Extremal density for sparse minors and subdivisions

We prove an asymptotically tight bound on the extremal density guaranteeing subdivisions of bounded-degree bipartite graphs with a mild separability condition. As corollaries, we answer several questions of Reed and Wood on embedding sparse minors. Among others, $\bullet$ $(1+o(1))t^2$ average degree is sufficient to force the $t\times t$ grid as a topological minor; $\bullet$ $(3/2+o(1))t$ average degree forces every $t$-vertex planar graph as a minor, and the constant $3/2$ is optimal, furthermore, surprisingly, the value is the same for $t$-vertex graphs embeddable on any fixed surface; $\bullet$ a universal bound of $(2+o(1))t$ on average degree forcing every $t$-vertex graph in any nontrivial minor-closed family as a minor, and the constant 2 is best possible by considering graphs with given treewidth.


Introduction
Classical extremal graph theory studies sufficient conditions forcing the appearance of substructures. A seminal result of this type is the Erdős-Stone-Simonovits theorem [11,10], determining the asymptotics of the average degree needed for subgraph containment. It reads as The study of such problem has a long history. An initial motivation was Hadwiger's conjecture that every graph of chromatic number t has K t as a minor, which is a far-reaching generalisation of the four-colour theorem. Since every graph of chromatic number k contains a subgraph of average degree at least k − 1, a natural angle of attack is to find bounds on the average degree which will guarantee a K t -minor. The first upper bound for general t was given by Mader [21,22], who subsequently improved this bound to O(t log t). In celebrated work of Kostochka [19] and, independently, Thomason [37], it was improved to the best possible bound O(t √ log t), Thomason subsequently determining the asymptotic [38]. Denoting he proved that d ≻ (K t ) = (α + o t (1))t √ log t, where α = 0.6382 . . . is explicitly defined. This remained the best order of magnitude bound even for the chromatic number question until very recent breakthrough by Norin, Postle and Song [28] and by Postle [29].

Main result
We study in this paper the problem of finding subdivisions of a natural class of sparse bipartite graphs. In particular, our main result offers the asymptotics of the average degree needed to force subdivisions of such graphs, showing that a necessary bound is already sufficient. It reads as follows.
Theorem 1.1. For given ε > 0 and ∆ ∈ N, there exist α 0 and d 0 satisfying the following for all 0 < α < α 0 and d ≥ d 0 . If H is an α-separable bipartite graph with at most (1 − ε)d vertices and ∆(H) ≤ ∆, and G is a graph with average degree at least d, then G contains a subdivision of H.
Here a graph H is α-separable if there exists a set S of at most α|H| vertices such that every component of H − S has at most α|H| vertices. Graphs in many well-known classes are o(1)-separable. For example, any large graphs in nontrivial minor-closed family of graphs is o(1)-separable [1,23].
As an immediate corollary, our main result answers the above question of Reed and Wood in a strong sense by showing that any β > 1 is sufficient to force the k-dimensional grid G k t,...,t not only as a minor but as a topological minor, and so d T (G k t,...,t ) = d ≻ (G k t,...,t ) = (1 + o t (1))t k .
We remark that the optimal constant 1 in Theorem 1.1 is no longer sufficient if H is not bipartite. Indeed, if e.g. H is the disjoint union of triangles, then the Corrádi-Hajnal theorem [5] implies that d ≻ (H) = 4 3 |H| − 2. We shall elaborate more on the leading constant in the next section.
Our proof utilises both pseudorandomness from Szemerédi's regularity lemma and expansions for sparse graphs. The particular expander that we shall make use of is an extension of the one introduced by Komlós and Szemerédi, which has played an important role in some recent developments on sparse graph embedding problems, see e.g. [20,13].
Organisation. Applications of our main result will be given in Section 2. Preliminaries on expanders and basic building blocks are given in Section 3. In Section 4, we outline the strategies for proving Theorem 1.1. Its proof is then given in Sections 5-7. Lastly, concluding remarks and some open problems are given in Section 8.
k > ∆, replace it with two adjacent vertices of degrees k − (∆ − 1) and ∆; when doing this, allocate the neighbours of the original vertex to the two vertices in such a way as to preserve the genus. Continue until no vertices with degree bigger than ∆ remain. Thus we obtain a graph H ′ of genus at most g with at most t + v∈V (H)
If g = 0 (i.e. H ′ is planar) then take a four-colouring of H ′ , and set aside the two largest colour classes A and B, which cover at least (1 + ε/3)t/2 vertices between them. If g > 0, using a result of Djidjev [7] we may find a planar induced subgraph on |H ′ |−O( |H ′ |) vertices. Thus, by extending a four-colouring of this subgraph, we may colour H ′ in such a way that the two largest colour classes, A and B, cover at least |H ′ |/2 − o(t) vertices. So, in either case we have |V (H ′ ) \ (A ∪ B)| ≤ (1 + ε)t/2 for t sufficiently large.
We may assume that every other vertex has neighbours in both A and B, since otherwise we could use a different colouring for which the total size of A ∪ B is larger. We define a bipartite graph H ′′ as follows. Vertices in A ∪ B, and edges between them, are unchanged. The genus of H ′′ may be greater than g, but the bounded genus of H ′ ensures that H ′ is α/2separable for t sufficiently large. Since any subset of V (H ′ ) of size at most α|H ′ |/2 corresponds to a subset of V (H ′′ ) of size at most α|H ′′ |, H ′′ is α-separable. Now H ′′ is a bipartite graph with at most (3/2 + ε)t vertices, so by Theorem 1.1 we can find a subdivision of H ′′ , which in turn contains H as a minor, in any graph with average degree at least (3/2 + ε)t provided t is sufficiently large.

A universal bound for nontrivial minor-closed families
Many important classes of graphs are naturally closed under taking minors, for example, graphs embeddable on a given surface considered in Theorem 2.1. The seminal graph minor theorem of Robertson and Seymour (proved in a sequence of papers culminating in [33]) shows that every minor-closed family can be characterised by a finite list of minimal forbidden minors. For example, the linklessly-embeddable graphs are defined by a minimal family of seven forbidden minors, including K 6 and the Petersen graph [36]. The existence of a forbidden-minor characterisation has far-reaching algorithmic implications, since for any fixed graph F there exists an algorithm to determine whether an n-vertex graph contains F as a minor in O(n 3 ) time [32], and hence there is a cubic-time algorithm (since improved to quadratic [12]) to check for membership of any given minor-closed family; prior to these results it was not even known that the property of having a linkless embedding was decidable. However, the constants concealed by the asymptotic notation are typically prohibitively large. Furthermore, for many families a complete forbidden minor classification, and hence a specific algorithm, is not known, and the number of minimal forbidden minors can be extremely large, even for families that may be very naturally and simply defined. For example, there are over 68,000,000,000 minimal forbidden minors for the class of Y ∆Y -reducible graphs [40].
We can extend the methods of the previous subsection to minor-closed families more generally. For each k ∈ N, define α k (G) := max{|U | : U ⊆ V (G), χ(G[U ]) = k}. So α 1 (G) is the usual independence number and α 2 (G) is the maximum size of the union of two independent sets. Theorem 2.2. Let F be a nontrivial minor-closed family. For each F ∈ F with t vertices, we have Proof. It is well known that the t-vertex graphs in F are o(1)-separable with at most C F t edges for some constant C F [1].
To prove the upper bound, take two disjoint independent sets A and B with α 2 (F ) = |A∪B|. By the same argument as in Theorem 2.1, we can define a bipartite graph F ′′ containing F as a minor with 2t − α 2 (F ) + o(t) vertices having bounded maximum degree. Apply Theorem 1.1 to F ′′ to obtain the upper bound.
To prove the lower bound, consider K s,n−s where s = t − α(F ) − 1. If it contains an F -minor, let V 1 , . . . , V |F | be the vertex sets corresponding to the vertices of F . By our choice of s, it is easy to see that at least α(F )+ 1 of them have to completely reside in the independent set of size n − s in K s,n−s , which is impossible. Thus, K s,n−s does not contain an F -minor. By choosing large n, we have d( Theorem 2.2 yields the following universal bound for all nontrivial minor closed families. Corollary 2.3. For any nontrivial minor closed family F, we have We remark that the constant 2 above cannot be improved as we shall see in Corollary 2.4.

