arXiv subjects

Julia Böttcher

Publications and source records attributed to Julia Böttcher.

At least 37 records · Page 2Linked to original sources

arXiv2020

The Bandwidth Theorem in Sparse Graphs

The bandwidth theorem [Mathematische Annalen, 343(1):175--205, 2009] states that any $n$-vertex graph $G$ with minimum degree $\big(\tfrac{k-1}{k}+o(1)\big)n$ contains all $n$-vertex $k$-colourable graphs $H$ with bounded maximum degree and bandwidth $o(n)$. We provide sparse analogues of this statement in random graphs as well as pseudorandom graphs. More precisely, we show that for $p\gg \big(\tfrac{\log n}{n}\big)^{1/Δ}$ asymptotically almost surely each spanning subgraph $G$ of $G(n,p)$ with minimum degree $\big(\tfrac{k-1}{k}+o(1)\big)pn$ contains all $n$-vertex $k$-colourable graphs $H$ with maximum degree $Δ$, bandwidth $o(n)$, and at least $C p^{-2}$ vertices not contained in any triangle. A similar result is shown for sufficiently bijumbled graphs, which, to the best of our knowledge, is the first resilience result in pseudorandom graphs for a rich class of spanning subgraphs. Finally, we provide improved results for $H$ with small degeneracy, which in particular imply a resilience result in $G(n,p)$ with respect to the containment of spanning bounded degree trees for $p\gg \big(\tfrac{\log n}{n}\big)^{1/3}$.

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arXiv2019

A spanning bandwidth theorem in random graphs

The bandwidth theorem [Mathematische Annalen, 343(1):175--205, 2009] states that any $n$-vertex graph $G$ with minimum degree $(\frac{k-1}{k}+o(1))n$ contains all $n$-vertex $k$-colourable graphs $H$ with bounded maximum degree and bandwidth $o(n)$. In [arXiv:1612.00661] a random graph analogue of this statement is proved: for $p\gg (\frac{\log n}{n})^{1/Δ}$ a.a.s. each spanning subgraph $G$ of $G(n,p)$ with minimum degree $(\frac{k-1}{k}+o(1))pn$ contains all $n$-vertex $k$-colourable graphs $H$ with maximum degree $Δ$, bandwidth $o(n)$, and at least $C p^{-2}$ vertices not contained in any triangle. This restriction on vertices in triangles is necessary, but limiting. In this paper we consider how it can be avoided. A special case of our main result is that, under the same conditions, if additionally all vertex neighbourhoods in $G$ contain many copies of $K_Δ$ then we can drop the restriction on $H$ that $Cp^{-2}$ vertices should not be in triangles.

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arXiv2019

Embedding spanning bounded degree graphs in randomly perturbed graphs

We study the model $G_α\cup G(n,p)$ of randomly perturbed dense graphs, where $G_α$ is any $n$-vertex graph with minimum degree at least $αn$ and $G(n,p)$ is the binomial random graph. We introduce a general approach for studying the appearance of spanning subgraphs in this model using absorption. This approach yields simpler proofs of several known results. We also use it to derive the following two new results. For every $α>0$ and $Δ\ge 5$, and every $n$-vertex graph $F$ with maximum degree at most $Δ$, we show that if $p=ω(n^{-2/(Δ+1)})$ then $G_α\cup G(n,p)$ with high probability contains a copy of $F$. The bound used for $p$ here is lower by a $\log$-factor in comparison to the conjectured threshold for the general appearance of such subgraphs in $G(n,p)$ alone, a typical feature of previous results concerning randomly perturbed dense graphs. We also give the first example of graphs where the appearance threshold in $G_α\cup G(n,p)$ is lower than the appearance threshold in $G(n,p)$ by substantially more than a $\log$-factor. We prove that, for every $k\geq 2$ and $α>0$, there is some $η>0$ for which the $k$th power of a Hamilton cycle with high probability appears in $G_α\cup G(n,p)$ when $p=ω(n^{-1/k-η})$. The appearance threshold of the $k$th power of a Hamilton cycle in $G(n,p)$ alone is known to be $n^{-1/k}$, up to a $\log$-term when $k=2$, and exactly for $k>2$.

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arXiv2019

Perfectly packing graphs with bounded degeneracy and many leaves

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

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arXiv2019

Universality for bounded degree spanning trees in randomly perturbed graphs

We solve a problem of Krivelevich, Kwan and Sudakov [SIAM Journal on Discrete Mathematics 31 (2017), 155-171] concerning the threshold for the containment of all bounded degree spanning trees in the model of randomly perturbed dense graphs. More precisely, we show that, if we start with a dense graph $G_α$ on $n$ vertices with $δ(G_α)\ge αn$ for $α>0$ and we add to it the binomial random graph $G(n,C/n)$, then with high probability the graph $G_α\cup G(n,C/n)$ contains copies of all spanning trees with maximum degree at most $Δ$ simultaneously, where $C$ depends only on $α$ and $Δ$.

