arXiv subjects

Julia Böttcher

Publications and source records attributed to Julia Böttcher.

At least 19 recordsLinked to original sources

arXiv2026

Making spanning graphs

We prove that for each $D\ge 2$ there exists $c>0$ such that whenever $b\le c\big(\tfrac{n}{\log n}\big)^{1/D}$, in the $(1:b)$ Maker-Breaker game played on $E(K_n)$, Maker has a strategy to guarantee claiming a graph $G$ containing copies of all graphs $H$ with $v(H)\le n$ and $Δ(H)\le D$. We show further that the graph $G$ guaranteed by this strategy also contains copies of any graph $H$ with bounded maximum degree and degeneracy at most $\tfrac{D-1}{2}$. This lower bound on the threshold bias is sharp up to the $\log$-factor when $H$ consists of $\tfrac{n}{3}$ vertex-disjoint triangles or $\tfrac{n}{4}$ vertex-disjoint $K_4$-copies.

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arXiv2026

Partition universality for graphs of bounded degeneracy and degree

We prove asymptotically optimal bounds on the number of edges a graph $G$ must have in order that any $r$-colouring of $E(G)$ has a colour class which contains every $D$-degenerate graph on $n$ vertices with bounded maximum degree. We also improve the upper bounds on the number of edges $G$ must have in order that any $r$-colouring of $E(G)$ has a colour class which contains every $n$-vertex graph with maximum degree $Δ$, for each $Δ\ge 4$. In both cases, we show that a binomial random graph with $Cn$ vertices and a suitable edge probability is likely to provide the desired $G$.

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arXiv2026

Local Resilience for Containment of Bounded Degree Spanning Subgraphs

We prove that for all $Δ\geq 2$ and $γ> 0$, there exists a constant $C = C(Δ, γ)$ such that for $p\geq C(\log n/n)^{1/Δ}$, asymptotically almost surely, every spanning subgraph $G$ of $G(n,p)$ with minimum degree at least $(1-1/(2Δ)+γ)pn$ contains every $n$-vertex graph $H$ with maximum degree at most $Δ$ and with at least $Cp^{-2}$ vertices not in any triangles of $H$. This is a 'sparse local resilience version' of a classical theorem of Sauer and Spencer. The condition that $H$ should contain some vertices not in triangles is necessary, and in fact, the quantity $p^{-2}$ is asymptotically best possible. A key feature of our result is that $H$ is allowed to be an expander graph, distinguishing it from previous results of similar nature, which dealt with, e.g., graphs of sublinear bandwidth. Our proof makes use of regularity arguments, with the sparse blow-up lemma for random graphs being a key tool.

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arXiv2026

The tree packing conjecture for trees of almost linear maximum degree

We prove that there is $c>0$ such that for all sufficiently large $n$, if $T_1,\dots,T_n$ are any trees such that $T_i$ has $i$ vertices and maximum degree at most $cn/\log n$, then $\{T_1,\dots,T_n\}$ packs into $K_n$. Our main result actually allows to replace the host graph $K_n$ by an arbitrary quasirandom graph, and to generalize from trees to graphs of bounded degeneracy that are rich in bare paths, contain some odd degree vertices, and only satisfy much less stringent restrictions on their number of vertices.

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arXiv2026

A generalised transference principle

The last two decades have witnessed a growing trend towards proving sparse random analogues of combinatorial theorems. One unified approach to proving such theorems, formalised by Conlon and Gowers [Ann. of Math. 2016], involves establishing a 'transference principle' which allows one to translate between robust properties in the dense setting and the sparse $p$-random setting, provided $p$ is not too small. Our results provide a more general transference theorem, extending the results of Conlon and Gowers and also those of Schacht [Ann. of Math. 2016]. Among a variety of other applications, we use this to obtain a sparse counting lemma for graphs and hypergraphs which are not necessarily strictly balanced. Our method achieves asymptotically optimal bounds on the probability $p$, and the probability of success.