Graphs with given treewidth or clique minor
Reed and Wood asked the following questions for specific minor-closed families.
Problem B ( [30]). What is the least function g 1 such that every graph with average degree at least g 1 (k) · t contains every graph with t vertices and treewidth at most k as a minor?
Problem C ( [30]). What is the least function g 2 such that every graph with average degree at least g 2 (k) · t contains every K k -minor-free graph with t vertices as a minor?
Graphs with treewidth at most k are k-degenerate, and hence have at most kt edges, and graphs without a K k -minor have average degree O(k √ log k). Consequently the result of Reed and Wood [30] showed that g i (k) = O( √ log k) for i ∈ {1, 2}. As a corollary of Theorem 2.2, we get the following optimal bound of g i (k) = 2 + o k (1), showing that somewhat surprisingly, when k is sufficiently large, both the treewidth and the size of a largest clique minor play negligible roles in the leading coefficient.
Corollary 2.4. Every graph with average degree (2+o k (1))t contains every graph with t vertices, which either has treewidth at most k or is K k -minor-free, as a minor.
Proof. The upper bound follows immediately from Theorem 2.2. Note that a disjoint union of copies of K k+1 has treewidth k. Then the unbalanced complete bipartite graph K (1−1/(k+1))t−1,n provides the matching lower bound 2(1 − 1/(k + 1))t = (2 + o k (1))t. For graphs without K k -minor, consider instead a disjoint union of copies of K k−1 .

Beyond minor-closed classes
In Section 2.2, the two properties of minor-closed families that we needed were o(1)-separability and bounded average degree. Many other sparse graph classes have these properties. In particular, any class which obeys a strongly-sublinear separator theorem is o(1)-separable, see [23]. A k-shallow minor of a graph G is a minor for which each contracted subgraph has radius at most k. We say that a graph class C has bounded expansion if the average degree of k-shallow minors of graphs in C is bounded by a function of k; in particular, since 0-shallow minors are just subgraphs, C itself has bounded average degree. If the bound is a polynomial function, we say that C has polynomial expansion. These definitions were introduced by Nešetřil and Ossona de Mendez [26].
Classes of polynomial expansion have strongly-sublinear separator theorems, and for hereditary classes the two notions are equivalent [9]. Thus we may extend Theorem 2.2 to classes of polynomial expansion. Such classes include the 1-planar graphs, that is, the graphs which may be embedded in the plane with each edge crossing at most one other edge once [27], and intersection graphs of systems of balls with only a bounded number meeting any point [24].
Polynomial expansion is a much weaker property than being minor-closed. It is easy to see, for example, that any graph can be suitably subdivided to obtain a 1-planar graph. However, Borodin [3] showed that all 1-planar graphs are 6-colourable, and since they include disjoint unions of K 6 , we obtain the following tight result from the extension of Theorem 2.2 to classes of polynomial expansion.

Preliminaries
For n ∈ N, let [n] := {1, . . . , n}. Given a set X and k ∈ N, let X k the family of all k-sets in X. For brevity, we write v for a singleton set {v} and xy for a set of pairs {x, y}.
If we claim that a result holds whenever we have 0 < a ≪ b, c ≪ d < 1, it means that there exist positive functions f, g such that the result holds as long as a < f (b, c) and b < g(d) and c < g(d). We will not compute these functions explicitly. In many cases, we treat large numbers as if they are integers, by omitting floors and ceilings if it does not affect the argument. We write log for the base-e logarithm.

Graph notations
Given graphs H and G, in a copy of H-subdivision in G, we call the vertices that correspond to V (H) the anchor vertices of the subdivision. For a given path P = x 1 . . . x t , we write Int(P ) = {x 2 , . . . , x t−1 } to denote the set of its internal vertices. Given a graph H, a set of vertices S ⊆ V (H) and a subgraph F ⊆ H, denote by H − S = H[V (H) \ S] the subgraph induced on V (H) \ S and by H \ F the spanning subgraph obtained from H by removing edges in F . Given a graph G, denote its average degree 2e(G)/|G| by d(G). For two sets X, Y ⊆ V (G), the (graph) distance between them is the length of a shortest path from X to Y . For two graphs G, H, we write G ∪ H to denote the graph with vertex set V (G) ∪ V (H) and the edge set E(G) ∪ E(H). A k-star denotes a copy of K 1,k which is a star with k edges. Given a collection of graphs F = {F i : i ∈ I}, we write V (F) = i∈I V (F i ) and |F| = |I|. For path P and a vertex set U , we write P | U for the induced subgraph of P on vertex set V (P ) ∩ U .
For a set of vertices X ⊆ V (G) and i ∈ N, denote N i (X) := {u ∈ V (G) : the distance between X and u is exactly i} the i-th sphere around X, and write N 0 (X) = X, N (X) := N 1 (X), and for i ∈ N ∪ {0}, let B i (X) = i j=0 N j (X) be the ball of radius i around X. We write ∂(X) for the edge-boundary of X, that is, the set of edges between X and V (G) \ X in G. Given another set Z ⊆ V (G), we write N (X, Z) = N (X) ∩ Z for the set of neighbours of X in Z.

Robust expander
To define the robust graph expansion, we need the following function. For ε 1 > 0 and t > 0, let ρ(x) be the function where, when it is clear from context we will not write the dependency on ε 1 and t of ρ(x). Note that when x ≥ t/2, while ρ(x) is decreasing, ρ(x) · x is increasing.
Komlós and Szemerédi [17,18] introduced a notion of expander G in which any set X of reasonable size expands by a sublinear factor, that is, |N G (X)| ≥ ρ(|X|)|X|. We shall extend this notion to a robust one such that similar expansion occurs even after removing a relatively small set of edges.
We shall use the following version of expander lemma, stating that every graph contains a robust expander subgraph with almost the same average degree.
Then every graph G with d(G) = d has a subgraph H such that We remark that, though almost retaining the average degree, the robust expander subgraph H in Lemma 3.2 could be much smaller than G. For instance, if G is a union of many vertex disjoint small cliques, then H could be just one of those cliques. Such drawback often makes it difficult to utilise expanders iteratively within graphs. We include the proof of the robust expander lemma, Lemma 3.2, in the appendix of the online version.
A key property of the robust expanders that we shall use is that it has small (logarithmic) diameter.

