arXiv subjects

Jaehoon Kim

Publications and source records attributed to Jaehoon Kim.

At least 73 records · Page 4Linked to original sources

arXiv2021

Embedding clique-factors in graphs with low $\ell$-independence number

The following question was proposed by Nenadov and Pehova and reiterated by Knierim and Su: Given integers $\ell,r$ and $n$ with $n\in r\mathbb{N}$, is it true that every $n$-vertex graph $G$ with $δ(G) \ge \max \{ \frac{1}{2},\frac{r - \ell}{r} \}n + o(n)$ and $α_{\ell}(G) = o(n) $ contains a $K_{r}$-factor? We give a negative answer for the case when $\ell\ge \frac{3r}{4}$ by giving a family of constructions using the so-called cover thresholds and show that the minimum degree condition given by our construction is asymptotically best possible. That is, for all integers $r,\ell$ with $r > \ell \ge \frac{3}{4}r$ and $μ>0$, there exist $α> 0$ and $N$ such that for every $n\in r\mathbb{N}$ with $n>N$, every $n$-vertex graph $G$ with $δ(G) \ge \left( \frac{1}{2-\varrho_{\ell}(r-1)} + μ\right)n $ and $α_{\ell}(G) \le αn$ contains a $K_{r}$-factor. Here $\varrho_{\ell}(r-1)$ is the Ramsey--Turán density for $K_{r-1}$ under the $\ell$-independence number condition.

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arXiv2021

Nested cycles with no geometric crossings

In 1975, Erdős asked the following question: what is the smallest function $f(n)$ for which all graphs with $n$ vertices and $f(n)$ edges contain two edge-disjoint cycles $C_1$ and $C_2$, such that the vertex set of $C_2$ is a subset of the vertex set of $C_1$ and their cyclic orderings of the vertices respect each other? We prove the optimal linear bound $f(n)=O(n)$ using sublinear expanders.

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arXiv2021

$2$-complexes with unique embeddings in 3-space

A well-known theorem of Whitney states that a 3-connected planar graph admits an essentially unique embedding into the 2-sphere. We prove a 3-dimensional analogue: a simply-connected $2$-complex every link graph of which is 3-connected admits an essentially unique locally flat embedding into the 3-sphere, if it admits one at all. This can be thought of as a generalisation of the 3-dimensional Schoenflies theorem.

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arXiv2021

Fractional Helly theorem for Cartesian products of convex sets

Helly's theorem and its variants show that for a family of convex sets in Euclidean space, local intersection patterns influence global intersection patterns. A classical result of Eckhoff in 1988 provided an optimal fractional Helly theorem for axis-aligned boxes, which are Cartesian products of line segments. Answering a question raised by Bárány and Kalai, and independently Lew, we generalize Eckhoff's result to Cartesian products of convex sets in all dimensions. In particular, we prove that given $α\in (1-\frac{1}{t^d},1]$ and a finite family $\mathcal{F}$ of Cartesian products of convex sets $\prod_{i\in[t]}A_i$ in $\mathbb{R}^{td}$ with $A_i\subset \mathbb{R}^d$ if at least $α$-fraction of the $(d+1)$-tuples in $\mathcal{F}$ are intersecting then at least $(1-(t^d(1-α))^{1/(d+1)})$-fraction of sets in $\mathcal{F}$ are intersecting. This is a special case of a more general result on intersections of $d$-Leray complexes. We also provide a construction showing that our result on $d$-Leray complexes is optimal. Interestingly the extremal example is representable as a family of cartesian products of convex sets, implying the bound $α>1-\frac{1}{t^d}$ and the fraction $(1-(t^d(1-α))^{1/(d+1)})$ above are also best possible. The well-known optimal construction for fractional Helly theorem for convex sets in $\mathbb{R}^d$ does not have $(p,d+1)$-condition for sublinear $p$. Inspired by this we give constructions showing that, somewhat surprisingly, imposing additional $(p,d+1)$-condition has negligible effect on improving the quantitative bounds in neither the fractional Helly theorem for convex sets nor Cartesian products of convex sets. Our constructions offer a rich family of distinct extremal configurations for fractional Helly theorem, implying in a sense that the optimal bound is stable.

