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1. Introduction


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PETER ALLEN, JULIA BÖTTCHER, JAN HLADKÝ, AND DIANA PIGUET We prove that if G is a family of graphs with at most n vertices each, with constant degeneracy, with maximum degree at most O (n/ log n), and with total number of edges at most (1 − o(1)) n2 , then G packs into the complete graph Kn . This strengthens recent results of BöttcherHladkýPiguetTaraz, MessutiRödlSchacht, FerberLeeMousset, KimKühnOsthus Tyomkyn, and FerberSamotij related to the Tree Packing Conjecture. In this extended abstract we describe the main steps of our proof.


1. Introduction A packing of a family G = {G1 , . . . , Gk } of graphs into a host graph H is a colouring of the edges of H with the colours 0, 1, . . . , k such that the edges of colour i form an isomorphic copy of Gi for each 1 ≤ i ≤ k . Graph packing problems can be considered as a common generalisation of a number of the

most important lines of investigation in Extremal Graph Theory: Turán-type problems, Dirac-type problems, and Ramsey-type problems. The focus of this research announcement is on packings of large connected graphs that either exhaust all (so-called perfect packings ) or almost all the edges of the host graph H (so-called nearperfect packings). Historically the rst and still the most famous problems in this direction concern the packing of trees. In 1963 Ringel [9] conjectured that if T is any n + 1-vertex tree, then 2n + 1 copies of T pack into Kn , and in 1976 Gyárfás [5] proposed the Tree Packing Conjecture, stating that, if Ti is an i-vertex tree  P for each n1≤ i ≤ n, then {T1 , . . . , Tn } packs into Kn . Since we have (2n + 1) · e(T ) = n2 and e(Ti ) = 2 , both conjectures ask for perfect packings. Despite many partial results (which mostly deal with very restricted classes of trees) both these problems were wide open until quite recently. The rst near-perfect packing result in the direction of these packing conjectures for trees was obtained by Böttcher, Hladký, Piguet and Taraz [1], who showed that one can pack into Kn any family of trees whose maximum degree is at most ∆, whose order is at most (1 − δ)n, and whose total number of edges is at most (1 − δ) n2 , provided that n is suciently large given the constants ∆ ∈ N and δ > 0. This approximately answers both Ringel's Conjecture and the Tree Packing Conjecture for bounded degree trees. Various generalisations of this result were obtained in quick succession. Messuti, Rödl and Schacht [8] showed that one can replace trees with graphs from any nontrivial minor-closed family (satisfying all other conditions). Then, Ferber, Lee and Mousset [2] improved on this result by allowing the graphs to be packed to be spanning (but still only providing a near-perfect packing). Kim, Kühn, Osthus and Tyomkyn [7] proved a near-perfect packing result for families of graphs with bounded maximum degree which are otherwise unrestricted. Joos, Kim, Kühn and Osthus [6] solved both Ringel's conjecture and the Tree Packing conjecture for trees of bounded maximum degree. Relaxing the restriction on the maximum degree, Ferber and Samotij [3] gave two near-perfect packing results for trees, one for spanning trees, and one for  almost spanning  trees. In these results, they allow the maximum degrees to be as big O n1/6 / log6 n , and O n/ log n , respectively.

The research leading to these results has received funding from the People Programme (Marie Curie Actions) of the European Union's Seventh Framework Programme (FP7/2007-2013) under REA grant agreement number 628974. Hladký and Piguet were supported by the Czech Science Foundation, grant number GJ16-07822Y. 1



Our new result is a near-perfect packing theorem in the complete graph for spanning graphs with  bounded degeneracy and maximum degrees up to O n/ log n , extending the mentioned packing results in [1, 2, 3, 7, 8].1 For a graph G, a linear ordering of its vertex set V (G) is D-degenerate if every vertex has at most D neighbours preceding it (we call these the left-neighbours ). The graph G is D-degenerate if V (G) has a D-degenerate ordering. We remark that such graphs G may be expanding and may have very high maximum degree. Theorem 1.