Lemma 3.3 ([18] Corollary 2.3).
If G is an n-vertex (ε 1 , t)-robust-expander, then for any two vertex sets X 1 , X 2 each of size at least x ≥ t/2, and a vertex set W of size at most ρ(x)x/4, there exists a path in G − W between X 1 and X 2 of length at most 2 ε 1 log 3 15n t .

Exponential growth for small sets
In an (ε 1 , t)-robust-expander graph, for a set X with size at least t/2, the ball B i (X) grows with the radius i. For our purpose, we need to quantify how resilient this growth is to deletion of some thin set around X.
Definition 3.4. For a set X ⊆ W of vertices, the paths P 1 , . . . , P q are consecutive shortest paths from X within W if the following holds. For each i ∈ [q], P i | W is a shortest path from X to some vertex v i ∈ W \ X in the graph restricted to W − j∈[i−1] V (P j ).
In particular, the following proposition shows that the rate of expansion for small sets is almost exponential in a robust expander even after deleting a few consecutive shortest paths. Proposition 3.5. Let 0 < 1/d ≪ ε 1 , ε 2 ≪ 1. Suppose G is an n-vertex (ε 1 , ε 2 d)-robustexpander and X, Y are sets of vertices with |X| = x ≥ ε 2 d and |Y | ≤ 1 4 · ρ(x) · x. Let P 1 , . . . , P q be consecutive shortest paths in G − Y from X within B r G−Y (X), where 1 ≤ r ≤ log n and q < x log −8 x, and let P = i∈[q] V (P i ). Then for each i ∈ [r], we have Proof. For each i ≥ 0, let Z i = B i G−P −Y (X). As P 1 , . . . , P q are consecutive shortest paths from X, for each i ≥ 0, each path P j , j ∈ [q], can intersect with the set N G−Y (Z i ) on at most i + 2 vertices. Indeed, otherwise we can replace the initial segment of P j with a path in We first use induction on i to show that for each 0 We may then assume i > log 4 x, as f (i) ≤ g(i) ≤ |Z i | when i ≤ log 4 x. Now, as i > log 4 x, as desired.

Basic building structures
The following structures will serve as basic building blocks for our constructions of subdivisions. Note that a (2a, 0)-sun is just an even cycle of length 2a.  head M Figure 2: A (4, s, r, τ )-nakji (Definition 3.9).
} is a collection of pairwise internally vertex disjoint paths, each of length at most h 3 , such that P i is a u, x i -path.
For every vertex w ∈ Ext(F ), let P (F, w) be the unique path from the core vertex u to w in F .
} is a collection of pairwise internally vertex disjoint paths such that Q i is a v, u i -path of length at most h 3 .
Define the exterior Ext(W ) : For every vertex w ∈ Ext(W ), let P (W, w) be the unique path from the core vertex v to w in W .
, each having size at most s, and paths • P i is an M, D i -path with length at most 10r, and all paths P i , i ∈ [t], are pairwise internally disjoint; • D i has diameter at most r, and all D i , i ∈ [t], are a distance at least τ in G from each other and from M , and they are disjoint from internal vertices of i∈[t] P i .
We call M the head of the nakji and each

Outline of the proofs
To prove Theorem 1.1, we first use Lemma 3.2 to pass to a robust expander subgraph without losing much on the average degree. Depending on the density of the expander, we use different approaches. Roughly speaking, when the expander has positive edge density, we will utilise pseudorandomness via the machinery of the graph regularity lemma and the blow-up lemma (Lemma 4.1); while if the expander is not dense, then we exploit its sublinear expansion property (Lemmas 4.2 and 4.3).
and H be a graph with at most (1 − ε)d vertices with ∆(H) ≤ ∆ and G is an n-vertex graph and m = 2 Our main theorem readily follows, assuming these three lemmas.
In the rest of this section, we outline the ideas in our constructions. Let G be an expander and H be the bounded-degree bipartite graph whose subdivision we want to embed in G.

Embeddings in dense graphs
The regularity lemma essentially partitions our graph G into a bounded number of parts, in which the bipartite subgraphs induced by most of the pairs of parts behave pseudorandomly. The information of this partition is then stored to a (weighted) fixed-size so-called reduced graph R which inherits the density of G.
We seek to embed H in G using the blow-up lemma, which boils down to finding a 'balanced' bounded-degree homomorphic image of H in R. This is where the additional separable assumption on H kicks in. The separability of H (in the form of sublinear bandwidth) enables us to cut H into small pieces to offer suitable 'balanced' homomorphic images. Now, if the reduced graph R has chromatic number at least three, the density of R inherited from G is just large enough to guarantee an odd cycle long enough in R to serve as our boundeddegree homomorphic image of H.
It is, however, possible that R is a bipartite graph, hence all cycles within are even cycles, which might not be long enough for our purpose. Indeed, in the worst case scenario that H is an extremely asymmetric bipartite graph, an even cycle has to be twice as long as that of an odd one to be useful. This is because in the odd cycle we can circumvent the asymmetry of H by 'breaking the parity' via wrapping around the odd cycle twice.
To handle the case when R is bipartite, instead of cycles, we will make use of the sun structure (Definition 3.6), in which the leaves attaching to the main body of sun help in balancing out the asymmetry of H.

Embeddings in robust expanders with medium density
The robust expansion underpins all of our constructions of H-subdivisions when the graph G is no longer dense. At a high level, in G, we anchor on some carefully chosen vertices and embed paths between anchors (corresponding to the edge set of H) one at a time.
As these paths in the subdivision need to be internally vertex disjoint, to realise this greedy approach we will need to build a path avoiding a certain set of vertices. This set of vertices to avoid contains previous paths that we have already found and often some small set of 'fragile' vertices that we wish to keep free.
To carry out such robust connections, we use the small-diameter property of expanders (Lemma 3.3). Let m be the diameter of G. Recall that H is of order at most d with bounded degree and we need to embed e(H) = O(d) paths. Thus, all in all, the set of vertices involved in all connections, say W , is of size O(dm). To enjoy Lemma 3.3, we want to anchor at vertices with large 'boundary' compared to W , that is, being able to access many (dm 10 say) vertices within short distance.
With that being said, if there are now d vertices of high degree (at least dm 10 ), we can easily finish the embedding anchoring on these high degree vertices. This almost enables us to view G as if it is a 'relatively regular' graph. In reality, what happens is that we can assume the set of high degree vertices, L, is small: |L| < d; and deduce from the robust expansion property that, G − L is still dense. It is worth pointing out that this is where we need to extend the original notion of expander to this robust one.
Without high degree vertices, we turn to the web structure (Definition 3.8), in which the core vertex has a large 'boundary' (the web's exterior) of size about dm 9 . If there are d webs of suitable size, we can then anchor on their core vertices and connect pairs via the exteriors of the corresponding webs. One thing to be careful about is that a web will become useless if the (few) vertices in its centre are involved in previous connections. To prevent this, when constructing paths between exteriors, we protect the 'fragile' centres of all webs.
Lastly, as G is 'relatively regular', we can pull out many large stars (size roughly d) and link them up to find the webs one by one. The construction of webs is one of the places we require G to be not too sparse (d being a large power of m suffices).