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arXiv2021

The Erdős-Hajnal property for graphs with no fixed cycle as a pivot-minor

We prove that for every integer $k$, there exists $\varepsilon > 0$ such that for every n-vertex graph $G$ with no pivot-minor isomorphic to $C_k$, there exist disjoint sets $A,B \subseteq V(G)$ such that $|A|,|B| \geq \varepsilon n$, and $A$ is either complete or anticomplete to $B$. This proves the analog of the Erdős-Hajnal conjecture for the class of graphs with no pivot-minor isomorphic to $C_k$.

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arXiv2021

Resolution of the Oberwolfach problem

The Oberwolfach problem, posed by Ringel in 1967, asks for a decomposition of $K_{2n+1}$ into edge-disjoint copies of a given $2$-factor. We show that this can be achieved for all large $n$. We actually prove a significantly more general result, which allows for decompositions into more general types of factors. In particular, this also resolves the Hamilton-Waterloo problem for large $n$.

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arXiv2020

Asymptotic Structure for the Clique Density Theorem

The famous Erdős-Rademacher problem asks for the smallest number of $r$-cliques in a graph with the given number of vertices and edges. Despite decades of active attempts, the asymptotic value of this extremal function for all $r$ was determined only recently, by Reiher [Annals of Mathematics, 184 (2016) 683--707]. Here we describe the asymptotic structure of all almost extremal graphs. This task for $r=3$ was previously accomplished by Pikhurko and Razborov [Combinatorics, Probability and Computing, 26 (2017) 138--160].

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arXiv2020

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.

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arXiv2020

$K_{r+1}$-saturated graphs with small spectral radius

For a graph $H$, a graph $G$ is $H$-saturated if $G$ does not contain $H$ as a subgraph but for any $e \in E(\overline{G})$, $G+e$ contains $H$. In this note, we prove a sharp lower bound for the number of paths and walks on length $2$ in $n$-vertex $K_{r+1}$-saturated graphs. We then use this bound to give a lower bound on the spectral radii of such graphs which is asymptotically tight for each fixed $r$ and $n\to\infty$.

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arXiv2019

Resilient degree sequences with respect to Hamilton cycles and matchings in random graphs

Pósa's theorem states that any graph $G$ whose degree sequence $d_1 \le \ldots \le d_n$ satisfies $d_i \ge i+1$ for all $i < n/2$ has a Hamilton cycle. This degree condition is best possible. We show that a similar result holds for suitable subgraphs $G$ of random graphs, i.e. we prove a `resilience version' of Pósa's theorem: if $pn \ge C \log n$ and the $i$-th vertex degree (ordered increasingly) of $G \subseteq G_{n,p}$ is at least $(i+o(n))p$ for all $i<n/2$, then $G$ has a Hamilton cycle. This is essentially best possible and strengthens a resilience version of Dirac's theorem obtained by Lee and Sudakov. Chvátal's theorem generalises Pósa's theorem and characterises all degree sequences which ensure the existence of a Hamilton cycle. We show that a natural guess for a resilience version of Chvátal's theorem fails to be true. We formulate a conjecture which would repair this guess, and show that the corresponding degree conditions ensure the existence of a perfect matching in any subgraph of $G_{n,p}$ which satisfies these conditions. This provides an asymptotic characterisation of all degree sequences which resiliently guarantee the existence of a perfect matching.

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arXiv2019

Rainbow structures in locally bounded colourings of graphs

We prove several results on approximate decompositions of edge-coloured quasirandom graphs into rainbow spanning structures. More precisely, we say that an edge-colouring of a graph is locally $\ell$-bounded if no vertex is incident to more than $\ell$ edges of any given colour, and that it is (globally) $g$-bounded if no colour appears more than $g$ times in the colouring. Note that every proper colouring of an $n$-vertex graph is locally $1$-bounded, and (globally) $n/2$-bounded. Our results imply the following: (i) The existence of approximate decompositions of edge-coloured $K_n$ into rainbow almost-spanning cycles, provided that the colouring is $\frac{n}{2}$-bounded and locally $o(n)$-bounded. (ii) The existence of approximate decompositions of edge-coloured $K_n$ into rainbow Hamilton cycles, provided that the colouring is $(1-o(1))\frac n2$-bounded and locally $o\big(\frac{n}{\log^4 n}\big)$-bounded. (iii) A bipartite version of our results implies that every $n\times n$ array, where each symbol appears $(1-o(1))n$ times in total and appears only $o\big(\frac{n}{\log^2 n}\big)$ times in each row or column, has an approximate decomposition into full transversals. We also prove analogues of (i) and (ii) for $F$-factors, where $F$ is any fixed graph. Apart from the logarithmic factor in (ii), all these bounds are essentially best possible. (i) can be viewed as a generalization of a recent result of Alon, Pokrovskiy and Sudakov, who showed the existence of an almost spanning cycle in a properly coloured complete graph. Both (i) and (ii) imply approximate versions of a conjecture of Brualdi and Hollingsworth, stating that every properly edge-coloured complete graph can be decomposed into rainbow spanning trees.

