for Hypergraphs

TL;DR

This paper establishes new bounds for hypergraph partitioning, proving that the minimal number of complete r-partite r-graphs needed grows sublinearly for large r, including an explicit result for r=295.

Contribution

The authors prove that the constant c_r is less than 1 for r=295 and show that c_r approaches zero as r increases, advancing understanding of hypergraph partition bounds.

Findings

Proved c_{295}<1 for the first odd r.

Demonstrated c_r approaches zero as r increases.

Improved bounds for hypergraph partitioning problem.

Abstract

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.