Embeddings in sparse robust expanders
The current way of building and connecting webs breaks down if the expander is too sparse, say with average degree at most log log n. We will have to rely on other structures to build subdivisions in this case.
Let us first look at the easier problem of finding H-minors, in which case we just need to find d large balls (and contract them afterwards) and find O(d) internally disjoint paths between them. Note that now |L| < d ≤ m 100 is quite small. Suppose additionally that G−L has average degree Ω(d) and within it we can find d vertices, v 1 , . . . , v d , pairwise a distance √ log n apart, such that for each v i , the ball B i of radius say (log log n) 20 around it has size at least m 200 . So each B i is large enough to enjoy exponential growth (Proposition 3.5) avoiding all paths previously built. Now to get, say, a v i , v j -path, we first expand B i , B j to larger balls with radius say (log n) 1/10 . These larger balls are so gigantic that we can connect them avoiding all the smaller balls i∈[d] B i . It is left to find such v i and B i . We can find them one by one, by collectively growing a set U of pairwise far apart vertices past L and using an averaging argument to locate the next v i that expands well in G − L.
Coming back to embedding H-subdivisions, we shall follow the general strategy as that of finding minors. However, an immediate obstacle we encounter is the following. To get a subdivision instead of a minor, we need to be able to lead up to ∆(H) = O(1) many paths arriving at B i disjointly to v i . In other words, each anchor vertex v i has to expand even after removing O(1) disjoint paths starting from itself. Here comes the problem: in the minor case, we just need to expand U ignoring a smaller set L; whereas now U is asked to expand past a larger set of Θ(|U |) vertices that are used in the previous connections. Our expansion property is simply too weak for this. This is where the nakji structure (Definition 3.9) comes into play. It is designed precisely to circumvent this problem by doing everything in reverse order. Basically, instead of looking for anchor vertices that expand robustly, we rather anchor on nakjis and link them via their legs first and then extend the paths from the legs in each nakji's head using connectivity.
Why do we require that the head and legs of a nakji are stretched far apart? This is so that, before linking nakjis together to a subdivision, we can expand each leg without bumping into any other part to an enormous size, so that each connection made leaves irrelevant structures untouched.
The remaining task is then to find many nakjis in G − L. This is done essentially by linking small subexpanders within G − L. A subtlety here worth pointing out is that we only have |L| < d, while each of the subexpanders, though having size Ω(d), could be smaller than L. This keeps us from expanding and linking each subexpander in G − L. Intuitively, given that L is not large, one would like to take a huge set of subexpanders, whose union is so large and thus grows easily past L. However, as the expanding function ρ(·) is sublinear, if there are too many subexpanders to begin with, after averaging, the expansion rate of each subexpander in G − L is too weak to be useful. To overcome this difficulty, instead, we shall average over a set of subexpanders of appropriate size that is just big enough to ignore L and on the other hand just small enough that ρ(·) does not decay too much.

Separable bipartite graphs in dense graphs
In this section, we prove Lemma 4.1. We will use Szemerédi's regularity lemma; for a detailed survey of this lemma and its numerous applications, see [15,16] |A||B| be the density of a bipartite graph G with vertex classes A and B. For a positive number ε > 0, we say that a bipartite graph G with vertex classes A and B is ε-regular if every X ⊆ A and Y ⊆ B with |X| ≥ ε|A| and |Y | ≥ ε|B| satisfy |d The following is a version of the regularity lemma suitable for our purpose.
Lemma 5.1. Suppose 0 < 1/n ≪ 1/r 1 ≪ ε ≪ τ < 1 with n ∈ N and let G be an n-vertex graph with d(G) ≥ dn for some d ∈ (0, 1). Then there is a partition of the vertex set of G into V 1 , . . . , V r and a graph R on the vertex set [r] such that the following holds: The graph R above is often called the reduced graph of G (with respect to the partition V 1 , . . . , V r ). One of the reasons why the regularity lemma is useful is that it can be combined with the blow-up lemma of Komlós, Sárközy and Szemerédi (see [14,Remark 8]). Here, we only need the following weaker version of the blow-up lemma which only yields a non-spanning subgraph in a given graph.
Then G contains H as a subgraph.

Balanced homomorphic image of H
We now show how to find a suitable partition of H to invoke the blow-up lemma. This will use the separability property; however, we find it more convenient to work with bandwidth. A graph H has bandwidth b if we can order its vertices In general, small bandwidth is a stronger notion than small separability. However the following result of Böttcher, Pruessmann, Taraz and Würfl shows that for bounded-degree graphs the two notions are roughly equivalent.
If H is an α-separable graph with at most d vertices and ∆(H) ≤ ∆, then H has bandwidth at most βd.
We want to partition H into almost equal sized sets X 1 ∪ · · · ∪ X r for an appropriate r where all edges of H lie between two consecutive sets X i and X i+1 where X r+1 = X 1 . In other words, we want to find a 'balanced' homomorphism from H into C r .
Later we will apply regularity lemma to G and find a cycle C in the reduced graph we obtain from the regularity lemma, and apply the blow-up lemma with the above partition of H where r = |C|. If |C| is odd, then the value r is large enough for us to fit H into G[ i∈C V i ]. However, if |C| is even and H is an unbalanced bipartite graph then our strategy does not work. For such a case, we need to consider a sun instead.
The following two lemmas provide partitions of H suitable for our purpose. First one finds a 'balanced' homomorphism of H into an odd cycle C r and the second finds a 'balanced' homomorphism of H into a suitable sun. As two proofs are similar, we omit the proof of the first lemma here. It will be available in the appendix of the online version of this paper.
• For each i ∈ [r], we have |X i | ≤ d r .
• Every edge of H is between X i and X i+1 for some i ∈ [r] where X r+1 = X 1 .
• For each u ∈ V (R 0 ), we have |X u | ≤ d r • Every edge of H is between X u and X v for some uv ∈ E(R 0 ).
Proof. As H has bandwidth at most βd, there exists an ordering . . u 2s be the cycle in the sun R 0 and assume u p 1 , . . . , u pq be the vertices which has a leaf-neighbour such that p 1 , . . . , p q are all even. We consider all indices in sun (respectively in partition of H) up to modulo 2s (respectively r), i.e. u a+2s = u a and X a+r = X r for all a ∈ N. Let v p i be the leaf neighbour of u p i for each i ∈ [q].
Let t = ⌈β −1/2 ⌉ and we divide the vertices of H according to the ordering as follows: for each i ∈ [t 2 ] and C ∈ {A, B}, let Note that this guarantees that no edge of H is between Y i and Y j with |i − j| > 1. For each i ∈ [t] and C ∈ {A, B}, let In the claim below, we will decide to which part X ℓ we assign the vertices in Z C i . To make such an assignment possible while keeping the edges only between two consecutive parts, we allow the vertices in W C i to be assigned to some other parts. As each set W C i is much smaller than Z C i , the uncontrolled assignments of W C i will not harm us too much. Indeed, the set W = i∈[t] (W A i ∪ W B i ) has size at most rβ 1/2 d < δ 2 d/r. Now, we will decide to which part X u we will assign Z C i . For this, we partition the set [t] into I 1 , . . . , I s , J 1 , . . . , J q as in the following claim. If i ∈ I ℓ , then we will later assign the vertices in Z A i to X u 2ℓ−1 and the vertices in Z B i to X u 2ℓ . If i ∈ J ℓ ′ , then we will later assign the vertices in Z A i to X vp ℓ ′ and Z B i to X up ℓ ′ . As we do not know how unbalanced two sets Z A i and Z B i are for each i, we prove the following claim using random assignments.
With these sets I 1 , . . . , I s , J 1 , . . . , J q , we can distribute the vertices as desired. Assume we already distributed vertices in i∈[kt] Y i to X 1 , . . . , X 2s and X ′ p 1 , . . . , X ′ pq in such a way that . . , X 2ℓ−1 , X 2ℓ , respectively and we allocate the remaining vertices in W A k+1 ∪Z A k+1 to X 2ℓ−1 and the remaining vertices in . . , X p ℓ −1 , X p ℓ respectively and we allocate the remaining vertices in W A k+1 ∪ Z A k+1 to X ′ p ℓ and the remaining vertices in W B k+1 ∪ Z B k+1 to X p ℓ . By repeating this for k = 0, . . . , t, we distribute all vertices. For each i ∈ [s] and j ∈ [q], let X u i = X i and X vp j = X ′ p j . Then, by the bandwidth condition, all edges of H are between Y i and Y i+1 , so we know that each edge of H is between X u and X v for some uv ∈ E(R 0 ). As Again, as |W | < δ 2 d/r, for each u = u 2ℓ , u ′ = u p ℓ ′ ∈ V (R 0 ) where 2ℓ / ∈ {p 1 , . . . , p q }, we have This proves the lemma.

