On Packing Trees into Complete Bipartite Graphs

: Let B n ( X  Y ) denote the complete bipartite graph of order n with vertex partition sets X and Y . We prove that for each tree T of order n , there is a packing of k copies of T into a complete bipartite graph B n + m ( X  Y ) . The ideal of the work comes from the "Tree packing conjecture" made by Gyr á f á s and Lehel. Bollob á s confirmed the "Tree packing conjecture" for many small trees, who showed that one can pack T 1  T 2  T n 2 into K n and that a better bound would follow from a famous conjecture of Erd o ̈ s. In a similar direction, Hobbs, Bour ‐ geois and Kasiraj made the following conjecture: Any sequence of trees T 2 , … , T n , with T i having order i , can be packed into K n - 1  é ù n 2 . Fur ‐ ther Hobbs, Bourgeois and Kasiraj proved that any two trees can be packed into a complete bipartite graph K n - 1  é ù n 2 . Motivated by these re ‐ sults, Wang Hong proposed the conjecture: For each tree T of order n , there is a k - packing of T in some complete bipartite graph B n + k - 1 ( X , Y ) . In this paper, we prove a weak version of this conjecture.


Introduction
For graphs G and H, an embedding of G into H is an injection ϕ:V (G)® V (H) such that ϕ(a)ϕ(b)Î E(H) whenever ab Î E(G).A packing of p graphs G 1 G 2  G p into H is a p-tuple Φ = (ϕ 1 ϕ 2 ϕ p ) such that, for i = 12p, ϕ i is an embedding of G i into H and the p sets ϕ i (E(G i )) are mutually disjoint.When all G i are isomorphic to G, we call it a k-parking of G.A bipartite graph G with the vertex partition Packing problems are central to combinatorics.Many classical problems can be stated as packing problems, such as Mantel  s Theorem which can be formulated by saying that if G is an n-vertex graph with less than ( ) edges, then the two graphs K 3 and G can be packed into K n .The packing problem has received a lot of attention.Many interesting results and elegant proofs of these results were obtained.For a survey, see Refs.[1, 2].Among the best known packing problems, the famous tree packing conjecture of Gyráfás and Lehel has driven a large amount of research in the area.Conjecture 1 (Gyráfás and Lehel [3] ) Given n Î  and trees T 1  T n with T i having order i, the graphs T 1 T n can be packed into complete graph K n .
A packing of many of the small trees from Conjecture 1 was obtained by Bollobȧs [4] , who showed that one can pack T 1 T n 2 into K n and that a better bound would follow from a famous conjecture of Erdös.In a similar direction, Hobbs, Bourgeois and Kasiraj made the following conjecture.
Conjecture 2 (Hobbs, Bourgeois and Kasiraj [5] ) Any sequence of trees T 2 T n , with T i having order i, can be packed into K n -1é ù n 2 .The conjecture has been verified for several very special classes of trees.Hobbs, Bourgeois and Kasiraj [5] proved that any two trees of order m and n with m < n can be packed into a complete bipartite graph K n -1é ù n 2 .Yuster [6] proved that any sequence of trees T 2 T s , s < 5 8 n can be packed into K n -1é ù n 2 .Motivated by these results, Wang proposed the following conjecture.
Conjecture 3 (Wang [7] ) For each tree T of order n, there is a k-packing of T in some complete bipartite graph This conjecture is true for k = 2 and k = 3 (see Theorem 1 and Theorem 2).
Theorem 1 [8] Let S(U 0 U 1 ) and T(V 0 V 1 ) be two trees of order n with |U i | = |V i | (i = 01).Then there ex- ists a complete bipartite graph B n + 1 (X 0  X 1 ) such that there is a packing of S(U 0 U 1 ) and T(V 0 V 1 ) in B n + 1 (X 0  X 1 ).
Theorem 2 [7] For each tree T of order n, there is a 3-packing of T in some complete bipartite graph B n + 2 (XY ).
In this paper we prove the following theorem.Theorem 3 For each tree T of order n, whose bipartite vertex classes are of size k 1 and k 2 , there is a kpacking of T in some complete bipartite graph

Proof of Theorem 3
We recall the following lemma due to Yuster [6] .Lemma 1 [6] Let H be a bipartite graph with vertex classes H 1 and H 2 of sizes h 1 and h 2 , respectively, h 1 ≤ h 2 .Let T be a tree whose bipartite vertex classes are of size k 1 and k 2 .If k 1 ≤ h 1 and k 2 ≤ h 2 and e (H ) Proof of Theorem 3 Let T be a tree of order n, whose bipartite vertex classes are of size k 1 and k 2 , where k 1 + k 2 = n.Let B n (X Y ) be a complete bipartite graph of order n with vertex partition sets X and Y of sizes k 1 and k 2 , respectively.Now we add some vertices into X and Y such that . So we get a complete bipartite graph B n + m (XY ) of order n + m, where m = (h 1 -k 1 ) + (h 2 -k 2 ).Clearly, B n + m (XY ) contains a copy of T. Suppose that we have already packed k -1 copies of T in B n + m (XY ).Let H be the spanning subgraph of B n + m (XY ) which contains all the edges that do not appear in the packing.It is easy to see that e (H ) = h 1 h 2 -( k 1 + k 2 -1 )( k -1).Since (h 1 -k 1 )(h 2 -k 2 )+(h 1 - k 1 ) + (h 2 -k 2 )≥(k -1)(n -1), we have e (H ) By Lemma 1, we find a copy of T in H, and add T to the packing.So there is a k-packing of T in the complete bipartite graph B n + m (XY ).
The proof is completed.