Open Access
 Issue Wuhan Univ. J. Nat. Sci. Volume 28, Number 3, June 2023 221 - 222 https://doi.org/10.1051/wujns/2023283221 13 July 2023

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## 0 Introduction

For graphs G and H, an embedding of G into H is an injection such that whenever . A packing of p graphs into is a p-tuple such that, for , is an embedding of into H and the p sets are mutually disjoint. When all are isomorphic to G, we call it a k-parking of G. {L-End} A bipartite graph G with the vertex partition is denoted as or . For a packing of into a bipartite graph , we mean that is a packing such that , ,

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 and G can be packed into . 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 and trees with having order , the graphs can be packed into complete graph .

A packing of many of the small trees from Conjecture 1 was obtained by Bollobs [4], who showed that one can pack into and that a better bound would follow from a famous conjecture of Erds. In a similar direction, Hobbs, Bourgeois and Kasiraj made the following conjecture.

Conjecture 2 (Hobbs, Bourgeois and Kasiraj[5]) Any sequence of trees , with having order i, can be packed into .

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 can be packed into a complete bipartite graph . Yuster[6] proved that any sequence of trees , can be packed into . 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 and (see Theorem 1 and Theorem 2).

Theorem 1[8] Let and be two trees of order with . Then there exists a complete bipartite graph such that there is a packing of and in .

Theorem 2[7] For each tree T of order n, there is a 3-packing of T in some complete bipartite graph .

In this paper we prove the following theorem.

Theorem 3   For each tree T of order n, whose bipartite vertex classes are of size and , there is a k-packing of T in some complete bipartite graph , where , and .

## 1 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 and of sizes and , respectively, . Let T be a tree whose bipartite vertex classes are of size and . If and and then H contains a subgraph isomorphic to T.

Proof of Theorem 3   Let T be a tree of order n, whose bipartite vertex classes are of size and , where . Let be a complete bipartite graph of order n with vertex partition sets X and Y of sizes and , respectively. Now we add some vertices into X and Y such that , , and . So we get a complete bipartite graph of order , where . Clearly, contains a copy of T. Suppose that we have already packed copies of T in . Let H be the spanning subgraph of which contains all the edges that do not appear in the packing. It is easy to see that Since , we have . 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 .

The proof is completed.

## References

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