Proof of Lemma 4.1
The last ingredient for the dense case is the following result of Voss and Zuluaga providing a long cycle in 2-connected graphs.
Lemma 5.7. [39] Suppose that a graph R is a 2-connected graph on at least 2d vertices with δ(G) ≥ d. If R is a bipartite graph with vertex partition A ∪ B, then R has an even cycle of length at least min{2|A|, 2|B|, 4d − 4}. If R is not a bipartite graph, then it has an odd cycle with length at least min{|R| − 1, 2d − 1}.
Proof of Lemma 4.1. Let d ′ := d/n > η. We can choose numbers r 1 ∈ N and β, ε ′ , τ > 0 such that We apply Lemma 5.1 to G with ε ′ , τ, d ′ playing the roles of ε, τ, d to obtain a partition V 1 ∪ · · · ∪ V r of V (G) and a corresponding reduced graph R with V (R) = [r], 1/ε ′ ≤ r ≤ r 1 , such that One easy way to see such a graph R ′ exists is to apply Lemma 3.2 to R to obtain R ′ with τ, 1/4 playing the roles of ε 1 , ε 2 . Let r 0 := (d ′ − 4τ )r. Now we will find a graph R 0 in R ′ which is either an odd cycle of length at least r 0 or an (2a, b)-sun with a + b ≥ r 0 . This will provides a structure in G suitable for us to use the blow-up lemma.
If R ′ is not a bipartite graph, we let R 0 be an odd cycle of length at least min{|R ′ | − 1, 2δ(R ′ ) − 1} ≥ r 0 in R ′ guaranteed by Lemma 5.7.
If R ′ is bipartite with vertex bipartition A ∪ B and |A| ≤ |B|, then we let R 0 be a (2a, b)-sun in R ′ for some a, b with a + b ≥ r 0 . We claim that such a sun exists. If |A| ≥ r 0 , then Lemma 5.7 yields an even cycle C of length at least min{2|A|, 4δ(R ′ ) − 4} ≥ 2r 0 in R ′ . This is a (2r 0 , 0)-sun as claimed.
If |A| < r 0 , then let p = r 0 − |A|. As |A| ≤ |B|, we have Thus, we have p ≤ r 0 2 = 1 2 (d ′ − 4τ )r and r 0 ≤ (d ′ − 3τ )r ≤ |B|. Now, as |A| = r 0 − p, we use Lemma 5.7 to find an even cycle C of length at least min{2|A|, 4δ . This matching together with the cycle C forms a (2|A|, p)-sun with |A| + p = r 0 as claimed. Now, using Lemma 5.4 or Lemma 5.5 with ε ′ , R 0 , (1 − ε/2)d playing the roles of δ, R 0 , d, respectively, we can partition V (H) into {X u : u ∈ R 0 } where for each i ∈ [p] we have and every edge of H is between X u and X v for some uv ∈ E(R 0 ). Hence, (i)-(iii) in Theorem 5.2 are all satisfied with ε ′ , τ, R 0 playing the roles of ε, τ, R 0 , respectively, to conclude that G contains H as a subgraph. This proves the lemma.

Subdivisions in robust expanders with medium density
In this section, our goal is to prove Lemma 4.2, which finds an H-subdivision in a robust expander with medium density. We first prove the following lemma, which bounds the number of high degree vertices in our expanders. Take a set Z = {v 1 , . . . , v h } of h distinct vertices in L. Then, for each i ∈ [h], the set X i := N (v i ) has size at least dm 10 . Assume that we have pairwise internally disjoint paths P 1 , . . . , P ℓ with 0 ≤ ℓ < h ′ where P j is a path between v a j and v b j of length at most 2m.
Proof of Lemma 4.2. As d n ≤ η ≪ ε 1 , ε 2 , we have that m 100 = 2 ε 1 log 3 15n Hence To derive a contradiction, we assume that G does not contain an H-subdivision. Then by Lemma 6.1, we have |L| < d.
For a contradiction, we will find an H-subdivision in G − L. For this purpose, we need not only that L is small, but also that the graph G − L is still relatively dense to contain an H-subdivision. We claim that This follows essentially from the robust expansion property of G. Indeed, otherwise a random vertex set X of size dm 2 chosen uniformly at random from G − L has expected degree sum Note that, by definition of F , once we delete the edges of F from G, the external neighbourhood of X lies entirely in L, that is, N G\F (X) ⊆ L. However, this implies |N G\F (X)| ≤ |L| (2) ≤ d ≤ ρ(dm 2 ) · dm 2 = ρ(|X|) · |X|, contradicting that G is an (ε 1 , ε 2 d)-robust-expander. Hence we have (3). Now that G − L is still relatively dense and no vertex in G − L has too large degree as ∆(G − L) ≤ dm 10 , we can find in G − L many webs with disjoint interiors. To find such webs, we follow the strategy as [13,Lemma 5.7]. We include the proof in the online appendix.
Before moving on, we set up two pieces of notations. For a path Q with endvertices a ∈ Ext(W i ) and b ∈ Ext(W j ), for some For a given set Z, we say a web W is Z-good if |Int(W )∩Z| ≤ m 12 /2. We define (X, I, I ′ , Q, f ) be a good path system if the followings hold. 2 , Q ij is an a i,j , a j,i -path of length at most m where a i,j ∈ Ext(W i ) and a j,i ∈ Ext(W j ) with Q ij ∩ (Int(W i ) ∪ Int(W j )) = ∅.
We shall show that there is a good path system with X = V (H), which would finish the proof as by A1-A4, e∈E(H) Q * e is an H-subdivision. We proceed as follows.