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arXiv2019

On 1-factors with prescribed lengths in tournaments

Kühn, Osthus, and Townsend asked whether there exists a constant $C$ such that every strongly $Ct$-connected tournament contains all possible $1$-factors with at most $t$ components. We answer this question in the affirmative. This is best possible up to constant. In addition, we can ensure that each cycle in the $1$-factor contains a prescribed vertex. Indeed, we derive this result from a more general result on partitioning digraphs which are close to semicomplete. More precisely, we prove that there exists a constant $C$ such that for any $k\geq 1$, if a strongly $Ck^4t$-connected digraph $D$ is close to semicomplete, then we can partition $D$ into $t$ strongly $k$-connected subgraphs with prescribed sizes, provided that the prescribed sizes are $Ω(n)$. This result improves the earlier result of Kühn, Osthus, and Townsend. Here, the condition of connectivity being linear in $t$ is best possible, and the condition of prescribed size being $Ω(n)$ is also best possible.

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arXiv2019

Optimal packings of bounded degree trees

We prove that if $T_1,\dots, T_n$ is a sequence of bounded degree trees so that $T_i$ has $i$ vertices, then $K_n$ has a decomposition into $T_1,\dots, T_n$. This shows that the tree packing conjecture of Gyárfás and Lehel from 1976 holds for all bounded degree trees (in fact, we can allow the first $o(n)$ trees to have arbitrary degrees). Similarly, we show that Ringel's conjecture from 1963 holds for all bounded degree trees. We deduce these results from a more general theorem, which yields decompositions of dense quasi-random graphs into suitable families of bounded degree graphs. Our proofs involve Szemerédi's regularity lemma, results on Hamilton decompositions of robust expanders, random walks, iterative absorption as well as a recent blow-up lemma for approximate decompositions.

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arXiv2018

On the rational Turán exponents conjecture

The extremal number $\mathrm{ex}(n,F)$ of a graph $F$ is the maximum number of edges in an $n$-vertex graph not containing $F$ as a subgraph. A real number $r \in [1,2]$ is realisable if there exists a graph $F$ with $\mathrm{ex}(n , F) = Θ(n^r)$. Several decades ago, Erdős and Simonovits conjectured that every rational number in $[1,2]$ is realisable. Despite decades of effort, the only known realisable numbers are $0,1, \frac{7}{5}, 2$, and the numbers of the form $1+\frac{1}{m}$, $2-\frac{1}{m}$, $2-\frac{2}{m}$ for integers $m \geq 1$. In particular, it is not even known whether the set of all realisable numbers contains a single limit point other than two numbers $1$ and $2$. In this paper, we make progress on the conjecture of Erdős and Simonovits. First, we show that $2 - \frac{a}{b}$ is realisable for any integers $a,b \geq 1$ with $b>a$ and $b \equiv \pm 1 ~({\rm mod}\:a)$. This includes all previously known ones, and gives infinitely many limit points $2-\frac{1}{m}$ in the set of all realisable numbers as a consequence. Secondly, we propose a conjecture on subdivisions of bipartite graphs. Apart from being interesting on its own, we show that, somewhat surprisingly, this subdivision conjecture in fact implies that every rational number between 1 and 2 is realisable.

