Packing tetrahedrons in edge-weighted graphs

Abstract

We prove that for all $\hat{I}¼>0,t\in (0,1)$ and sufficiently large $n\in 4\mathbb{N}$, if $G$ is an edge-weighted complete graph on $n$ vertices with a weight function $w:E(G)\to [0,1]$ and the minimum weighted degree $\hat{I}{´}^w(G)\ge (\frac{1+3t}{4}+\hat{I}¼)n$, 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.

20 pages+3 page appendix, 4 figures