Tree decompositions of graphs without large bipartite holes
A recent result of Condon, Kim, Kühn and Osthus implies that for any , an -vertex almost -regular graph has an approximate decomposition into any collections of -vertex bounded degree trees. In this paper, we prove that a similar result holds for an almost -regular graph with any and a collection of bounded degree trees on at most vertices if does not contain large bipartite holes. This result is sharp in the sense that it is necessary to exclude large bipartite holes and we cannot hope for an approximate decomposition into -vertex trees.
Moreover, this implies that for any and an -vertex almost -regular graph , with high probability, the randomly perturbed graph has an approximate decomposition into all collections of bounded degree trees of size at most simultaneously. This is the first result considering an approximate decomposition problem in the context of Ramsey-Turán theory and the randomly perturbed graph model.
Finding sufficient conditions for the existence of a subgraph of isomorphic to a specific graph is a central theme in extremal graph theory. The earliest results of this type are Mantel’s theorem  and Turán’s theorem  stating that an -vertex graph contains a complete graph on vertices whenever contains at least edges. Erdős-Stone-Simonovits theorem [17, 18] further generalises this into any small graph .
On the other hand, the nature of problems changes if we consider a ‘large’ graph whose number of vertices is comparable (or equal) to that of . One important cornerstone in this direction is Dirac’s theorem  which shows that whenever we have , the -vertex graph contains a Hamilton cycle. Komlós, Sárközy and Szemerédi  proved that the condition of ensures the containment of every -vertex bounded degree tree as a subgraph, and in , they extended this result to the trees with maximum degree . Furthermore, Böttcher, Schacht and Taraz  found a minimum degree condition guaranteeing the containment of an -vertex graph with sublinear bandwidth and bounded maximum degree.
Another important research direction in extremal graph theory concerns with decomposition of graphs. We say that a collection of graphs packs into if contains pairwise edge-disjoint copies of as a subgraph. If packs into and (where ), then we say that the graph has a decomposition into . If a packing covers almost all edges of the host graph , then we informally say that has an approximate decomposition. The history of graph decomposition problems dates back to 19th century when Kirkman characterised all such that decomposes into triangles and when Walecki characterised all such that decomposes into Hamilton cycles. The latter was extended to Hamilton decompositions of regular graphs of high degree in a seminal work of Csaba, Kühn, Lo, Osthus and Treglown . Yet another generalisation, the famous Oberwolfach conjecture states that for any -vertex graph consisting of vertex-disjoint cycles, has a decomposition into , except a finitely many values of . After many partial results, this was finally resolved very recently for all large by Glock, Joos, Kim, Kühn and Osthus .
Further famous open problems in the area are the tree packing conjecture of Gyárfás and Lehel, which says that for any collection of trees with , the complete graph has a decomposition into , and Ringel’s conjecture which says that for any -vertex tree , the complete graph has a decomposition into copies of . Lots of research has been done regarding these conjectures, [9, 19, 26, 35]. Recently, Joos, Kim, Kühn and Osthus  proved both conjectures for trees with bounded degree and larger . A key ingredient of their proof is a blow-up lemma for approximate decompositions of -regular graphs developed by Kim, Kühn, Osthus and Tyomkyn . Allen, Böttcher, Hladkỳ and Piguet  later proved an approximate decomposition result for degenerate graphs with maximum degree . Montgomery, Pokrovskiy and Sudakov  found an approximate decomposition of into any -vertex tree , proving an approximate version of Ringel’s conjecture.
In , Condon, Kim, Kühn and Osthus determined the degree threshold for an almost regular graph to have an approximate decomposition into a collection of separable graphs with bounded degree. In particular, one corollary of their result is that for any collection of -vertex bounded degree trees, any almost-regular -vertex graph with degree at least has an approximate decomposition into .
Most of the aforementioned results are sharp as there are graphs which do not satisfy the conditions and do not have a desired subgraph or a desired (approximate)-decomposition. For example, regarding the corollary on approximate tree decomposition, a complete balanced bipartite graph or disjoint union of two complete graphs shows that the degrees of has to be at least to contain a single copy of an -vertex tree with unbalanced bipartition, let alone an approximate decomposition. However, such examples have very special structures. Hence it is natural to ask how the degree conditions change if we exclude graphs with such special structures.