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arXiv2019

Regularity inheritance in pseudorandom graphs

Advancing the sparse regularity method, we prove one-sided and two-sided regularity inheritance lemmas for subgraphs of bijumbled graphs, improving on results of Conlon, Fox and Zhao [Adv. Math. 256 (2014), 206--290]. These inheritance lemmas also imply improved $H$-counting lemmas for subgraphs of bijumbled graphs, for some $H$.

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arXiv2017

Large-scale structures in random graphs

In recent years there has been much progress in graph theory on questions of the following type. What is the threshold for a certain large substructure to appear in a random graph? When does a random graph contain all structures from a given family? And when does it contain them so robustly that even an adversary who is allowed to perturb the graph cannot destroy all of them? I will survey this progress, and highlight the vital role played by some newly developed methods, such as the sparse regularity method, the absorbing method, and the container method. I will also mention many open questions that remain in this area.

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arXiv2016

Chromatic thresholds in dense random graphs

The chromatic threshold $δ_χ(H,p)$ of a graph $H$ with respect to the random graph $G(n,p)$ is the infimum over $d > 0$ such that the following holds with high probability: the family of $H$-free graphs $G \subset G(n,p)$ with minimum degree $δ(G) \ge dpn$ has bounded chromatic number. The study of the parameter $δ_χ(H) := δ_χ(H,1)$ was initiated in 1973 by Erdős and Simonovits, and was recently determined for all graphs $H$. In this paper we show that $δ_χ(H,p) = δ_χ(H)$ for all fixed $p \in (0,1)$, but that typically $δ_χ(H,p) \ne δ_χ(H)$ if $p = o(1)$. We also make significant progress towards determining $δ_χ(H,p)$ for all graphs $H$ in the range $p = n^{-o(1)}$. In sparser random graphs the problem is somewhat more complicated, and is studied in a separate paper.

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arXiv2016

Chromatic thresholds in sparse random graphs

The chromatic threshold $δ_χ(H,p)$ of a graph $H$ with respect to the random graph $G(n,p)$ is the infimum over $d > 0$ such that the following holds with high probability: the family of $H$-free graphs $G \subset G(n,p)$ with minimum degree $δ(G) \ge dpn$ has bounded chromatic number. The study of $δ_χ(H) :=δ_χ(H,1)$ was initiated in 1973 by Erdős and Simonovits. Recently $δ_χ(H)$ was determined for all graphs $H$. It is known that $δ_χ(H,p) =δ_χ(H)$ for all fixed $p \in (0,1)$, but that typically $δ_χ(H,p) \ne δ_χ(H)$ if $p = o(1)$. Here we study the problem for sparse random graphs. We determine $δ_χ(H,p)$ for most functions $p = p(n)$ when $H\in\{K_3,C_5\}$, and also for all graphs $H$ with $χ(H) \not\in \{3,4\}$.

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arXiv2015

Triangle-free subgraphs of random graphs

Recently there has been much interest in studying random graph analogues of well known classical results in extremal graph theory. Here we follow this trend and investigate the structure of triangle-free subgraphs of $G(n,p)$ with high minimum degree. We prove that asymptotically almost surely each triangle-free spanning subgraph of $G(n,p)$ with minimum degree at least $\big(\frac{2}{5} + o(1)\big)pn$ is $\mathcal O(p^{-1}n)$-close to bipartite, and each spanning triangle-free subgraph of $G(n,p)$ with minimum degree at least $(\frac{1}{3}+\varepsilon)pn$ is $\mathcal O(p^{-1}n)$-close to $r$-partite for some $r=r(\varepsilon)$. These are random graph analogues of a result by Andrásfai, Erdős, and Sós [Discrete Math. 8 (1974), 205-218], and a result by Thomassen [Combinatorica 22 (2002), 591--596]. We also show that our results are best possible up to a constant factor.

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arXiv2014

Tight cycles and regular slices in dense hypergraphs

We study properties of random subcomplexes of partitions returned by (a suitable form of) the Strong Hypergraph Regularity Lemma, which we call regular slices. We argue that these subcomplexes capture many important structural properties of the original hypergraph. Accordingly we advocate their use in extremal hypergraph theory, and explain how they can lead to considerable simplifications in existing proofs in this field. We also use them for establishing the following two new results. Firstly, we prove a hypergraph extension of the Erdős-Gallai Theorem: for every $δ>0$ every sufficiently large $k$-uniform hypergraph with at least $(α+δ)\binom{n}{k}$ edges contains a tight cycle of length $αn$ for each $α\in[0,1]$. Secondly, we find (asymptotically) the minimum codegree requirement for a $k$-uniform $k$-partite hypergraph, each of whose parts has $n$ vertices, to contain a tight cycle of length $αkn$, for each $0<α<1$.