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arXiv2026

Breaking the Bollobás-Eldridge-Catlin Barrier for Bipartite Graphs

The celebrated Bollobás-Eldridge-Catlin packing conjecture states that every $n$-vertex graph $G$ with minimum degree at least $\big(1-\frac{1}{Δ+1}\big) n$ contains every $n$-vertex graph $H$ of maximum degree at most $Δ$. Despite considerable attention, the conjecture remains widely open. We show that for bipartite $H$ this threshold can be greatly improved: there is an absolute constant $c>0$ such that every $n$-vertex graph $G$ with minimum degree at least $ \big(1-c\frac{\logΔ}Δ\big)n $ contains every $n$-vertex bipartite graph $H$ of maximum degree at most $Δ$, provided $Δ$ is not too large compared to $n$. Moreover, we prove that this logarithmic improvement is best possible up to the value of the constant.

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arXiv2026

A robust Corrádi--Hajnal Theorem

For a graph $G$ and $p\in[0,1]$, we denote by $G_p$ the random sparsification of $G$ obtained by keeping each edge of $G$ independently, with probability $p$. We show that there exists a $C>0$ such that if $p\geq C(\log n)^{1/3}n^{-2/3}$ and $G$ is an $n$-vertex graph with $n\in 3\mathbb{N}$ and $δ(G)\geq \tfrac{2n}{3}$, then with high probability $G_p$ contains a triangle factor. Both the minimum degree condition and the probability condition, up to the choice of $C$, are tight. Our result can be viewed as a common strengthening of the seminal theorems of Corrádi and Hajnal, which deals with the extremal minimum degree condition for containing triangle factors (corresponding to $p=1$ in our result), and Johansson, Kahn and Vu, which deals with the threshold for the appearance of a triangle factor in $G(n,p)$ (corresponding to $G=K_n$ in our result). It also implies a lower bound on the number of triangle factors in graphs with minimum degree at least $\tfrac{2n}{3}$ which gets close to the truth.

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arXiv2026

On Ramsey-type problems for paths and cycles with few colour changes

In 1967, Gerencser and Gyárfás determined the exact values of the two-colour Ramsey numbers of paths. In a footnote, they made the following observation: Every $2$-edge-coloured complete graph contains a Hamilton path with at most one colour change. Later, this led to a challenging and still wide open conjecture about covering edge-coloured complete graphs with monochromatic paths. Inspired by the original statement, we study paths and cycles with few colour changes in $3$-edge-coloured complete graphs. For this, we introduce a new Ramsey-type parameter: For $q,k \in \mathbb{N}$ and a graph $G$, let $R_q^k(G)$ denote the smallest $N \in \mathbb{N}$ such that every $q$-edge-coloured complete graph on $N$ vertices contains a copy of $G$ with at most $k$ vertices that are incident to edges in $G$ of different colours. For paths, we show that $R_3^1(P_n) = \frac{3n}{2} + O(1)$, and for even cycles, we show that $R_3^2(C_n) = \frac{3n}{2} + o(n)$.

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arXiv2025

Bounds for Hypergraph Universality

A graph $Γ$ is said to be universal for a class of graphs $\mathcal{H}$ if $Γ$ contains a copy of every $H \in \mathcal{H}$ as a subgraph. The number of edges required for a host graph $Γ$ to be universal for the class of $D$-degenerate graphs on $n$ vertices has been shown to be $O(n^{2-1/D}(\log n)^{2/D}(\log\log n)^{5})$. We generalise this result to $r$-uniform hypergraphs, showing the following. Given $D, r \ge 2$ and $n$ sufficiently large, there exists a constant $C = C(D, r)$ such that there exists a graph with at most \[Cn^{r-1/D}(\log n)^{2/D}(\log\log n)^{2r+1}\] edges which is universal for the class of $D$-degenerate $r$-uniform hypergraphs on $n$ vertices. This is tight up to the polylogarithmic term.

