Packings in complete graphs (Q2715943)

A packing in a graph is a set of pairwise edge-disjoint subgraphs of \(G\). The density of a packing is the ratio of the total number of edges in graphs of the packing and the number of edges of \(G\); its maximum is looked for. A complete graph \(K_n\) whose vertices are labelled by \(x_1,\ldots ,x_n\) is considered. The cyclic length of the edge \(x_ix_j\) is defined as \(\min (|i-j|, n-|i-j|)\). A linear forest is a graph whose connected components are paths; the maximum number of vertices of its connected component is the height \(h\) of the forest. The paper studies packings of \(K_n\) by edges of pairwise different cyclic lengths, by linear forests with \(h=n\), with \(2h = n\), with \(h=2\) or with \(h^2 = n\) (in the last case such forests are called balanced). A connection with finite projective geometries is mentioned.