Packing and Covering of the Complete Graph, $V$: The Forests of Order Six and Their Multiple Copies

Abstract

It is shown that the maximal number of pairwise edge disjoint forests, $F$, of order six in the complete graph $K_n$, and the minimum number of forests of order six, whose union is $K_n$ are $\lfloor \frac{n(n-1)}{2e(F)}\rfloor$ and $\lceil \frac{n(n-1)}{2e(F)}\rceil$, $n\ge 6$, respectively and $e(F)$ is the number of edges of $F$. ($\lfloor x\rfloor$ denotes the largest integer not exceeding $x$ and $\lceil x\rceil$ the least integer not less than $x$). Some generalizations to multiple copies of these forests and of paths are also given.