Proceed to
Step (i + 1) with the new good path system (X i+1 , I i+1 , I ′ i+1 , Q i+1 , f ). Let (X, I, I ′ , Q, f ) be the good path system obtained from the above process. Note that the sequence |X 1 |, |X 2 |, . . . might not be an increasing sequence, as we may delete some elements when updating the list of bad webs in each step. However, we will show that this procedure will eventually end with X = V (H) as desired.
First, we claim that |I ′ | ≤ d/m. Note that Q might contain some paths which connects W i with i ∈ I ′ . However, as at most ∆(H) ≤ ∆ paths are added at each step, Recall that {Int(W j ) : j ∈ [2d]} are pairwise disjoint and so, by definition of I ′ , |I ′ |·m 12 /2 ≤ |Q|. Thus, |I ′ | ≤ |Q| m 12 /2 < d/m as claimed. As in each step a new vertex is added before updating I ′ i+1 , the above claim guarantees that this process terminates when |I ∪ I ′ | ≤ |H| + |I ′ | < 2d, implying that X = V (H). To finish the proof, it is left to show that all connections in each step, i.e. paths in Q i+1 \ Q i , can indeed be constructed to keep the process running.
Thus, there are at least (m 2 − ∆)m 10 − m 12 /2 ≥ m 12 /4 available paths in Int(W j ) \ Ctr(W j ) (and their corresponding paths in Ctr(W j )) disjoint from V (Q * i ); let A j ⊆ Ext(W j ) be the union of the leaves of the stars corresponding to these available paths. Then |A j | ≥ a := dm 9 /4. Now, for each j ∈ [s], since using Lemma 3.3, we can find the desired path Q xy j with length at most m connecting A i * and . This finishes the proof of Lemma 4.2.

Subdivisions in sparse robust expanders
In this section, we prove Lemma 4.3. First, we show in Section 7.1 that we cannot have too many edges sticking out of a small set of vertices, for otherwise we can obtain H even as a subgraph in G. In particular, G − L is still dense (Claim 7.1), and consequently it must contain a subexpander F with d(F ) = Ω(d).
Note that this subexpander could be a lot smaller and hence we have no control on its density. Suppose for a moment that F is sparse, and additionally F is large, then it inherits the 'bounded' maximum degree from G− L. This case is handled in Section 7.2. For such an 'almost regular' sparse expander, we can work entirely within it to find an H-subdivision (Lemma 7.3). Now, if F has medium edge density, then we can invoke Lemma 4.2 on F and we are done. Therefore, F must either be sparse and small, or very dense. In both cases, F is small in order.
To this end, we may assume that all subexpanders in G − L are small. In Section 7.3, we shall iteratively pull out many small expanders in G − L that are pairwise far apart (C1-C4), most of which expand well inside G − L (Claim 7.4).
We then in Section 7.4 link these nicely expanding small expanders to construct nakjis (Claim 7.5). The strategy here is to extract nakjis iteratively. Every time, we step far away in G − L from previously built nakjis and expand an appropriate collection of expanders to find the next one.
Finally, the finishing blow is delivered in Section 7.5, in which we anchor on the nakjis and connect them to build an H-subdivision.
We first establish the following claim, stating that the density does not drop much upon removing a small set of vertices. The idea is that if lots of edges are incident to a small set, then we will see a dense and skewed bipartite subgraph, which contains a copy of H.

Proof of claim. Let
Let f : Z → L ∆ be a function with maximum t(f ) among all functions satisfying f (v) ⊆ N G (v, L) for every v ∈ Z. As |N G (v, L)| ≥ ∆, such a function f must exist. However, as |Z| > ∆d ∆ > ∆ |L| ∆ , there exist ∆ + 1 vertices u 1 , . . . , u ∆+1 ∈ Z with f (u 1 ) = · · · = f (u ∆+1 ). Let X = N G (u 1 , L). If any set Y ∈ X ∆ satisfies |f −1 (Y )| < ∆, then we can redefine f (u 1 ) to be Y to increase t(f ). Hence, by the maximality of f , for every set . It is easy to see that X together with the vertices {v i Y : Y ∈ X ∆ , i ∈ [∆]} induces a graph containing any (1 − ε)d-vertex bipartite graph with maximum degree at most ∆. Hence G contains H as a subgraph, a contradiction. Thus |Z| ≤ ∆d ∆ .

'Bounded' degree sparse expander
As outlined at the start, since G − L is still dense, it contains a subexpander F at our disposal. We first take care of the case when this subexpander is sparse and 'almost regular'. We will use the following proposition to take many vertices that are pairwise far apart ([20, Proposition 5.3] taking s = 2000).
As d ≤ δ(F )/ε 2 ≤ ∆(F )/ε 2 ≤ log 30001 n, Proposition 7.2 implies that we can find vertices v 1 , . . . , v h , where h = |H| ≤ d, such that the distance between any two of them is at least 2r + 2r ′ . Let x 1 , . . . , x h be the vertices of H and Suppose that we have Q 1 , . . . , Q ℓ for some 0 ≤ ℓ < h ′ such that: B2 for distinct i, j ∈ [ℓ], the paths Q i and Q j are internally vertex disjoint; B3 for each i ∈ [h], for the edges {e k 1 , . . . , e ks } = {e k j : k j ∈ [ℓ], x i ∈ e k j } with k 1 < · · · < k s , the paths Q k 1 , . . . , Q ks form consecutive shortest paths from v i in B r (v i ); and Note that v 1 , . . . , v h are pairwise a distance at least 2r + 2r ′ apart, so by B4 we have . By B3, we can apply Proposition 3.5 with B 1 F −W 1 (v k ℓ+1 ), W 1 , ∅, ∆ playing the roles of X, P, Y, q, and then for each k ∈ {a, b} we have where the last inequality follows from d ≤ log 30001 n. This implies that Hence, by applying Proposition 3.5, now with B r F −W (v k ℓ+1 ), ∅, W 1 playing the roles of X, P, Y for each k ∈ {a, b}, we similarly have As ∆(F ) ≤ log 30000 n and d ≤ log 30001 n, we then have |W | ≤ |W 1 | + |W 2 | ≤ ∆d log 4 n + d · 2(log 30000 n) r < 1 4 ρ exp( 9 log n) exp( 9 log n).
Therefore, by Lemma 3.3, there is a path in F −W of length at most log 4 n between B r+r ′ F −W (v a ℓ+1 ) and B r+r ′ F −W (v b ℓ+1 ). So we can let Q ℓ+1 be a shortest path between v a ℓ+1 and v b ℓ+1 in F −W , which has length at most log 4 n + 2r + 2r ′ ≤ 2 log 4 n. Hence, Q 1 , . . . , Q ℓ+1 satisfy B1-B4. Repeating this for ℓ = 0, 1, . . . , h ′ − 1, then the union of all paths i∈[h ′ ] Q i is an H-subdivision.