Another active line of research is to study these changes on the degree conditions when we exclude a large independent set. Balogh, Molla and Sharifzadeh  initiated this by proving that if an -vertex does not contain any linear-sized independent set and , then contains a triangle-factor. This weakens the bound from the Corrádi-Hajnal theorem . Nenadov and Pehova  further generalised this into a -factor.
However, excluding large independent sets is not sufficient to guarantee a large connected subgraph, e.g. does not contain an independent set of size three, and clearly it does not contain any tree with more than vertices. This example suggests that it is necessary to exclude large bipartite holes, rather than independent sets. An -bipartite hole in a graph consists of two disjoint vertex sets with such that there are no edges between and in . The bi-independence number of a graph denotes the largest number such that contains an -bipartite hole for every pair of non-negative integers and with . Note that implies that there is at least one edge between any two disjoint vertex sets of size , i.e. . McDiarmid and Yolov  proved the existence of Hamilton cycle on a graph satisfying .
Our main theorem states that if has sublinear bi-independence number and consists of bounded degree trees with at most vertices, then the degree threshold for an approximate tree-decomposition of Condon, Kim, Kühn and Osthus can be significantly lowered. There is an obvious analogy between this theme and the Ramsey-Turán theory in which one studies Turán type problmes for graphs with sublinear independence number. See e.g.  for more of Ramsey-Turán theory. Here we replace a Turán-type conclusion with one along the lines of approximate decomposition of into large graphs.
For all , , there exist and such that the following holds for all . Suppose that is an -vertex graph such that for all vertices except at most vertices and . Then any collection of trees satisfying the following conditions packs into .
and for all
Note that by considering a collection of paths of length , it is easy to see that the almost regular degree condition on is necessary. Theorem 1.1 is sharp in several point of views. First, the condition on is necessary as embedding even a single copy of -vertex tree into is impossible. Hence, is the correct parameter to consider. Second, the trees in having at most vertices is also best possible. To see this, we consider a copy of slightly unbalanced complete bipartite graph with parts of size and of size . We put a copy of random graphs in and , respectively. Let be the resulting graph and let be a collection of copies of -vertex paths. Then satisfies all the conditions in Theorem 1.1 except that the trees are now spanning. Even more, it satisfies a stronger condition that and all vertices in has degree . However, as each path has the unique bipartition which is almost balanced, each path uses at least edges inside the bigger part . Thus, we need at least edges inside the bigger part in order to pack into . Since only contains at most edges, does not pack into if .
As the last example contains two vertices with degree difference at least , one might speculate that it is plausible to obtain a packing of spanning trees into if one additionally assume that is much closer to being regular. However, the following example shows that we still need more conditions. Consider a graph obtained from by putting a copy of in each part , respectively. It is easy to see that . Let be the collection of copies of -vertex complete ternary tree of height . Csaba, Levitt, Nagy-Győrgy and Szemerédi  showed that any embedding of such complete ternary tree must use at least non-crossing edges inside parts of . Thus, we need at least non-crossing edges to obtain a packing of into . However, contains at most non-crossing edges. Hence, this shows that it is necessary that the trees in have at most vertices. It is not difficult to modify the above example to obtain a regular graph (rather than just close to being regular) with which does not admit an approximate decomposition into complete ternary trees.
Our theorem has a corollary in randomly perturbed graph model which combines extremal and probabilistic aspects in one graph model. Bohman, Frieze and Martin  introduced the concept of randomly perturbed graph model by proving that given any fixed there exists a constant such that for any -vertex graph with , the graph contains a Hamilton cycle with high probability. This sparks numerous research see e.g. [4, 5, 6, 8, 10, 21, 24, 29, 30, 31, 34].
The following corollary is a direct consequence of Theorem 1.1. It is easy to see that for large constant , the random graph does not contain a -bipartite hole, hence .
