$H$-Decomposition of $r$-graphs when $H$ is an $r$-graph with exactly $k$ independent edges

Abstract

Let ${\varphi }_H^r(n)$ be the smallest integer such that, for all $r$-graphs $G$ on $n$ vertices, the edge set $E(G)$ can be partitioned into at most ${\varphi }_H^r(n)$ parts, of which every part either is a single edge or forms an $r$-graph isomorphic to $H$. The function ${\varphi }_H^2(n)$ has been well studied in literature, but for the case $r\ge 3$, the problem that determining the value of ${\varphi }_H^r(n)$ is widely open. Sousa (2010) gave an asymptotic value of ${\varphi }_H^r(n)$ when $H$ is an $r$-graph with exactly 2 edges, and determined the exact value of ${\varphi }_H^r(n)$ in some special cases. In this paper, we first give the exact value of ${\varphi }_H^r(n)$ when $H$ is an $r$-graph with exactly 2 edges, which improves Sousa's result. Second we determine the exact value of ${\varphi }_H^r(n)$ when $H$ is an $r$-graph consisting of exactly $k$ independent edges.