Packing pentagons into complete graphs: how clumsy can you get?

A pentagonal packing PP(n;t) is a family of t edge-disjoint pentagons in the complete graph Kn. A pentagonal packing is maximal if the complement of the union of its pentagons is pentagon-free. The spectrum S(5)(n) for maximal pentagonal packings is the set of sizes t such that there exists a maximal PP(n;t). We determine the extremes of the spectrum S(5)(n) for all n. Our results may be viewed as an extension of similar results for maximal …