For all , , there exist and such that the following holds for all and . Suppose that is an -vertex graph such that for all vertices except at most vertices and is a binomial random graph on the vertex set . Then the following holds with high probability. Any collection of trees satisfying the following conditions packs into .
and for all
Note that the above statement is universal in the sense that with high probability, this holds for every collection simultaneously. Corollary 1.2 is sharp in the following senses. By considering a disconnected graph , it is easy to see that the probability is best possible. Also the trees having size is best possible. The above first example obtained from slightly unbalanced complete bipartite graph show that we need in order to obtain an approximate decomposition of almost -regular graphs into spanning trees with bounded maximum degree, and the second example with complete ternary trees shows that at least is required for obtaining an approximate decomposition of regular graphs into spanning trees. Motivated by this second example, we ask the following question.
Determine the optimal function satisfying the following. For given , if is an -vertex -regular graph and . Let be a collection trees satisfying the following conditions.
and for all
Then pack into .
We can consider the same question of finding the optimal function by replacing the graph with and an arbitrary -vertex -regular graph . Since the regularity lemma does not distinguish between an -regular graph and an -regular graph, the example we obtained from shows that Question 1.3 may not be proved by the approach in this paper which is based on the regularity lemma.
Our theorem also has further applications on tree packing conjectures, such as Ringel’s conjecture in the setting of almost regular graphs. It implies that if and is an almost -regular -vertex graph with , and is an -vertex tree with bounded maximum degree, then has an approximate decomposition into copies of . Same statement also holds for with any almost -regular -vertex graph .
Denote If we claim that a result holds for , this mean that there exist non-decreasing functions and such that the result holds for all and all with and . We may omit floors and ceilings when they are not essential. In this paper, graphs are simple undirected finite graphs and multigraphs are graphs with potentially parallel edges without loops.
Given collection of trees , denote by the number of trees in and . Let be a graph and satisfying . Denote by the set of edges in between and . Let . For sets , we define . In particular, we have , and let . Let . We write for . Denoted by the set of vertices of distance at most from a vertex in a set . Note that, in this definition, and are in general different for . For a tree and a vertex , let be the unique vertex partition into two independent sets satisfying . Denote by the induced subgraph on , and by the spanning subgraph with edge set , where and . For a graph and two disjoint vertex subsets and , the density of is defined as
For a rooted tree with the root , let be the subtree of consisting of all vertices such that the path between and contains . For a vertex , denoted by the parent of . Denoted by the set of all descendents of with distance exactly in the tree , and by be the set of descendents of with distance at most . We write . For two functions and with , we define as a function from to such that for each ,
We will use well-known Chernoff’s inequality and Azuma’s inequality. As our applications are very simple and standard, we will omit the detailed computation. See [2, 22, 23] for the statements of Chernoff’s inequality and Azuma’s inequality. The concept of -regularity and Szemerédi’s regularity lemma will be useful for us. A bipartite graph with vertex partition is -regular if for all sets , with , , we have
A bipartite graph is -regular if is -regular for some . Additionally, a bipartite graph is -super-regular if is -regular with for and for . The following three well-known lemmas will be useful when we modify a given -regular partition.
Let . Suppose that is an -regular bipartite graph with vertex partition . Let be a set with . Then contains at least vertices satisfying .
Let . Suppose that is an -regular bipartite graph with vertex partition . Then, there exists sets with and such that is an -super-regular bipartite graph with vertex partition .
Let . Suppose that is an -regular bipartite graph with vertex partition . Let be a set of edges with . Then is -regular.
The following two lemmas will be useful for finding some edge/vertex partition of graphs. We omit the proofs as they easily follow from a standard random splitting argument.
Let . Suppose that is an -regular bipartite graph with vertex partition satisfying . Let be values such that . Then there exist edge-disjoint spanning subgraphs of such that is -regular for each .
Let . Suppose that is an -super-regular bipartite graph with vertex partition satisfying . Let be numbers such that , , and . Then there exists a partition of and of such that for any , we have and the graph is -super-regular.
The following is a version of well-known Szemerédi’s regularity lemma.
Lemma 2.6 (Szemerédi’s regularity lemma).