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arXiv2018

A bandwidth theorem for approximate decompositions

We provide a degree condition on a regular $n$-vertex graph $G$ which ensures the existence of a near optimal packing of any family $\mathcal H$ of bounded degree $n$-vertex $k$-chromatic separable graphs into $G$. In general, this degree condition is best possible. Here a graph is separable if it has a sublinear separator whose removal results in a set of components of sublinear size. Equivalently, the separability condition can be replaced by that of having small bandwidth. Thus our result can be viewed as a version of the bandwidth theorem of Böttcher, Schacht and Taraz in the setting of approximate decompositions. More precisely, let $δ_k$ be the infimum over all $δ\ge 1/2$ ensuring an approximate $K_k$-decomposition of any sufficiently large regular $n$-vertex graph $G$ of degree at least $δn$. Now suppose that $G$ is an $n$-vertex graph which is close to $r$-regular for some $r \ge (δ_k+o(1))n$ and suppose that $H_1,\dots,H_t$ is a sequence of bounded degree $n$-vertex $k$-chromatic separable graphs with $\sum_i e(H_i) \le (1-o(1))e(G)$. We show that there is an edge-disjoint packing of $H_1,\dots,H_t$ into $G$. If the $H_i$ are bipartite, then $r\geq (1/2+o(1))n$ is sufficient. In particular, this yields an approximate version of the tree packing conjecture in the setting of regular host graphs $G$ of high degree. Similarly, our result implies approximate versions of the Oberwolfach problem, the Alspach problem and the existence of resolvable designs in the setting of regular host graphs of high degree.

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arXiv2018

Tree decompositions of graphs without large bipartite holes

A recent result of Condon, Kim, Kühn and Osthus implies that for any $r\geq (\frac{1}{2}+o(1))n$, an $n$-vertex almost $r$-regular graph $G$ has an approximate decomposition into any collections of $n$-vertex bounded degree trees. In this paper, we prove that a similar result holds for an almost $αn$-regular graph $G$ with any $α>0$ and a collection of bounded degree trees on at most $(1-o(1))n$ vertices if $G$ does not contain large bipartite holes. This result is sharp in the sense that it is necessary to exclude large bipartite holes and we cannot hope for an approximate decomposition into $n$-vertex trees. Moreover, this implies that for any $α>0$ and an $n$-vertex almost $αn$-regular graph $G$, with high probability, the randomly perturbed graph $G\cup \mathbf{G}(n,O(\frac{1}{n}))$ has an approximate decomposition into all collections of bounded degree trees of size at most $(1-o(1))n$ simultaneously. This is the first result considering an approximate decomposition problem in the context of Ramsey-Turán theory and the randomly perturbed graph model.

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arXiv2018

Two conjectures in Ramsey-Turán theory

Given graphs $H_1,\ldots, H_k$, a graph $G$ is $(H_1,\ldots, H_k)$-free if there is a $k$-edge-colouring $ϕ:E(G)\rightarrow [k]$ with no monochromatic copy of $H_i$ with edges of colour $i$ for each $i\in[k]$. Fix a function $f(n)$, the Ramsey-Turán function $\textrm{RT}(n,H_1,\ldots,H_k,f(n))$ is the maximum number of edges in an $n$-vertex $(H_1,\ldots,H_k)$-free graph with independence number at most $f(n)$. We determine $\textrm{RT}(n,K_3,K_s,δn)$ for $s\in\{3,4,5\}$ and sufficiently small $δ$, confirming a conjecture of Erdős and Sós from 1979. It is known that $\textrm{RT}(n,K_8,f(n))$ has a phase transition at $f(n)=Θ(\sqrt{n\log n})$. However, the values of $\textrm{RT}(n,K_8, o(\sqrt{n\log n}))$ was not known. We determined this value by proving $\textrm{RT}(n,K_8,o(\sqrt{n\log n}))=\frac{n^2}{4}+o(n^2)$, answering a question of Balogh, Hu and Simonovits. The proofs utilise, among others, dependent random choice and results from graph packings.

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arXiv2018

Spanning trees in randomly perturbed graphs

A classical result of Komlós, Sárközy and Szemerédi states that every $n$-vertex graph with minimum degree at least $(1/2+ o(1))n$ contains every $n$-vertex tree with maximum degree $O(n/\log{n})$ as a subgraph, and the bounds on the degree conditions are sharp. On the other hand, Krivelevich, Kwan and Sudakov recently proved that for every $n$-vertex graph $G_α$ with minimum degree at least $αn$ for any fixed $α>0$ and every $n$-vertex tree $T$ with bounded maximum degree, one can still find a copy of $T$ in $G_α$ with high probability after adding $O(n)$ randomly-chosen edges to $G_α$. We extend their results to trees with unbounded maximum degree. More precisely, for a given $n^{o(1)}\leq Δ\leq cn/\log n$ and $α>0$, we determine the precise number (up to a constant factor) of random edges that we need to add to an arbitrary $n$-vertex graph $G_α$ with minimum degree $αn$ in order to guarantee a copy of any fixed $n$-vertex tree $T$ with maximum degree at most~$Δ$ with high probability.

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