Many small subexpanders
With Lemma 7.3 at hand, we can now proceed to show that all subexpanders in G ′ = G − L must be small and that we can find many of them pairwise far apart.
Let F be a maximal collection of subgraphs of G ′ satisfying the following.
For each F ∈ F, let If some F ∈ F satisfies m 100 ≤ m 220 or d(F ) ≤ m 100 F . If the latter case holds, we claim that n F ≤ exp( 50 √ log n). Indeed, if n F > exp( 50 √ log n) holds, then we have So we can apply Lemma 7.3 on F to get an H-subdivision, a contradiction. Thus we have Moreover, if |F| < n 0.99 , then by (4) this implies that Hence, Claim 7.1 implies that G ′ − U has average degree at least εd/6. Thus Lemma 3.2 finds another expander F satisfying C1 in G ′ − U . Then C2 also holds, contradicting the maximality of F. Hence, we have C4 |F| ≥ n 0.99 .
Futhermore, the following claim states that most of the expanders in F expand well in G ′ .
Claim 7.4. There exist at least |F| − d log n graphs F in F such that for each 1 ≤ r ≤ log n, Proof of claim. Fix a choice of 1 ≤ r ≤ log n and consider a set I ⊆ F with |I| = d and let X = F ∈I V (F ). Note that by C1, |X| ≥ εd 2 /10, and so |L| < d ≤ 1 4 ρ(|X|)|X|. By Proposition 3.5 with L playing the role of Y , we have |B r G ′ (X)| ≥ exp(r 1/4 ). Then by the pigeonhole principal, there exists F ∈ I such that Therefore, whenever there are d members left in F, we can keep picking out one, the r-ball around which expands nicely in G ′ . Now varying r, we see that there are at least |F| − d log n graphs F in F as claimed.
By losing a factor of 2 in size, i.e. instead of C4, |F| ≥ n 0.99 /2, we may assume now every member in F expands well in G ′ as in Claim 7.4

Linking subexpanders to nakjis
We will now connect these far apart nicely expanding subexpanders in F to obtain d separate (∆, exp( 50 √ log n), m, 2 √ log n)-nakjis to anchor.
More precisely, take the following collections with maximum possible p ∈ N.
} is a collection of pairwise vertex disjoint paths, each of length at most 10m, such that P i,j is a u i,j , v i,j -path where u i,j ∈ C i,0 and v i,j ∈ S i,j and the internal vertices of P i,j are not in So, by C1 and D1-D4, we see that for each i ∈ [p], 0≤j≤∆ C i,j and j∈[∆] P i,j form a (∆, exp( 50 √ log n), m, 2 √ log n)-nakji, in which C i,0 is the head and each C i,j , j ∈ [∆], is a leg.
Claim 7.5. We have p ≥ d.
Proof of claim. Suppose to the contrary that p < d. Let W be the set of vertices involved in C and all the paths P i,j , and W ′ be a √ log n-ball in G ′ around W . That is, Then, by D1-D3 and (4), we see that |W | ≤ (∆ + 1)d · exp 50 log n + 20m ≤ exp 49 log n , and |W ′ | ≤ exp 49 log n · 2∆(G ′ ) √ log n ≤ n 0.1 .
We shall find a nakji in G ′ − W ′ , which will lead to a contradiction to the maximality of p.
Let F ′′ be the graphs in We shall connect X and U . First we expand X as follows. As |X| ≥ εd 2 /10, |L| ≤ d and ℓ ≤ ∆d ≤ |X| log 8 |X| , E2 implies that we can apply Proposition 3.5 with X, Q, L playing the roles of X, P, Y to obtain that where the first equality follows from X being far from W ∪ W * owing to X ∩ W ′ = ∅ and C2. As and |U | ≥ n 0.9 , Lemma 3.3 implies that there exists a path of length at most m between the sets U and B , let the resulting path be Q ℓ+1 . The choices of W * , F ′ ℓ+1 and the path Q ℓ+1 ensure that E1-E3 hold for Q 1 , . . . , Q ℓ+1 . Repeating this for ℓ = 0, . . . , d∆, we obtain paths Q 1 , . . . , Q d∆ . Recall that |F 0 | = d and so by the pigeonhole principle, there exists a graph F 0 ∈ F 0 such that at least ∆ paths among {Q 1 , . . . , Q ∆d } are incident to F 0 . Relabelling and keeping the relative ordering, let those paths be Q 1 , . . . , Q ∆ connecting F 0 and F ′ 1 , . . . , F ′ ∆ ∈ F ′′ . Let C p+1,0 = F 0 and C p+1, . By E3 and the definition of and W , and furthermore, by Claim 7.4, we can find a connected subgraph S p+1,i ⊆ Let v p+1,i ∈ S p+1,i be its first contact point with Q i , and let P p+1,i = Q i − S p+1,i \ {v p+1,i } be the truncated path. It is routine to check that D1-D4 still hold with the additions of C p+1,0 , and C p+1,i ⊆ S p+1,i and P p+1,i , i ≤ [∆]. This contradicts the maximality of p. Hence, we have p ≥ d, proving the claim.

The finishing blow
It is time now to complete the game: we will wire nakjis together to build an H-subdivision. Assume we have pairwise disjoint paths R 1 , . . . , R ℓ for some 0 ≤ ℓ < h ′ satisfying the following for each i ∈ [ℓ], where (i ′ , j ′ ) = (s i , f s i (e i )) and (i ′′ , j ′′ ) = (t i , f t i (e i )).
F1 R i is a path between v i ′ ,j ′ and v i ′′ ,j ′′ with length at most 10m.
F2 R i does not intersect with any C i * ,0 for i * ∈ [h] and does not intersect with any Then by C3, D2 and d ≤ m 100 , we have |Y | ≤ (∆ + 1)d exp( 50 √ log n) ≤ exp( 49 √ log n). Let As |R ∪ L| ≤ dm 2 , then by D2 we can apply Proposition 3.5 to X and X ′ , with R ∪ L playing the role of Y to obtain that |X|, |X ′ | ≥ exp( 10 log n).
By C2 and D1, D2, we know that X, X ′ does not intersect with Y . As |Y ∪ R ∪ L| ≤ 2 exp( 49 √ log n) < 1 4 ρ(|Z|)|Z| for each Z ∈ {X, X ′ }, by Lemma 3.3, we can find a path R from is connected due to D2, and C i,j is an (ε 1 , ε 2 d)-robust-expander with diameter at most m due to C1 and D1, using the definition of X, X ′ , we can extend R inside X, X ′ to get a path from v i ⊢ ,j ⊢ to v i ⊣ ,j ⊣ satisfying F1 and F2. Repeating this for each ℓ = 0, . . . , h ′ − 1, we obtain paths R 1 , . . . , R h ′ satisfying F1 and F2.
Note that for each i ∈ [h ′ ] and (i ′ , j ′ ) = (s i , f s i (e i )), (i ′′ , j ′′ ) = (t i , f t i (e i )), the path R i and P i ′ ,j ′ ∪ P i ′′ ,j ′′ might intersect at their interiors. However, Moreover, D3 and F2 imply that each P * e i only intersects C s i ,0 and C t i ,0 at u i ′ ,j ′ and u i ′′ ,j ′′ respectively, and P * e 1 , . . . , P * e h ′ are all pairwise vertex disjoint. Let P * = i∈[h ′ ] V (P * e i ). Now, for each i ∈ [h], we consider C i,0 and set d i = d H (x i ). Note that C i,0 intersects with P * only at the distinct vertices u i,1 , . . . ,

