Tree Packing Conjecture Research Articles

Packing 4-Partite Tree into Complete 4-Partite Graph

For graphs G and H, an embedding of G into H is an injection [see formula in PDF] such that [see formula in PDF] whenever [see formula in PDF]. A packing of p graphs [see formula in PDF] into [see formula in PDF]is a p-tuple [see formula in PDF] such that, for [see formula in PDF], [see formula in PDF] is an embedding of [see formula in PDF] into H and the p sets [see formula in PDF] are mutually disjoint. Motivated by the "Tree Packing Conjecture" made by Gy[see formula in PDF]rf[see formula in PDF]s and Lehel, Wang Hong conjectured that for each k-partite tree, there is a packing of two copies of [see formula in PDF] into a complete k-partite graph [see formula in PDF], where [see formula in PDF]. In this paper, we confirm this conjecture for [see formula in PDF].

Seven largest trees pack

The Tree Packing Conjecture (TPC) by Gyárfás states that any set of trees \(T_2,\dots,T_{n-1}, T_n\) such that \(T_i\) has \(i\) vertices pack into \(K_n\). The conjecture is true for bounded degree trees, but in general, it is widely open. Bollobás proposed a weakening of TPC which states that \(k\) largest trees pack. The latter is true if none tree is a star, but in general, it is known only for \(k=5\). In this paper we prove, among other results, that seven largest trees pack.

On Packing Trees into Complete Bipartite Graphs

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.

Perfectly packing graphs with bounded degeneracy and many leaves

We prove that one can perfectly pack degenerate graphs into complete or dense n-vertex quasirandom graphs, provided that all the degenerate graphs have maximum degree $$o\left( {{n \over {\log \,n}}} \right)$$ , and in addition Ω(n) of them have at most (1 − Ω(1))n vertices and Ω(n) leaves. This proves Ringel’s conjecture and the Gyárfás Tree Packing Conjecture for all but an exponentially small fraction of trees (or sequences of trees, respectively).

Optimal packings of bounded degree trees

We prove that if T_1,\dots, T_n is a sequence of bounded degree trees such that T_i has i vertices, then K_n has a decomposition into T_1,\dots, T_n . This shows that the tree packing conjecture of Gyárfás and Lehel from 1976 holds for all bounded degree trees (in fact, we can allow the first o(n) trees to have arbitrary degrees). Similarly, we show that Ringel's conjecture from 1963 holds for all bounded degree trees. We deduce these results from a more general theorem, which yields decompositions of dense quasi-random graphs into suitable families of bounded degree graphs. Our proofs involve Szemerédi's regularity lemma, results on Hamilton decompositions of robust expanders, random walks, iterative absorption as well as a recent blow-up lemma for approximate decompositions.

A blow-up lemma for approximate decompositions

We develop a new method for constructing approximate decompositions of dense graphs into sparse graphs and apply it to long-standing decomposition problems. For instance, our results imply the following. Let G G be a quasi-random n n -vertex graph and suppose H 1 , … , H s H_1,\dots ,H_s are bounded degree n n -vertex graphs with ∑ i = 1 s e ( H i ) ≤ ( 1 − o ( 1 ) ) e ( G ) \sum _{i=1}^{s} e(H_i) \leq (1-o(1)) e(G) . Then H 1 , … , H s H_1,\dots ,H_s can be packed edge-disjointly into G G . The case when G G is the complete graph K n K_n implies an approximate version of the tree packing conjecture of Gyárfás and Lehel for bounded degree trees, and of the Oberwolfach problem. We provide a more general version of the above approximate decomposition result which can be applied to super-regular graphs and thus can be combined with Szemerédi’s regularity lemma. In particular our result can be viewed as an extension of the classical blow-up lemma of Komlós, Sárkőzy, and Szemerédi to the setting of approximate decompositions.

A bandwidth theorem for approximate decompositions

We provide a degree condition on a regular n ‐vertex graph G which ensures the existence of a near optimal packing of any family H of bounded degree n ‐vertex k ‐chromatic separable graphs into G . In general, this degree condition is best possible. Here a graph is separable if it has a sublinear separator whose removal results in a set of components of sublinear size. Equivalently, the separability condition can be replaced by that of having small bandwidth. Thus our result can be viewed as a version of the bandwidth theorem of Bottcher, Schacht and Taraz in the setting of approximate decompositions. More precisely, let δ k be the infimum over all δ ⩾ 1 / 2 ensuring an approximate K k ‐decomposition of any sufficiently large regular n ‐vertex graph G of degree at least δ n . Now suppose that G is an n ‐vertex graph which is close to r ‐regular for some r ⩾ ( δ k + o ( 1 ) ) n and suppose that H 1 , ⋯ , H t is a sequence of bounded degree n ‐vertex k ‐chromatic separable graphs with ∑ i e ( H i ) ⩽ ( 1 − o ( 1 ) ) e ( G ) . We show that there is an edge‐disjoint packing of H 1 , ⋯ , H t into G . If the H i are bipartite, then r ⩾ ( 1 / 2 + o ( 1 ) ) n is sufficient. In particular, this yields an approximate version of the tree packing conjecture in the setting of regular host graphs G of high degree. Similarly, our result implies approximate versions of the Oberwolfach problem, the Alspach problem and the existence of resolvable designs in the setting of regular host graphs of high degree.

