A note on the 1-factorization of non-uniform complete hypergraph

Abstract

Given $G=(g_1,\cdots ,g_t)$ with $g_i\in N$ for $1\le i\le t$, let $G{K}_s^{\le t}$ denote the non-uniform complete hypergraph on s vertices, whose edge set contains $g_i$ copies of every i-subset of vertex set for $1\le i\le t$. Let $K_s^{\le t}$ denote $G{K}_s^{\le t}$ for $g_i=1$ for $1\le i\le t$. Recently, He et al. determined all $s,t$ such that $K_s^{\le t}$ has a 1-factorization. In this manuscript, we consider the 1-factorization of $G{K}_s^{\le t}$ and obtain the following results. (1) If $2g_t\ge g_j$ for $1\le j\le t-1$ and $s\equiv 0\left(modt\right)$, then $G{K}_s^{\le t}$ has a 1-factorization for sufficiently large s. (2) If $G{K}_s^{\le t}$ has a 1-factorization for sufficiently large s, then $s\equiv 0,-1\left(modt\right)$.