Bounded degree planar graphs as subdivision
We might ask whether we can in fact guarantee subdivisions rather than minors in Theorem 2.1. Our methods for finding subdivisions only cover bounded-degree graphs, however. So a natural question might be: what is the best constant c such that every graph of average degree (c+o(1))t contains a subdivision of every bounded-degree planar graph of order t?
First, it is easy to see that we can achieve c = 5/2: a planar graph with t vertices has fewer than 3t edges, and any graph has a bipartite subgraph with at least half the edges, so after subdividing at most 3t/2 edges we obtain a bipartite subdivision with at most 5t/2 vertices. Theorem 1.1 then allows us to find a subdivision of this subdivision in any graph of average degree (5/2 + o(1))t. In fact we can do better. Proof. We may assume H is maximal planar, so every face is a triangle and there are 2t − 4 faces. Consider the dual graph H * . This is a 2-connected 3-regular graph, and so, by Petersen's theorem, has a 1-factor. Each edge of this 1-factor corresponds to an edge of the original graph H. We subdivide each of these edges once. Suppose there is an odd cycle C in the original graph H. By double-counting edges bordering faces surrounded by C, there are an odd number of such faces. Thus the 1-factor in H * contains an odd number of edges crossing C. Thus we have subdivided an odd number of edges of C, so C becomes an even cycle in the subdivision. By exactly the same argument, we can see that any even cycle in H remains even in the subdivision, so the subdivision is bipartite. It has exactly |H| + |H * |/2 = 2t − 2 vertices, as required. Lemma 8.1 is best possible, since in any maximal planar graph we must subdivide at least one edge of each face, and this requires at least t − 2 extra vertices. Together with Theorem 1.1, it immediately gives the following. Our lower bound from Theorem 2.1 shows only that we cannot improve the constant 2 in Proposition 8.2 below 3/2. However, no similar example will give a stronger lower bound, for the following reason. Any t-vertex planar graph H has a bipartite subdivision with at most 3t/4 vertices in one part (to see this, take a largest independent set X and subdivide all edges which do not meet X; this is bipartite with one part being V (H) \ X, which has size at most 3t/4 by the four-colour theorem), and this subdivision is a subgraph of K 3t/4,n for sufficiently large n.
Problem D. What is the right value for the constant in Proposition 8.2? Is it 2, 3/2, or something in between?
Problem E. What can we say about d T (H) for k-degenerate graphs H for general k ∈ N?

Better bounds for minor closed families
As α 2 (F ) ≥ 2t/χ(F ), writing χ(F) = max{χ(F ) : F ∈ F}, Theorem 2.2 implies the following Erdős-Simonovits-Stone type bound for any minor-closed family F: However, this is in general not tight. It is sharp for the disjoint union of cliques of order χ(F), if such a graph is in F.
In fact, the Hadwiger conjecture would imply d ≻ (F, t) = 2(1 − 1/χ(F) + o(1))t for F closed under disjoint union. To see this, consider the maximum s such that K s ∈ F. Then the Hadwidger conjecture would give χ(F) = s. As the disjoint union of K s belongs to F, the above discussion implies d ≻ (F, t) = 2(1 − 1/χ(F) + o(1))t. However, a minor-closed family does not have to be closed under disjoint union (for example, the class of graphs embeddable in a fixed surface other than the plane), and for such families we may have d ≻ (F, t) < 2(1 − 1/χ(F) − c)t for some absolute constant c > 0, as we have seen in Theorem 2.1.
Another interesting question is to see when the upper and lower bounds arising from Theorem 2.2 coincide, which motivates the following problem. The case k = 6 of the Hadwiger conjecture, proved by Robertson, Seymour and Thomas [34], gives the above conclusion for any minor-closed family for which the minimal forbidden minors are all connected and include K 6 . In particular, this applies to the linklessly embeddable graphs discussed in Section 2.2. Since the Y ∆Y -reducible graphs form a subfamily of L containing the extremal example rK 5 , the corollary also applies to this class.
A Proof of the robust expander lemma, Lemma 3.2 We first set up some functions as follows. For x ≥ 1, define Note that γ(x) is a decreasing function. We will make use of the following two inequalities: and, for C ′ > 1 and x ≥ t/2, since ρ(x) is decreasing and ρ(x) · x is increasing for x ≥ t/2, We say that G is φ-maximal if φ(G) = max H⊆G φ(H).
The fact that H is νd-connected was shown in [13,Lemma 5.2(iv)]. This finishes the proof of Lemma 3.2.
B Balanced homomorphism to an odd cycle: proof of Lemma 5.4 As H has bandwidth at most βd, there exists an ordering x 1 , . . . , x |H| of V (H) such that x i x j ∈ E(H) implies |i − j| ≤ βd. Let A and B be a bipartition of H. Let t = ⌈β −1/2 ⌉ and we divide the vertices of H according to the ordering as follows: for each i ∈ [t 2 ] and C ∈ {A, B}, let Note that this guarantees that no edge of H is between Y i and Y j with |i − j| > 1. For each i ∈ [t] and C ∈ {A, B}, let In the claim below, we will decide to which part X j we assign the vertices in Z C i . To make such an assignment possible while keeping the edges only between two consecutive parts, we allow the vertices in W C i to be assigned to some other parts. As each set W C i is much smaller than Z C i , the uncontrolled assignments of W C i will not harm us too much. Indeed, the set W = i∈[t] (W A i ∪ W B i ) has size at most rβ 1/2 d < δ 2 d/r. Now, we will decide to which part X j we will assign Z C i . For this, we partition the set [t] into I 1 , . . . , I r as follows where I 0 = I r . If i ∈ I ℓ , then we will later assign the vertices in Z A i to X ℓ and the vertices in Z B i to X ℓ+1 . Note that the vertices in the set i∈I ℓ Z A i ∪ i∈I ℓ−1 Z B i below will be later assigned to X ℓ .
Claim B.1. There exists a partition I 1 , . . . , I r of [t] satisfying the following.
Proof of claim. We add each ℓ ∈ [t] independently to one of I 1 , . . . , I r uniformly at random. Standard concentration inequalities (e.g. Azuma's inequality) easily show that with positive probability, we can ensure that for each ℓ ∈ [r] we have as β ≪ δ, 1/r ≪ 1.
With these sets I 1 , . . . , I r , we distribute the vertices in Y 1 , . . . , Y t 2 in order as follows. Assume that we have distributed the vertices in Y 1 , . . . , Y st for some 0 ≤ s < t among the sets X 1 , . . . , X r as follows.
G1 For each ℓ ∈ [r] and i ∈ I ℓ ∩ [s], we have Z A i ⊆ X ℓ and Z B i ⊆ X ℓ+1 .
G2 Each edge of H[ i∈ [st] Y i ] lies between X ℓ and X ℓ+1 for some ℓ ∈ [r].
G3 Y B st ⊆ X ℓ ′ for some ℓ ′ ∈ [r]. For s = 0, the trivial partition X 1 = . . . X r = ∅ trivially satisfies the above (G3 is vacuous as Y 0 does not exist. For technical reason, we assume ℓ ′ = r in this case).
We write X r+i = X i for each i ∈ [r]. Now we describe how one can distribute vertices in Y st+1 , . . . , Y (s+1)t . Let s + 1 ∈ I ℓ for some ℓ ∈ [r]. Let ℓ * be a nonnegative integer less than 2r such that ℓ ′ + ℓ * ≡ ℓ (mod r) and ℓ * is odd. Such an ℓ * exists as r is odd.
C Webs with disjoint interiors: proof of Claim 6.2 In this section, we prove Claim 6.2. We will use (1)-(3) in Section 6. We first prove that the following holds.
For any set X of size at most dm 90 , the graph G − L − X contains vertex disjoint stars S 1 , . . . , S m 85 where each S i has exactly 2d/m 3 leaves.
For any set X of size at most dm 30 , the graph G − L − X contains vertex disjoint (m 10 , d/m 3 , m + 2)-units F 1 , . . . , F m 30 .