Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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PACKING TREES INTO COMPLETE K-PARTITE GRAPH

  • Peng, Yanling (Department of Mathematics Suzhou University of Science and Technology) ;
  • Wang, Hong (Department of Mathematics University of Idaho)
  • Received : 2021.02.16
  • Accepted : 2021.12.06
  • Published : 2022.03.31
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Abstract

In this work, we confirm a weak version of a conjecture proposed by Hong Wang. The ideal of the work comes from the tree packing conjecture made by Gyárfás and Lehel. Bollobás confirms the tree packing conjecture for many small tree, who showed that one can pack T1, T2, …, $T_{n/\sqrt{2}}$ into Kn and that a better bound would follow from a famous conjecture of Erdős. In a similar direction, Hobbs, Bourgeois and Kasiraj made the following conjecture: Any sequence of trees T1, T2, …, Tn, with Ti having order i, can be packed into Kn-1,[n/2]. Further Hobbs, Bourgeois and Kasiraj [3] proved that any two trees can be packed into a complete bipartite graph Kn-1,[n/2]. Motivated by the result, Hong Wang propose the conjecture: For each k-partite tree T(𝕏) of order n, there is a restrained packing of two copies of T(𝕏) into a complete k-partite graph Bn+m(𝕐), where $m={\lfloor}{\frac{k}{2}}{\rfloor}$. Hong Wong [4] confirmed this conjecture for k = 2. In this paper, we prove a weak version of this conjecture.

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Acknowledgement

We are very grateful to the referee for helpful suggestions and comments.

References

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