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

TL;DR

This paper determines the exact values of the function _H^r(n) for decomposing r-graphs into parts, especially when H has exactly k independent edges, extending previous results for the case of 2 edges.

Contribution

It provides the exact values of _H^r(n) for r-graphs with exactly k independent edges, generalizing prior work and improving known results for the case of 2 edges.

Findings

Exact value of _H^r(n) for H with 2 edges

Exact value of _H^r(n) for H with k independent edges

Extension of previous asymptotic and special case results

Abstract

Let $\phi_H^r(n)$ 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 $\phi_H^r(n)$ parts, of which every part either is a single edge or forms an $r$-graph isomorphic to $H$. The function $\phi^2_H(n)$ has been well studied in literature, but for the case $r\ge 3$, the problem that determining the value of $\phi_H^r(n)$ is widely open. Sousa (2010) gave an asymptotic value of $\phi_H^r(n)$ when $H$ is an $r$-graph with exactly 2 edges, and determined the exact value of $\phi_H^r(n)$ in some special cases. In this paper, we first give the exact value of $\phi_H^r(n)$ when $H$ is an $r$-graph with exactly 2 edges, which improves Sousa's result. Second we determine the exact value of $\phi_H^r(n)$ when $H$ is an $r$-graph consisting of exactly $k$ independent edges.