Packing spanning graphs from separable families

Let $$\mathcal{G}$$ be a separable family of graphs. Then for all positive constants ϵ and Δ and for every sufficiently large integer n, every sequence G 1,..., G t ∈ $$\mathcal{G}$$ of graphs of order n and maximum degree at most Δ such that $$\left( {{G_1}} \right) + \cdots + e\left( {{G_t}} \right) \leqslant \left( {1 - \epsilon } \right)\left( {\begin{array}{*{20}{c}} n \\ 2 \end{array}} \right)$$ packs into K n . This improves results of Böttcher, Hladký, Piguet and Taraz when $$\mathcal{G}$$ is the class of trees and of Messuti, Rödl, and Schacht in the case of a general separable family. The result also implies approximate versions of the Oberwolfach problem and of the Tree Packing Conjecture of Gyárfás and Lehel (1976) for the case that all trees have maximum degree at most Δ. The proof uses the local resilience of random graphs and a special multi-stage packing procedure.

Packing large trees of consecutive orders

A conjecture by Bollobás from 1995 (which is a weakening of the famous Tree Packing Conjecture by Gyárfás from 1976) states that any set of k trees Tn,Tn−1,…,Tn−k+1, such that Tn−i has n−i vertices, pack into Kn, provided n is sufficiently large. We confirm Bollobás conjecture for trees Tn,Tn−1,…,Tn−k+1, such that Tn−i has k−1−i leaves or a pending path of order k−1−i. As a consequence we obtain that the conjecture is true for k≤5.

On Erdős-Sós Conjecture for Trees of Large Size

Erdős and Sós conjectured that every graph $G$ of average degree greater than $k-1$ contains every tree of size $k$. Several results based upon the number of vertices in $G$ have been proved including the special cases where $G$ has exactly $k+1$ vertices (Zhou), $k+2$ vertices (Slater, Teo and Yap), $k+3$ vertices (Woźniak) and $k+4$ vertices (Tiner). We further explore this direction. Given an arbitrary integer $c\geq 1$, we prove Erdős-Sós conjecture in the case when $G$ has $k+c$ vertices provided that $k\geq k_0(c)$ (here $k_0(c)=c^{12}{\rm polylog}(c)$). We also derive a corollary related to the Tree Packing Conjecture.

An approximate version of the tree packing conjecture

We prove that for any pair of constants ɛ > 0 and Δ and for n sufficiently large, every family of trees of orders at most n, maximum degrees at most Δ, and with at most ( 2 ) edges in total packs into $${K_{(1 + \varepsilon )n}}$$ . This implies asymptotic versions of the Tree Packing Conjecture of Gyárfás from 1976 and a tree packing conjecture of Ringel from 1963 for trees with bounded maximum degree. A novel random tree embedding process combined with the nibble method forms the core of the proof.

Packing trees into n-chromatic graphs

Let Ti denote a tree with i edges. The author’s tree packing conjecture [3] states that Kn has an edge disjoint decomposition into any given sequence T1, T2, . . . Tn−1 of trees. Gerbner, Keszegh and Palmer extended the conjecture by replacing Kn with an arbitrary n-chromatic graph (Conjecture 2 in [1]). Here we show that the extended conjecture follows from the original one. We need the following, perhaps folklore result (Theorem 1 in [4]) and for convenience we include its simple proof.

On the Tree Packing Conjecture

The Gyárfás tree packing conjecture states that any set of $n-1$ trees $T_{1},T_{2},\dots, T_{n-1}$ such that $T_i$ has $n-i+1$ vertices packs into $K_n$ (for $n$ large enough). We show that $t=\frac{1}{10}n^{1/4}$ trees $T_1,T_2,\dots, T_t$ such that $T_i$ has $n-i+1$ vertices packs into $K_{n+1}$ (for $n$ large enough). We also prove that any set of $t=\frac{1}{10}n^{1/4}$ trees $T_1,T_2,\dots, T_t$ such that no tree is a star and $T_i$ has $n-i+1$ vertices packs into $K_{n}$ (for $n$ large enough). Finally, we prove that $t=\frac{1}{4}n^{1/3}$ trees $T_1,T_2,\dots, T_t$ such that $T_i$ has $n-i+1$ vertices packs into $K_n$ as long as each tree has maximum degree at least $2n^{2/3}$ (for $n$ large enough). One of the main tools used in the paper is the famous spanning tree embedding theorem of Komlós, Sárközy, and Szemerédi [Combin. Probab. Comput., 10 (2001), pp. 397--416].

Generalizations of the tree packing conjecture

The Gyarfas tree packing conjecture asserts that any set of trees with 2,3,...,k vertices has an (edge-disjoint) packing into the complete graph on k vertices. Gyarfas and Lehel proved that the conjecture holds in some special cases. We address the problem of packing trees into k-chromatic graphs. In particular, we prove that if all but three of the trees are stars then they have a packing into any k-chromatic graph. We also consider several other generalizations of the conjecture.

Packing almost stars into the complete graph

We verify that the Tree Packing Conjecture is true for all sequences of trees T1, …, Tn such that there exists χi E V(Ti) and Ti − χi has at least isolated vertices. © 1997 John Wiley & Sons, Inc. J Graph Theory 25: 169–172, 1997

Two Packing Problems on k-Matroid Trees

In this paper we consider two packing problems on k -matroid trees (a generalization of trees from graphs to hypergraphs). The first problem is to decompose the complete k -uniform hypergraph into k -matroid trees and the second problem is to generalize the famous Tree Packing Conjecture to k -uniform hypergraphs.