holds for each

For each γ > 0 and each D ∈ N there exist c n > n0 . Suppose that (Gt )t∈[t∗ ] is a family of

> 0 and n0 ∈ N such that the following D-degenerate graphs, each of which has

cn n vertices and maximum degree at most log n . Suppose further that the total number of edges  n of (Gt )t∈[t∗ ] is at most (1 − γ) 2 . Then (Gt )t∈[t∗ ] packs into Kn .

at most

In the remainder we sketch our proof of Theorem 1. In order to make the presentation of this sketch more accessible we shall simplify the description of our proof. In particular, the auxiliary results required to prove Theorem 1 are technically more involved than the ones stated here. 2. Outline of the main idea We shall call the host graph H ; it turns out that it is convenient not to restrict ourselves to H = Kn but to a more general setting of quasirandom graphs (dened below). The main idea of our proof is as follows. We shall rst select almost spanning subgraphs G0i of the graphs Gi in the given family. Then we shall use a random embedding process to embed the graphs G0i one by one edge-disjointly into the host graph H (deleting any edges of H that we used). We shall show that this random embedding process with high probability preserves three invariants: quasirandomness of the host graph, and the diet and cover conditions (dened in Section 3). Further, we prove that, as long as these invariants are satised, the random embedding process can successfully continue. Finally, we complete the packing of the almost spanning G0i to a packing of the Gi by using a matching argument.  An n-vertex graph with p n2 edges is (ε, ∆)-quasirandom if the common neighbourhood N(S) of any set S of at most ∆ vertices has size (1 ± ε)p|S| n. We remark that this notion of quasirandomness is slightly stronger than the standard notion, in which the neighbourhoods of most, rather than all, sets S are controlled.  We now state formally underwhat conditions we can pack the almost-spanning graphs G0t . For the promised completion to Gt we actually need a slightly stronger statement, which we will sketch later; however the proof of Theorem 2 already contains all the diculties. Theorem 2.

holds for each

For each ν > 0 and each D ∈ N there exist c, ε > 0 and n0 ∈ N such that the following n > n0 . Suppose that (G0t )t∈[t∗ ] is a family of t∗ ≤ 2n many D-degenerate graphs,

cn (1 − ν)n vertices and maximum degree at most log n . Suppose that H is a (ε, 2D + 3)-quasirandom graph of order n. Suppose further that the total number of edges of (G0t )t∈[t∗ ]  n is at most e(H) − ν 2 . Then (G0t )t∈[t∗ ] packs into H .

each of which has at most

We outline the proof of Theorem 2 in Section 3. Let us now explain why in Theorem 2 we need to pack not into Kn but into any quasirandom subgraph of Kn . Suppose that (Gt )t∈[t∗ ] is a family as in Theorem 1. We rst deal with the possibility t∗ > 2n by modifying the family. We remove any isolated vertices from all graphs in the family, and so we obtain v(Gt ) ≤ 2e(Gt ) for each t ∈ [t∗ ]. Now, given Gt1 and Gt2 both of which have at most n/4 edges and hence at most n/2 vertices, we merge them into a single graph Gt1 t Gt2 with at most n vertices. Repeating this procedure until no further merging is possible, we end up with t∗ graphs each having at least n/4 edges; since the total number of edges in the family is at most n2 we have t∗ ≤ 2n, as is required in Theorem 2. Any 1While our result extends these results in the setting of complete host graphs, the main focus of [3] is on packing

into random graphs, and [7] provides a general packing result in the setting of the Regularity lemma.



packing of the modied family (which we still call (Gt )t∈[t∗ ] ) trivially gives a packing of the original family. Next, we want to nd a subgraph G0t ⊆ Gt for each t ∈ [t∗ ] of order at most (1 − ν)n. Theorem 2  0 gives us a packing of the Gt ; we want to choose G0t in order to make it easy to extend this packing to a packing of the Gt . It turns out to be convenient to nd an independent set It in Gt , all of whose vertices have the same degree, and that degree should be at most 2D, and set G0t = Gt − It . We obtain It with the following simple lemma. Let G be a D -degenerate n-vertex graph. Then there exists an integer 0 ≤ d ≤ 2D and a set I ⊆ V (G) with |I| ≥ (2D + 1)−3 n which is independent, and all of whose vertices have the same degree d in G.

