Packing tetrahedrons in edge-weighted graphs

TL;DR

This paper proves that large edge-weighted complete graphs with sufficiently high minimum weighted degree contain a perfect packing of tetrahedrons with total weight exceeding a certain threshold, confirming a specific conjecture.

Contribution

It establishes a weighted version of a tetrahedron packing conjecture in complete graphs, confirming the conjecture for the case of $K_4$-factors.

Findings

Proved the existence of $K_4$-factors with weight constraints under minimum degree conditions

Confirmed a conjecture by Balogh, Kemkes, Lee, and Young for tetrahedron packings

Applicable to large complete graphs with weighted edges

Abstract

We prove that for all $\mu>0, t\in (0,1)$ and sufficiently large $n\in 4\mathbb{N}$, if $G$ is an edge-weighted complete graph on $n$ vertices with a weight function $w: E(G)\rightarrow [0,1]$ and the minimum weighted degree $\delta^w(G)\geq (\tfrac{1+3t}{4}+\mu)n$, then $G$ contains a $K_4$-factor where each copy of $K_4$ has total weight more than $6t$. This confirms a conjecture of Balogh--Kemkes--Lee--Young for the tetrahedron case.