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arXiv2014

An approximate version of the Tree Packing Conjecture

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

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arXiv2014

An extension of Turán's Theorem, uniqueness and stability

We determine the maximum number of edges of an $n$-vertex graph $G$ with the property that none of its $r$-cliques intersects a fixed set $M\subset V(G)$. For $(r-1)|M|\ge n$, the $(r-1)$-partite Turan graph turns out to be the unique extremal graph. For $(r-1)|M|<n$, there is a whole family of extremal graphs, which we describe explicitly. In addition we provide corresponding stability results.

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arXiv2014

A density Corrádi-Hajnal Theorem

We find, for all sufficiently large $n$ and each $k$, the maximum number of edges in an $n$-vertex graph which does not contain $k+1$ vertex-disjoint triangles. This extends a result of Moon [Canad. J. Math. 20 (1968), 96-102] which is in turn an extension of Mantel's Theorem. Our result can also be viewed as a density version of the Corradi-Hajnal Theorem.

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arXiv2014

Powers of Hamilton cycles in pseudorandom graphs

We study the appearance of powers of Hamilton cycles in pseudorandom graphs, using the following comparatively weak pseudorandomness notion. A graph $G$ is $(\varepsilon,p,k,\ell)$-pseudorandom if for all disjoint $X$ and $Y\subset V(G)$ with $|X|\ge\varepsilon p^kn$ and $|Y|\ge\varepsilon p^\ell n$ we have $e(X,Y)=(1\pm\varepsilon)p|X||Y|$. We prove that for all $β>0$ there is an $\varepsilon>0$ such that an $(\varepsilon,p,1,2)$-pseudorandom graph on $n$ vertices with minimum degree at least $βpn$ contains the square of a Hamilton cycle. In particular, this implies that $(n,d,λ)$-graphs with $λ\ll d^{5/2 }n^{-3/2}$ contain the square of a Hamilton cycle, and thus a triangle factor if $n$ is a multiple of $3$. This improves on a result of Krivelevich, Sudakov and Szabó [Triangle factors in sparse pseudo-random graphs, Combinatorica 24 (2004), no. 3, 403--426]. We also extend our result to higher powers of Hamilton cycles and establish corresponding counting versions.

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arXiv2013

An Extension of the Blow-up Lemma to arrangeable graphs

The Blow-up Lemma established by Komlós, Sárközy, and Szemerédi in 1997 is an important tool for the embedding of spanning subgraphs of bounded maximum degree. Here we prove several generalisations of this result concerning the embedding of a-arrangeable graphs, where a graph is called a-arrangeable if its vertices can be ordered in such a way that the neighbours to the right of any vertex v have at most a neighbours to the left of v in total. Examples of arrangeable graphs include planar graphs and, more generally, graphs without a K_s-subdivision for constant s. Our main result shows that a-arrangeable graphs with maximum degree at most sqrt(n)/log(n) can be embedded into corresponding systems of super-regular pairs. This is optimal up to the logarithmic factor. We also present two applications. We prove that any large enough graph G with minimum degree at least ((r-1)/r+γ)n contains an F-factor of every a-arrangeable r-chromatic graph F with at most ξn vertices and maximum degree at most sqrt(n)/log(n), as long as ξ is sufficiently small compared to γ/(ar). This extends a result of Alon and Yuster [J. Combin. Theory Ser. B 66(2),269-282, 1996]. Moreover, we show that for constant p the random graph G(n,p) is universal for the class of a-arrangeable n-vertex graphs H of maximum degree at most ξn/log(n), as long as ξ is sufficiently small compared to p/a.

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arXiv2013

Spanning embeddings of arrangeable graphs with sublinear bandwidth

The Bandwidth Theorem of Böttcher, Schacht and Taraz [Mathematische Annalen 343 (1), 175-205] gives minimum degree conditions for the containment of spanning graphs H with small bandwidth and bounded maximum degree. We generalise this result to a-arrangeable graphs H with Δ(H)<sqrt(n)/log(n), where n is the number of vertices of H. Our result implies that sufficiently large n-vertex graphs G with minimum degree at least (3/4+γ)n contain almost all planar graphs on n vertices as subgraphs. Using techniques developed by Allen, Brightwell and Skokan [Combinatorica, to appear] we can also apply our methods to show that almost all planar graphs H have Ramsey number at most 12|H|. We obtain corresponding results for graphs embeddable on different orientable surfaces.

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arXiv2013

Almost spanning subgraphs of random graphs after adversarial edge removal

Let Delta>1 be a fixed integer. We show that the random graph G(n,p) with p>>(log n/n)^{1/Delta} is robust with respect to the containment of almost spanning bipartite graphs H with maximum degree Delta and sublinear bandwidth in the following sense: asymptotically almost surely, if an adversary deletes arbitrary edges in G(n,p) such that each vertex loses less than half of its neighbours, then the resulting graph still contains a copy of all such H.

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