Lemma 3.

0 Now Theorem  2 gives a packing of the (Gt ) into any n-vertex suciently quasirandom graph H n with nearly 2 edges. To complete the derivation of Theorem 1 we need to explain how we choose H inside Kn and how we complete the packing of the (G0t ) to a packing of the (Gt ). We choose H by taking away a random subgraph from Kn , and we let H ∗ = Kn − H . We choose the number of edges that Theorem 2 applies, but small enough that H ∗ contains P in H large enough 0 much more than t∈[t∗ ] e(Gt ) − e(Gt ) edges. We apply Theorem 2 to pack the (G0t ) into H , and then for each t ∈ [t∗ ] we nd a way to complete the copy of G0t in H to a copy of Gt in Kn using edges of H ∗ . The vertices of Gt remaining to embed are an independent set It . Each vertex x ∈ It has d ≤ 2D neighbours y1 , . . . , yd in Gt , which are all in G0t and hence already embedded to vertices v1 , . . . , vd of Kn . Now we complete the embeddings of the Gt , starting with t = 1. For t = 1, we only allow embedding x to vertices in the candidate set

 C(x) := u ∈ Kn : u 6∈ im(G0t ), uv1 , . . . , uvd ∈ H ∗ ,

and we simply need to match the vertices of It to the vertices of Kn such that each x is matched to a vertex of C(x). To see that this matching exists, we need to verify Hall's condition. Part of the strengthening of Theorem 2 that we need to do this roughly states that the sets C(x) are distributed in a random-like fashion. It is straightforward to argue from this that Hall's condition holds. Since all vertices of It have d neighbours, all the sets C(x) have about the same size, which makes this argument easier. For t ≥ 2, of course when we want to complete the embedding of Gt we should not use edges of H ∗ which were used to complete any of G1 , . . . , Gt−1 , and the denition of C(x) must change accordingly. The other strengthening of Theorem 2 that we require is that the vertices adjacent to those in It are embedded to sets distributed in a random-like fashion. This means that during the entire packing process we will use only a few edges of H ∗ at each vertex, and the verication of Hall's condition is robust enough to allow for such a change. 3. Proof of Theorem 2 In this section we outline the proof of Theorem 2. We will not explain how to obtain the strengthenings sketched in the previous section that we need for Theorem 1. However, this strengthening turns out not to require any fundamentally dierent ideas. The vertices of the graphs G0t will be always the rst v(G0t ) natural numbers, in a degeneracy order. We proceed by packing the graphs G01 , . . . , G0t∗ one by one in this order and call the randomised algorithm which embeds the graph G0t RandomEmbedding. The graphs H =: H0 ⊃ H1 ⊃ . . . ⊃ Ht∗ record the host graph edges remaining throughout the process. At a given stage t = 1, . . . , t∗ , we proceed as follows. We need to embed the graph G0t into Ht−1 . We embed the vertex 1 into Ht−1 uniformly at random. Having embedded vertices 1, . . . , j − 1 of G0t to Ht−1 , we need to embed the vertex j . We simply pick a valid choice uniformly at random. In other words, we choose uniformly an image for j from the set of vertices x ∈ V (Ht−1 ) to which we have not embedded any vertex 1, . . . , j of G0t , and which are adjacent to all of the embedded left-neighbours of j . If this set is ever empty then RandomEmbedding fails; if for each stage t ∈ [t∗ ] and j ∈ V (G0t ) it is not empty, then the sequence