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arXiv2025

Blow-up lemmas for sparse graphs

The blow-up lemma states that a system of super-regular pairs contains all bounded degree spanning graphs as subgraphs that embed into a corresponding system of complete pairs. This lemma has far-reaching applications in extremal combinatorics. We prove sparse analogues of the blow-up lemma for subgraphs of random and of pseudorandom graphs. Our main results are the following three sparse versions of the blow-up lemma: one for embedding spanning graphs with maximum degree $Δ$ in subgraphs of $G(n,p)$ with $p=C(\log n/n)^{1/Δ}$; one for embedding spanning graphs with maximum degree $Δ$ and degeneracy $D$ in subgraphs of $G(n,p)$ with $p=C_Δ\big(\log n/n\big)^{1/(2D+1)}$; and one for embedding spanning graphs with maximum degree $Δ$ in $(p,cp^{\max(4,(3Δ+1)/2)}n)$-bijumbled graphs. We also consider various applications of these lemmas.

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arXiv2025

Robustness of the Sauer-Spencer Theorem

We prove a robust version of a graph embedding theorem of Sauer and Spencer. To state this sparser analogue, we define $G(p)$ to be a random subgraph of $G$ obtained by retaining each edge of $G$ independently with probability $p \in [0,1]$, and let $m_1(H)$ be the maximum $1$-density of a graph $H$. We show that for any constant $Δ$ and $γ> 0$, if $G$ is an $n$-vertex host graph with minimum degree $δ(G) \geq (1 - 1/2Δ+ γ)n$ and $H$ is an $n$-vertex graph with maximum degree $Δ(H) \leq Δ$, then for $p \geq Cn^{-1/m_1(H)}\log n$, the random subgraph $G(p)$ contains a copy of $H$ with high probability. Our value for $p$ is optimal up to a log-factor. In fact, we prove this result for a more general minimum degree condition on $G$, by introducing an \emph{extension threshold} $δ_{\rm e}(Δ)$, such that the above result holds for graphs $G$ with ${δ(G) \geq (δ_{\rm e}(Δ) + γ)n}$. We show that $δ_{\rm e}(Δ) \leq (2Δ-1)/2Δ$, and further conjecture that $δ_{\rm e}(Δ)$ equals $Δ/(Δ+1)$, which matches the minimum degree condition on $G$ in the Bollobás-Eldridge-Catlin Conjecture. A main tool in our proof is a vertex-spread version of the blow-up lemma of Allen, Böttcher, Hàn, Kohayakawa, and Person, which we believe to be of independent interest.

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arXiv2024

The square of a Hamilton cycle in randomly perturbed graphs

We investigate the appearance of the square of a Hamilton cycle in the model of randomly perturbed graphs, which is, for a given $α\in (0,1)$, the union of any $n$-vertex graph with minimum degree $αn$ and the binomial random graph $G(n,p)$. This is known when $α> 1/2$, and we determine the exact perturbed threshold probability in all the remaining cases, i.e., for each $α\le 1/2$. We demonstrate that, as $α$ ranges over the interval $(0,1)$, the threshold performs a countably infinite number of `jumps'. Our result has implications on the perturbed threshold for $2$-universality, where we also fully address all open cases.

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arXiv2023

Universality for graphs of bounded degeneracy

Given a family $\mathcal{H}$ of graphs, a graph $G$ is called $\mathcal{H}$-universal if $G$ contains every graph of $\mathcal{H}$ as a subgraph. Following the extensive research on universal graphs of small size for bounded-degree graphs, Alon asked what is the minimum number of edges that a graph must have to be universal for the class of all $n$-vertex graphs that are $D$-degenerate. In this paper, we answer this question up to a factor that is polylogarithmic in $n.$

math.CO↗

arXiv2023

Graphs with large minimum degree and no small odd cycles are $3$-colourable

Answering a question by Letzter and Snyder, we prove that for large enough $k$ any $n$-vertex graph $G$ with minimum degree at least $\frac{1}{2k-1}n$ and without odd cycles of length less than $2k+1$ is $3$-colourable. In fact, we prove a stronger result that works with a slightly smaller minimum degree.

