Decomposition of Complete Graphs into Union of Stars

Let S k+1 denote a star with k edges. Tarsi and Yamamoto et al. have characterized the S k+1 –decomposability of K n the complete graph. In this paper we study the edge decomposition of both K n and the complete bipartite graph K m,n into copies of the union of two edge disjoint stars S p+1 and S q+1 where p ≠q and p,q ൒2 and obtain the necessary and sufficient conditions for the S p+1 ∪S q+1-decomposability of K n and K m,n .


Introduction
The graphs considered are finite and undirected.The terminology adopted agrees with that in any standard book on graph theory.[ 2 ] K n denotes the complete graph on n vertices.It has n C 2 edges.K m,n , the complete bipartite graph has m+n vertices and mn edges.A star S k+1 with k+1 vertices and k edges is nothing but the complete bipartite graph K 1,k .The star S k+1 with its centre at 1 and edges joining1 and i, i=2,3,…,k.is denoted by S{1:2,3,…,k}.A psubset of V is a subset of size p.The join G v H of two vertex disjoint graphs G and H is the graph with vertex set V(G) ∪V(H) and edge set E(G)∪ E(H) ∪{ uv : u ∈V(G) and v ∈V(H)} Let G be any graph and F={ G 1 ,G 2 ,…G k } be a family of subgraphs of G.An Fdecomposition of G is an edge-disjoint decomposition of G into copies of G i for i=1,2,…k, provided, no G i has isolated vertices.Obviously, k i is a positive integer.If each G i is isomorphic to a graph H then G is H-decomposable or G has an Hdecomposition, denoted by H│G.Clearly,when H│G ,then e(H)│e(G).If G is Fdecomposable, it necessarily satisfies the condition ∑ ݇ ୀଵ i e(H i ) =e(G).For a family F of stars, Lin and Shyu [ 1 ] have given a characterization of the Fdecomposability of K n .
Tarsi and Yamamoto et al. [4,6] have established the following result."Let k and n be positive integers.There exists a S k+1 decomposition of K ,n if and only if 2k ≤ n and n(n-1) ≡ 0 (mod 2k)." We concentrate on H-decompositions of K m,n , the complete bipartite graph and K ,n the complete graph where, H= S p+1 ∪S q+1 , the union of two vertex disjoint stars.
We assume that p ≠ q, for otherwise it reduces to star -decomposition.Even then, if either p or q=1, the problem reduces to decomposition of complete graphs into paths and stars.[5] Hence p>2and q>3.Regi T. [ 3]gives the conditions for the S 2 ∪S 3 decomposability of K n and K m,n .
In this paper the S p+1 ∪S q+1 , p 2, q 2, decomposability of complete graphs is characterized.The main results are given in (2.3) and (3.4), theorems A and B.

S q+1 , p ≠ q
The complete bipartite graph K m,n has m+n vertices and mn edges while S p+1 ∪S q+1 has p+q+2 vertices and p+q edges.The stars S p+1 and S q+1 are vertex disjoint and K m,n is connected.If K m,n is S p+1 ∪S q+1 -decomposable,then K m,n will be the edge disjoint union of at least two copies of S p+1 ∪S q+1 .For any value of p and q, Decomposition of complete graphs into union of stars 13 there exists the minimal S p+1 ∪S q+1 decomposable K m,n with the value two for either m or n .Hence , we fix m, n ≥2.The same condition is chosen in [3] also.There are S p+1 ∪S q+1 -non-decomposable complete bipartite graphs for m or n <.2.

Theorem A:
Let m, n, p, q be positive integers such that m,n ≥2 and p ≠ q.The necessary and sufficient condition that the complete bipartite graph K m,n is S p+1 ∪S q+1decomposable is (i) m+n ≥p+q+2 and (ii) mn ≡ 0(mod(p+q))
It is sufficient to consider any one of them.
Case I: n is even.n=2r, for some integer r ≥ 1.
Partition V 2 into r= ଶ subsets of size 2 each.
A p-subset of V 1 , a q=subset of V 1 and a 2-subset of V 2 span two disjoint copies of S p+1 ∪ S q+1 .
When all of the p,q,2-subsets are considered, they span a K m,n which is decomposed into 2×k×r = nk copies of S p+1 ∪S q+1 .
2-subsets and a 3-subset.As in case I,the 2-subsets span 2rk copies of S p+1 ∪S q+1 .From the 3-subset, two vertices are chosen in three ways, each pair spanning k copies of S p+1 ∪S q+1 .K m,n is decomposed into 2rk+3k = nk copies of S p+1 ∪S q+1 This completes the proof.

Note:
The conditions m,n ≥2 and m+n ≥p+q+2 guarantee the existence of a S p+1 ∪ S q+1 decomposable K m,n

Decomposition of K n , the complete graph into S p+1 ∪ ∪ ∪ ∪ S q+1 , p ≠ q
The proof methodology adopted here depends on the principle of mathematical induction.Given any p and q we have to establish the relation between p, q and n to check whether K n is S p+1 ∪S q+1 -decomposable.The proof for the initial values of p and q (2 and 3) is given separately, even though it is a part of the main theorem.Theorem A is used in proving Theorem B, the main result here.

Theorem:
The complete graph K n is S 3 ∪S 4 -decomposable if and only if n ≥ 10 and n ≡ 0,1 (mod5).
It is noted that both K 5 and K 6 are not S 3 ∪S 4 -decomposable .

Theorem B
The necessary and sufficient condition that the complete graph K n is S p+1 ∪S q+1decomposable, p ≠ q is n > p+q+2 and (ii) n ≡ 0,1 (mod 2(p+q)).
(sufficiency)The proof is by induction on n.The existence of such a K n is proved for p=2 and q=3 (Theorem3.3)Let p>2 and q>3 and satisfying (i) and (ii) and the graph K m is S p+1 ∪S q+1decomposable Let m ≡ 0, (mod 2(p+q)).
The next higher value of n is m+2(p+q) ≡ 0 (mod 2(p+q)) by the induction hypothesis.
The graphs on the r.h.s.are S p+1 ∪S q+1 -decomposable, by the induction hypothesis.
By the induction hypothesis, Theorem A and lemma 3.1, K m+1 is S p+1 ∪S q+1decomposable.

Conclusion:
We have given, in this paper, a characterization for the S p+1 ∪S q+1 , p 2,q2, decomposability of complete graphs .Similar results for other graphs will be given in another paper.