THE PACKING AND COVERING OF THE COMPLETE GRAPH

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


INTRODUCTION
Graphs in this paper are fiite with no multiple edges or loops, leineke [I] defined the general covering (respectively, packing) problem as follows" For a given graph G find the minimum (maximum) number of edge disjoint subgraphs of G such that each subgraph has a specified property P and the union of the subgraphs is G.
Solutions of these problems are known only for a few properties P, when G is arbitrary.In most cases G is taken to be the complete graph K n or the complete bipartite graph Km, n (for particular references one may look at Roditty [2]).
DEFINITION" The complete graph K n is said to have a GT.decomposition if it is the union of edge disjoint subgraphs each isomorphic to G. We denote such a decomposi- tion by G K n-The G-decomposition problem is to determine the set N(G) of natural numbers such that K n has a G-decomposition if and only if n N(G).Note that G-decomposi- tion is actually an exact packing and covering.In the proof of our problems of packing and covering, we make great use of the results obtained for the G-decomposi- tior, problem in cases when G has five vertices.As usual [x] will denote the largest integer not exceeding x and {x} the least integer not less than x.t We will let e(G) denote the number of edges of the graph G and H U G. i=I will show that the graph H is the union of t edg disjoint graphs Gi, i=1,2 t.The Theorem of this paper solves variations of the covering and packing problems for the four graphs below- (iv) F 4 y u denoted (x,y,z; u,v) denoted (z; x,y,u,v) Our theorem may now be states as THEOREM (Packing and Coveri,g).et F be F 1, F 2 or F 3 and n > 5 or F be F 4 and n > 7 then (i) The maximum number of edge disjoint graphs F which are suhgraphs of the complete graph K n is [e(Kn)/e(F)].
(i i) The minimum number of graphs F whose union is the complete graph K n is {e(K )/e(F)}.
The traph K8m has and F2-decomposition.Observe that K6,8m 3K2,8m.In the case k 2 we saw that F21K2,8m.Table shows that K 6 has F 2 packing leaving three non-packed edges as required, and these three can be icluded in one more F 2. k 7.
The graph '8m+] has an F2-decomposition, and F21K6,8m, as was shown above.By Table we know that the graph K has an F2-packing leaving one non-packed edge.
lhe Theorem has now been proved for F 2 since all cases have been considered.
F 3-The proof will consider the same cases as the proof for F 2.

Table
We still have to prove the theorem for the cases"