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arXiv2022

Triangles in randomly perturbed graphs

We study the problem of finding pairwise vertex-disjoint triangles in the randomly perturbed graph model, which is the union of any $n$-vertex graph $G$ satisfying a given minimum degree condition and the binomial random graph $G(n,p)$. We prove that asymptotically almost surely $G \cup G(n,p)$ contains at least $\min\{δ(G), \lfloor n/3 \rfloor\}$ pairwise vertex-disjoint triangles, provided $p \ge C \log n/n$, where $C$ is a large enough constant. This is a perturbed version of an old result of Dirac. Our result is asymptotically optimal and answers a question of Han, Morris, and Treglown [RSA, 2021, no. 3, 480--516] in a strong form. We also prove a stability version of our result, which in the case of pairwise vertex-disjoint triangles extends a result of Han, Morris, and Treglown [RSA, 2021, no. 3, 480--516]. Together with a result of Balogh, Treglown, and Wagner [CPC, 2019, no. 2, 159--176] this fully resolves the existence of triangle factors in randomly perturbed graphs. We believe that the methods introduced in this paper are useful for a variety of related problems: we discuss possible generalisations to clique factors, cycle factors, and $2$-universality.

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arXiv2022

Packing degenerate graphs

Given $D$ and $γ>0$, whenever $c>0$ is sufficiently small and $n$ sufficiently large, if $\mathcal{G}$ is a family of $D$-degenerate graphs of individual orders at most $n$, maximum degrees at most $\tfrac{cn}{\log n}$, and total number of edges at most $(1-γ)\binom{n}{2}$, then $\mathcal{G}$ packs into the complete graph $K_{n}$. Our proof proceeds by analysing a natural random greedy packing algorithm. This version of the manuscript corrects a small error that appeared in the published version [Adv Math, 354 (2019), 106739].

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arXiv2022

Partitioning a 2-edge-coloured graph of minimum degree $2n/3 + o(n)$ into three monochromatic cycles

Lehel conjectured in the 1970s that every red and blue edge-coloured complete graph can be partitioned into two monochromatic cycles. This was confirmed in 2010 by Bessy and Thomassé. However, the host graph $G$ does not have to be complete. It it suffices to require that $G$ has minimum degree at least $3n/4$, where $n$ is the order of $G$, as was shown recently by Letzter, confirming a conjecture of Balogh, Barát, Gerbner, Gyárfás and Sárközy. This degree condition is asymptotically tight. Here we continue this line of research, by proving that for every red and blue edge-colouring of an $n$-vertex graph of minimum degree at least $2n/3 + o(n)$, there is a partition of the vertex set into three monochromatic cycles. This approximately verifies a conjecture of Pokrovskiy and is essentially tight.

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arXiv2021

Cycle factors in randomly perturbed graphs

We study the problem of finding pairwise vertex-disjoint copies of the $\ell$-vertex cycle $C_\ell$ in the randomly perturbed graph model, which is the union of a deterministic $n$-vertex graph $G$ and the binomial random graph $G(n,p)$. For $\ell \ge 3$ we prove that asymptotically almost surely $G \cup G(n,p)$ contains $\min \{δ(G), \lfloor n/\ell \rfloor \}$ pairwise vertex-disjoint cycles $C_\ell$, provided $p \ge C \log n/n$ for $C$ sufficiently large. Moreover, when $δ(G) \geαn$ with $0<α\le 1/\ell$ and $G$ is not `close' to the complete bipartite graph $K_{αn,(1-α) n}$, then $p \ge C/n$ suffices to get the same conclusion. This provides a stability version of our result. In particular, we conclude that $p \ge C/n$ suffices when $α>1/\ell$ for finding $\lfloor n/\ell \rfloor$ cycles $C_\ell$. Our results are asymptotically optimal. They can be seen as an interpolation between the Johansson--Kahn--Vu Theorem for $C_\ell$-factors and the resolution of the El-Zahar Conjecture for $C_\ell$-factors by Abbasi.

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