Let and . Then for any -vertex graph , there exists a partition of into and a spanning subgraph satisfying the following:
For all with , the graph is either empty or -regular for some .
The following two lemmas will be useful to utilise the assumption on bi-independence number.
Let . Suppose is an -vertex graph with . If and are two (not necessarily disjoint) subsets of with size at least , then all but at most vertices in satisfies .
Suppose that the lemma does not hold, then there exists a set of exactly vertices satisfying . Then the set contains at least vertices. Hence , contradicting . This proves the lemma. ∎
Let . Suppose that is an -vertex graph with . Then there exists a spanning subgraph of with and .
For each edge of , we include it in independently at random with probability . A standard application of Chernoff’s inequality implies that, with probability at least , we have for all . We consider two disjoint sets with . By lemma 2.7, we have
A standard application of Chernoff’s inequality implies that with probability at least , we have . By a union bound, with probability at least , we have that for all disjoint sets with . This implies that . Hence, with probability at least , has the desired properties. ∎
The following proposition from  provides a useful partition of a tree.
 Let and . Then for any rooted tree on vertices with , there exists a collection of pairwise vertex-disjoint rooted subtrees such that the following holds.
 Let and . Suppose that with . Let be an -vertex balanced complete bipartite graph with the vertex partition and . Suppose that the graph is a subgraph of with the vertex partition and such that for each . Suppose that we have and sets satisfies for all and . Then there exists a -regular spanning subgraph of and a function which packs into such that and for distinct and .
Theorem 2.11 (Blow-up lemma for approximate decompositions ).
Let and . Let be a number such that . Suppose that the following properties hold.
is a -super-regular graph with the vertex partition and .
, where each is an -regular bipartite graph with the vertex partition and .
For all and there is a set with and for each , there is a set with
is a graph with and such that for each and , we have Moreover, for each and , we have
Then there exists a function packing into such that for all and
for all .
For all we have
3. Proof of Theorem 1.1
In this section, we prove our main theorem assuming the following lemma which will be proved in Section 4. This lemma states that if admits a certain -super-regularity partition and , then we can find an approximate decomposition of into arbitrary bounded degree -vertex trees. Here, consists of sets and which form an -super-regular matching structure. The reduced graph for this -regular partition is not connected, but the condition on provides a connection necessary to embed trees.
Suppose . Let be a graph with a vertex partition and let be a partition of and be a partition of . Let be a collection of trees on at most vertices with maximum degree at most . Assume that the following properties hold.
For each , each of four graphs , , and are -super-regular with and .
Then there exists a map packing into such that for each .
We start the proof of Theorem 1.1. For given and , we choose constants such that
Let . By deleting exactly vertices with degree furthest from , we can assume that is an -vertex graph such that every vertex satisfies . By Lemma 2.8, we can find a spanning subgraph of with and . By replacing with , assume that and are edge-disjoint graphs and
Suppose that is a collection of trees satisfying and . Now we aim to construct (not disjoint) sets and edge-disjoint subgraphs of . We will also partition into subcollections of trees , and pack the trees of into
Step 1. Partitioning . First, we will partition into graphs with appropriate structure which are suitable for applications of Lemma 3.1. We apply Szemerédi’s regularity lemma (Lemma 2.6) with playing the role of to obtain a partition of and a spanning subgraph satisfying the following.
for any , the graph is either empty or -regular for some .
Let be a reduced graph with
As if and only if is -regular with , for each , we have
Now we will find edge-disjoint subgraphs of each of which admits -regular matching structure. For each , letting , we use Lemma 2.4 with playing the roles of , respectively, to obtain edge-disjoint subgraphs of . For each and ,
We will take an appropriate unions of these graphs to form -regular matching structures. Let be a multi-graph obtained by replacing each edge of with edges between the vertices and . Let be a map from to such that . For each , we have
By applying Vizing’s theorem to , we obtain edge-disjoint (possibly empty) matchings covering all edges of . By (3.5) and the pigeonhole principle, at least matchings contain at least edges. Let be edge-disjoint matchings of of size at least , thus for each ,
For , we write if contains one of . For each , we define a graph with
For each and , apply Proposition 2.2 to obtain sets and