On Packing Trees into Complete Bipartite Graphs

Abstract

Let [see formula in PDF] denote the complete bipartite graph of order [see formula in PDF] with vertex partition sets [see formula in PDF] and [see formula in PDF]. We prove that for each tree [see formula in PDF] of order [see formula in PDF], there is a packing of [see formula in PDF] copies of [see formula in PDF] into a complete bipartite graph [see formula in PDF]. The ideal of the work comes from the "Tree packing conjecture" made by Gyráfás and Lehel. Bollobás confirmed the "Tree packing conjecture" for many small trees, who showed that one can pack [see formula in PDF] into [see formula in PDF] and that a better bound would follow from a famous conjecture of Erd[see formula in PDF]s. In a similar direction, Hobbs, Bourgeois and Kasiraj made the following conjecture: Any sequence of trees [see formula in PDF], … , [see formula in PDF], with [see formula in PDF] having order [see formula in PDF], can be packed into [see formula in PDF]. Further Hobbs, Bourgeois and Kasiraj proved that any two trees can be packed into a complete bipartite graph [see formula in PDF]. Motivated by these results, Wang Hong proposed the conjecture: For each tree [see formula in PDF] of order [see formula in PDF], there is a [see formula in PDF]⁃packing of [see formula in PDF] in some complete bipartite graph [see formula in PDF]. In this paper, we prove a weak version of this conjecture.