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

Abstract

The packing and covering problems have been considered for several classes of graphs. For instance, Bryant et. al. have investigated the packing problem for paths and cycles, and the packing and covering problems for 3-cubes. The packing and covering problems were settled for stars with up to six edges by Roditty. In this paper, for every possible leave graph (excess graph), we find a corresponding maximum packing (minimum covering) of the complete graph with stars with up to five edges.

Group divisible ‐packings with any minimum leave

“…When ≡ 0 (mod 5) or ≡ 0, 1 (mod 5), the conclusion follows from Theorem 1.3. When ∈ {2, 3,4,6,7,8,9,11,12,13, 14}, the conclusion follows from Examples 1.5 and 4.3, Lemmas 2.1-2.4, 5.1, 5.8 and Lemmas 5.12-5.19. When ≡ 1, 2, 3, 4 (mod 5) and ≥ 16, the conclusion follows from Lemma 5.20.…”

“…For = 6, the conclusion follows from Theorem 1.3. For ≥ 3 and ≠ 6, start from a ( 4 − )-HGDD of type ( , 3 2 6 1 ) (from Lemma 4.16 (2)), and apply Construction 4.12 with a ( 4 − )-IGDD of type (4, 1) (from Lemma 4.18) to obtain a ( 4 − )-IGDD of type (13,7) . Finally apply Construction 4.2 with a ( 4 − )-MGDP of type 7 (from Lemma 5.14).…”

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

“…The trees with four edges are S 4 , A, and P 5 , where A is a 3-star with one edge joined to one of its end vertices. The corresponding problem for S 4 was solved in [14]. Now consider A.…”

“…The spectrum problem for packing has been considered for trees with up to five edges [11]. In fact, it was proved that all possible leave graphs in packings of the complete graph with trees that have up to five edges are achievable.…”

“…In fact, it was proved that all possible leave graphs in packings of the complete graph with trees that have up to five edges are achievable. They also solved the spectrum problem for covering for stars with up to five edges and showed that all possible excess graphs in covering the complete graph with stars that have up to five edges can be obtained, except for the graph K 4 2 which is not achievable as the excess graph in any S 5 -covering of K 12 [12]. In this paper, we are considering the spectrum problem for covering for trees with up to four edges and we will prove that the spectrum of excess graphs for trees with up to four edges is the set of all possible excess graphs.…”