On Non-Uniform Baranyai Factorizations

For a fixed nonempty finite set $L$ of positive integers and $k=\max L\ge 2$, let $\left(\genfrac{}{}{0px}{}{[n]}{L}\right)={\bigcup }_{\mathrm{\ell }\in L}\left(\genfrac{}{}{0px}{}{[n]}{\mathrm{\ell }}\right)$ denote the non-uniform hypergraph whose edge set consists of all $\mathrm{\ell }$-subsets of $[n]$ for $\mathrm{\ell }\in L$. A $1$-factor of a hypergraph $H$ is a collection of pairwise disjoint edges whose union is $V(H)$. We say $H$ is $1$-factorable if $E(H)$ can be partitioned into $1$-factors. In this paper, we study the 1-factorability of $\left(\genfrac{}{}{0px}{}{[n]}{L}\right)$. Earlier work of He, Huang, and Ma shows that, for all sufficiently large $n$, $1$-factorability can occur only in the residue classes $0$ and $-1$ modulo $k$: it always occurs in class $0$, whereas it occurs in class $-1$ only if $k-1\in L$. We determine exactly when $1$-factorability occurs in this exceptional class. More precisely, we define a finite gcd-peeling algorithm on $L\setminus \{k-1,k\}$ and prove that $\left(\genfrac{}{}{0px}{}{[qk-1]}{L}\right)$ is $1$-factorable if and only if the algorithm accepts. As a consequence, our result provides a criterion for the $1$-factorability of $\left(\genfrac{}{}{0px}{}{[n]}{L}\right)$ and confirms the conjecture posed by He, Huang, and Ma.