RandomEmbedding s

gives an embedding of each G0t into Ht−1 , hence a packing of the G0t into H . Therefore we need to analyse the evolution of (Ht )t∈[t∗ ] and the run of RandomEmbedding at each stage t. In order to analyse the run of RandomEmbedding at stage t, we need Ht−1 to be very quasirandom; on the other hand, the graph Ht will be a little less quasirandom than Ht−1 . We set 

αx = C −1 exp

 C(x − 2n)  n

for some large constant C . The required quasirandomness for H = H0 is α0 ; note that this quantity does not depend on n. Our strategy is to prove that with high probability the sequence of RandomEmbedding s does not fail and each of the graphs Hi is (αi , 2D + 3)-quasirandom. The following two lemmas are key to the analysis. n

Suppose that an n-vertex graph H is (α, 2D + 3)-quasirandom with p 2 edges for some small α and p  α. The probability that RandomEmbedding fails when embedding a D -degenerate graph G of order at most (1 − ν)n into H is o(1/n). Lemma 4.

Suppose that we are in the setting described above Lemma 4. As the graphs G01 , G02 , . . . are embedded one by one, for each t the following holds. Provided that Hi is (αi , 2D + 3)-quasirandom for each 0 ≤ i < t and that RandomEmbedding does not fail before the end of stage t, the probability that Ht−1 fails to be (αt , 2D + 3)-quasirandom is o(1/n).

Lemma 5.

These two lemmas imply Theorem 2. Indeed, if some RandomEmbedding fails, then there must be a rst time t when either RandomEmbedding fails although Ht−1 is quasirandom, or RandomEmbedding succeeds but the resulting Ht s not quasirandom. Lemmas 4 and 5 respectively state that these two events have probability o(1/n); taking the union bound over times t, the probability of failure is o(1). 3.1. Sketch of the proof of Lemma 4. Recall that RandomEmbedding fails when it comes to a vertex j and there is no valid choice of an image for j . Since H is quasirandom, if j has d leftneighbours then about pd n vertices in H are adjacent to all these embedded left-neighbours, so failure can only occur if these vertices have been eaten up by the previous embeddings. We show that this is unlikely; in fact, we show that the following stronger diet condition for each t ∈ V (H) is likely to hold: (3.1) For each S ⊆ V (H) of size at most 2D +3, we have that |N(S)\im(G[1, . . . , t])| ≈ p|S| (n−t). We x S and aim to show that S is very unlikely to be a set which witnesses the diet condition failing at the rst time (since such an S must exist if the diet condition ever fails). In other words, assuming the diet condition holds up to time t − 1, we want to show that the sum t X

1 i is embedded to N(S)


is likely to be about p|S| t. If these Bernoulli random variables were independent, Hoeding's inequality would tell us that the sum is very likely to be close to its expectation. They are not independent, but nevertheless a martingale version of Hoeding's inequality shows that the sum is likely to be close to the sum of conditional expectations (3.2)

t t X X    X  E 1 i is embedded to N(S) Hi−1 = P i is embedded to w Hi−1 i=1

i=1 w∈N(S)

where Hi−1 denotes the history, that is, the choices for embedding vertices 1, . . . , i − 1, and the equality is by linearity of expectation. This sum is itself a random variable, but it turns out to be easier to control. To avoid a technical complication, let us pretend that each i has exactly d leftneighbours. Letting κ be a very small constant, for i in the interval j + 1, . . . , j + κn there is a chance to embed i to w each time w is in the candidate set of i; that is, each time that w is adjacent to all



the embedded neighbours of i. The following cover condition states that this happens about as often as one would expect: d (3.3) For each j ∈ V (G) and w ∈ V (H), there are about p κn vertices among [j+1, j+κn] ⊆ V (G) which contain w in their candidate set.

