On Packing Trees into Complete Bipartite Graphs

Yanling PENG; Hong WANG

doi:10.1051/wujns/2023283221

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Volume 28 / No 3 (June 2023)

Wuhan Univ. J. Nat. Sci., 28 3 (2023) 221-222

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Issue		Wuhan Univ. J. Nat. Sci. Volume 28, Number 3, June 2023


Page(s)		221 - 222
DOI		https://doi.org/10.1051/wujns/2023283221
Published online		13 July 2023

Wuhan University Journal of Natural Sciences, 2023, Vol.28 No.3, 221-222

Mathematics

CLC number: O157.5

On Packing Trees into Complete Bipartite Graphs

Yanling PENG¹ and Hong WANG²

¹ Department of Mathematics, Suzhou University of Science and Technology, Suzhou 215009, Jiangsu, China
² Department of Mathematics, University of Idaho, Moscow 83844-1103, Idaho, US

Received: 7 May 2022

Abstract

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 Gyráfás and Lehel. Bollobás confirmed the "Tree packing conjecture" for many small trees, who showed that one can pack $T_{1}, T_{2}, \dots, T_{n / \sqrt[]{2}}$ into $K_{n}$ and that a better bound would follow from a famous conjecture of Erd $\ddot{o}$ s. In a similar direction, Hobbs, Bourgeois and Kasiraj made the following conjecture: Any sequence of trees $T_{2}$ , … , $T_{n}$ , with $T_{i}$ having order $i$ , can be packed into $K_{n - 1, ⌈ n / 2 ⌉}$ . Further Hobbs, Bourgeois and Kasiraj proved that any two trees can be packed into a complete bipartite graph $K_{n - 1, ⌈ n / 2 ⌉}$ . Motivated by these results, 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.

Key words: packing of graphs / tree packing conjecture / embedding of graph

Biography: PENG Yanling, female, Professor, research direction: graph theory. E-mail: pengyanling@mail.usts.edu.cn

Fundation item: Supported by the National Natural Science Foundation of China (12071334)

© Wuhan University 2023

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

0 Introduction

For graphs G and H, an embedding of G into H is an injection $ϕ : V (G) \to V (H)$ such that $ϕ (a) ϕ (b) \in E (H)$ whenever $a b \in E (G)$ . A packing of p graphs $G_{1}, G_{2}, \dots,$ $G_{p}$ into $H$ is a p-tuple $Φ = (ϕ_{1}, ϕ_{2}, \dots, ϕ_{p})$ such that, for $i = 1,2, \dots, p$ , $ϕ_{i}$ 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. {L-End} A bipartite graph G with the vertex partition $X = (X_{1}, X_{2})$ is denoted as $G (X_{1}, X_{2})$ or $G (X)$ . For a packing $Φ = (ϕ_{1}, ϕ_{2}, \dots, ϕ_{p})$ of $G_{1} (X), G_{2} (X), \dots, G_{p} (X)$ into a bipartite graph $H (Y)$ , we mean that $Φ$ is a packing such that $ϕ_{i} (X_{j}) \subset Y_{j}$ , $i = 1, 2, \dots, p$ , $j = 1,2 .$

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 $(\begin{matrix} n \\ 2 \end{matrix}) - \frac{n^{2}}{4}$ 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 $n \in N$ and trees $T_{1}, \dots, T_{n}$ with $T_{i}$ having order $i$ , the graphs $T_{1}, \dots, T_{n}$ can be packed into complete graph $K_{n}$ .

A packing of many of the small trees from Conjecture 1 was obtained by Bollob $\dot{a}$ s^[4], who showed that one can pack $T_{1}, \dots, T_{n / \sqrt[]{2}}$ into $K_{n}$ and that a better bound would follow from a famous conjecture of Erd $\ddot{o}$ s. In a similar direction, Hobbs, Bourgeois and Kasiraj made the following conjecture.

Conjecture 2 (Hobbs, Bourgeois and Kasiraj^[5]) Any sequence of trees $T_{2}, \dots, T_{n}$ , with $T_{i}$ having order i, can be packed into $K_{n - 1, ⌈ n / 2 ⌉}$ .

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 $K_{n - 1, ⌈ n / 2 ⌉}$ . Yuster^[6] proved that any sequence of trees $T_{2}, \dots, T_{s}$ , $s < \sqrt[]{5 / 8} n$ can be packed into $K_{n - 1, ⌈ n / 2 ⌉}$ . 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 $B_{n + k - 1} (X, Y)$ .

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 exists 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 k-packing of T in some complete bipartite graph $B_{n + m} (X, Y)$ , where $m = (h_{1} - k_{1}) + (h_{2} - k_{2})$ , $h_{1} = | X |$ and $h_{2} = | Y |$ .

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 $H_{1}$ and $H_{2}$ of sizes $h_{1}$ and $h_{2}$ , respectively, $h_{1} \leq h_{2}$ . Let T be a tree whose bipartite vertex classes are of size $k_{1}$ and $k_{2}$ . If $k_{1} \leq h_{1}$ and $k_{2} \leq h_{2}$ and $e (H) \geq k_{2} h_{1} + k_{1} h_{2} + k_{1} + k_{2} - h_{1} - h_{2} - k_{1} k_{2},$ 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 $k_{1}$ 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 $| X | = h_{1}$ , $| Y | = h_{2}$ , $h_{1} \leq h_{2}$ and $(h_{1} - k_{1}) (h_{2} - k_{2}) + (h_{1} - k_{1}) + (h_{2} - k_{2}) \geq (k - 1) (n - 1)$ . 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 $(h_{1} - k_{1}) (h_{2} - k_{2}) + (h_{1} -$ $k_{1}) + (h_{2} - k_{2}) \geq (k - 1) (n - 1)$ , we have $e (H) \geq k_{2} h_{1} + k_{1} h_{2}$ $+ k_{1} + k_{2} - h_{1} - h_{2} - k_{1} k_{2}$ . 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.

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