The packing and covering of the complete graph I: the forests of order five

Abstract

The maximum number of pairwise edge disjoint forests of order five in the complete graph Kn, and the minimum number of forests of order five whose union is Kn, are determined

Packing spanning graphs from separable families

“…Although Conjecture 1 is still open, there are several partial results. The conjecture is known to be true when the trees satisfy various structural requirements, see [11,12,13,15,24]. For arbitrary trees, Bollobás [7] proved that one can at least pack the small trees T 1 , .…”

Constructing the Spectrum of Packings and Coverings for the Complete Graph with Stars with up to Five Edges

“…In 1986 Roditty solved the problem of packing the complete graph K n with 5-stars [12]. We prove that we can achieve all the non-isomorphic possible leave graphs, thus solving the spectrum problem.…”

“…, [11], [12], and [13]). For integers n and k, if n ≥ 2k − 1 and k ≤ 6, then the S k -packing number of the complete graph K n is n(n−1) 2k and if n ≥ 2k and k ≤ 6, then the S k -covering number of K n is n(n−1) 2k .…”

“…The packing and covering problems have been considered for many classes of graphs. These problems were solved for all trees of order seven or less by Roditty [10], [11], [12], and [13]. In particular, he solved the problems for stars with up to six edges.…”

The spectrum of optimal excess graphs for trees with up to four edges

“…The G-packing (G-covering) problem of a graph H is to determine the G-packing number (Gcovering number) of H . Roditty solved the problem for all trees with up to six edges [4][5][6][7].…”