For each i in the interval j + 1, . . . , j + κn whose candidate set contains w 6∈ im(G[1, . . . , i − 1]), the vertex i is embedded to a set of size about pd (n − i + 1) ≈ pd (n − j) by the diet condition, where the approximation is since κ is very small. So the probability of embedding i to w is about p−d (n − j)−1 . On the other hand, by the diet condition we have N(S) \ im(G[1, . . . , i − 1]) ≈ p|S| (n − j), which gives the number of vertices w contributing to the sum (3.2). Summing up, if the cover condition holds then the interval j + 1, . . . , j + κn contributes about p|S| κn to the sum (3.2). So the cover condition holding implies that the whole sum (3.2) comes to about p|S| t, as desired. This means that, provided the cover condition did not yet fail, the diet condition is unlikely to fail. We sketch why the cover condition is likely to hold provided the diet condition has not yet failed. When we embed a vertex i, provided the diet condition did not yet fail, the probability of embedding it to a neighbour of w is about p. Now a similar application of a martingale Hoeding inequality shows that the probability of a given w and j witnessing the failure of the cover condition, given that the diet condition did not yet fail, is very small. Consider the rst time at which one of the cover and diet conditions fails. Before this time both hold, so the probability that t is that rst time is by the above argument very small. Taking the union bound over t we conclude that with high probability no such rst time exists, and therefore RandomEmbedding succeeds, as desired. 3.2. Idea of the proof of Lemma 5. For lack of space, we will not explain any details of the proof of Lemma 5. We use similar ideas of showing that sums of dependent random variables concentrate using martingale inequalities. The interesting feature is that, because we allow the G0t to have vertices of very high degree but (because the G0t are D-degenerate) these must be very few, this time the random variables we are summing (such as the number of edges removed at a vertex of Ht−1 to form Ht ) have maximum values vastly larger than the expected value. In this situation Hoeding-type inequalities perform very poorly. We use Freedman's inequality [4] to obtain the desired concentration: this inequality performs better when the variance of each random variable is much smaller than its maximum value, and gives us the concentration we need.

References [1] J. Böttcher, J. Hladký, D. Piguet, and A. Taraz. An approximate version of the tree packing conjecture. Israel J. Math., 211(1):391446, 2016. [2] A. Ferber, C. Lee, and F. Mousset. Packing spanning graphs from separable families. arXiv:1512.08701, to appear in Israel J. Math. [3] A. Ferber and W. Samotij. Packing trees of unbounded degrees in random graphs. arXiv:1607.07342. [4] D. A. Freedman. On tail probabilities for martingales. Ann. Probability, 3:100118, 1975. [5] A. Gyárfás and J. Lehel. Packing trees of dierent order into Kn . In Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), volume 18 of Colloq. Math. Soc. János Bolyai, pages 463469. North-Holland, Amsterdam, 1978. [6] F. Joos, J. Kim, D. Kühn, and D. Osthus. Optimal packings of bounded degree trees. arXiv:1606.03953. [7] J. Kim, D. Kühn, D. Osthus, and M. Tyomkyn. A blow-up lemma for approximate decompositions. arXiv:1604.07282. [8] S. Messuti, V. Rödl, and M. Schacht. Packing minor-closed families of graphs into complete graphs. J. Combin. Theory Ser. B, 119:245265, 2016. [9] G. Ringel. Problem 25. In Theory of Graphs and its Applications (Proc. Int. Symp. Smolenice 1963). Czech. Acad. Sci., Prague, 1963.



Department of Mathematics, London School of Economics, Houghton Street, London, WC2A 2AE, UK

E-mail address :

{p.d.allen, j.boettcher}@lse.ac.uk

Fachbereich Mathematik, Institut für Geometrie, TU Dresden, 01062 Dresden, Germany.

E-mail address :

[email protected]

Institute of Computer Science, Czech Academy of Sciences, Pod Vodárenskou v¥ºí 2, 182 07 Prague, Czech Republic. With institutional support RVO:67985807.

E-mail address :

[email protected]