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

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 \(\lceil\frac{n(n-1)}{2e(F)}\rceil\), \(n\geq 6\), 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.