arXiv subjects
Publications and source records attributed to Stefan Glock.
For every fixed dimension $d$ and sufficiently large $n$, we determine the maximum possible diameter of a strongly connected $d$-dimensional simplicial complex on $n$ vertices. This improves on a sequence of previous results and settles a problem of Santos from 2013. On the way, as a special case, we also characterise the existence of an extra-tight Euler tour in the complete $d$-uniform hypergraph on $n$ vertices.
In 1963, Anton Kotzig famously conjectured that $K_{n}$, the complete graph of order $n$, where $n$ is even, can be decomposed into $n-1$ perfect matchings such that every pair of these matchings forms a Hamilton cycle. The problem is still wide open and here we consider a variant of it for the binomial random graph $G(n,p)$. We prove that, for every fixed $k$, there exists a constant $C=C(k)$ such that, when $p\ge \frac{C \log n}{n}$, with high probability, $G(n,p)$ contains $k$ edge-disjoint perfect matchings with the property that every pair of them forms a Hamilton cycle. In fact, our main result is a very precise counting result for $K_n$. We show that, given any $k$ edge-disjoint perfect matchings $M_1,\dots,M_k$, the probability that a uniformly random perfect matching $M^*$ in $K_n$ has the property that $M^*\cup M_i$ forms a Hamilton cycle for each $i\in [k]$ is $\Theta_k(n^{-k/2})$. This is proved by building on a variety of methods, including a random process analysis, the absorption method, the entropy method and the switching method. The result on the binomial random graph follows from a slight strengthening of our counting result via the recent breakthroughs on the expectation threshold conjecture.
In this paper, we initiate the study of discrepancy questions for combinatorial designs. Specifically, we show that, for every fixed $r\ge 3$ and $n\equiv 1,3 \pmod{6}$, any $r$-colouring of the triples on $[n]$ admits a Steiner triple system of order $n$ with discrepancy $\Omega(n^2)$. This is not true for $r=2$, but we are able to asymptotically characterise all $2$-colourings which do not contain a Steiner triple system with high discrepancy. The key step in our proofs is a characterization of 3-uniform hypergraphs avoiding a certain natural type of induced subgraphs, contributing to the structural theory of hypergraphs.
Confirming a conjecture of Erd\H{o}s on the chromatic number of Kneser hypergraphs, Alon, Frankl and Lov\'asz proved that in any $q$-colouring of the edges of the complete $r$-uniform hypergraph, there exists a monochromatic matching of size $\lfloor \frac{n+q-1}{r+q-1}\rfloor$. In this paper, we prove a transference version of this theorem. More precisely, for fixed $q$ and $r$, we show that with high probability, a monochromatic matching of approximately the same size exists in any $q$-colouring of a random hypergraph, already when the average degree is a sufficiently large constant. In fact, our main new result is a defect version of the Alon--Frankl--Lov\'asz theorem for almost complete hypergraphs. From this, the transference version is obtained via a variant of the weak hypergraph regularity lemma. The proof of the defect version uses tools from extremal set theory developed in the study of the Erd\H{o}s matching conjecture.
Let $f^{(r)}(n;s,k)$ denote the maximum number of edges in an $n$-vertex $r$-uniform hypergraph containing no subgraph with $k$ edges and at most $s$ vertices. Brown, Erd\H{o}s and S\'os [New directions in the theory of graphs (Proc. Third Ann Arbor Conf., Univ. Michigan 1971), pp. 53--63, Academic Press 1973] conjectured that the limit $\lim_{n\rightarrow \infty}n^{-2}f^{(3)}(n;k+2,k)$ exists for all $k$. The value of the limit was previously determined for $k=2$ in the original paper of Brown, Erd\H{o}s and S\'os, for $k=3$ by Glock [Bull. Lond. Math. Soc. 51 (2019) 230--236] and for $k=4$ by Glock, Joos, Kim, K\"uhn, Lichev and Pikhurko [Proc. Amer. Math. Soc., Series B, 11 (2024) 173-186] while Delcourt and Postle [Proc. Amer. Math. Soc., 152 (2024), 1881-1891] proved the conjecture (without determining the limiting value). In this paper, we determine the value of the limit in the Brown-Erd\H{o}s-S\'os Problem for $k\in \{5,6,7\}$. More generally, we obtain the value of $\lim_{n\rightarrow \infty}n^{-2}f^{(r)}(n;rk-2k+2,k)$ for all $r\geq 3$ and $k\in \{5,6,7\}$. In addition, by combining these new values with recent results of Bennett, Cushman and Dudek [arXiv:2309.00182] we obtain new asymptotic values for several generalised Ramsey numbers.
In this paper, we study discrepancy questions for spanning subgraphs of $k$-uniform hypergraphs. Our main result is that, for any integers $k \ge 3$ and $r \ge 2$, any $r$-colouring of the edges of a $k$-uniform $n$-vertex hypergraph $G$ with minimum $(k-1)$-degree $\delta(G) \ge (1/2+o(1))n$ contains a tight Hamilton cycle with high discrepancy, that is, with at least $n/r+\Omega(n)$ edges of one colour. The minimum degree condition is asymptotically best possible and our theorem also implies a corresponding result for perfect matchings. Our tools combine various structural techniques such as Tur\'an-type problems and hypergraph shadows with probabilistic techniques such as random walks and the nibble method. We also propose several intriguing problems for future research.
In his seminal 1976 paper, Pósa showed that for all $p\geq C\log n/n$, the binomial random graph $G(n,p)$ is with high probability Hamiltonian. This leads to the following natural questions, which have been extensively studied: How well is it typically possible to cover all edges of $G(n,p)$ with Hamilton cycles? How many cycles are necessary? In this paper we show that for $ p\geq C\log n/n$, we can cover $G\sim G(n,p)$ with precisely $\lceilΔ(G)/2\rceil$ Hamilton cycles. Our result is clearly best possible both in terms of the number of required cycles, and the asymptotics of the edge probability $p$, since it starts working at the weak threshold needed for Hamiltonicity. This resolves a problem of Glebov, Krivelevich and Szabó, and improves upon previous work of Hefetz, Kühn, Lapinskas and Osthus, and of Ferber, Kronenberg and Long, essentially closing a long line of research on Hamiltonian packing and covering problems in random graphs.
Let $f^{(r)}(n;s,k)$ be the maximum number of edges of an $r$-uniform hypergraph on $n$ vertices not containing a subgraph with $k$ edges and at most $s$ vertices. In 1973, Brown, Erdős and Sós conjectured that the limit $$\lim_{n\to \infty} n^{-2} f^{(3)}(n;k+2,k)$$ exists for all $k$ and confirmed it for $k=2$. Recently, Glock showed this for $k=3$. We settle the next open case, $k=4$, by showing that $f^{(3)}(n;6,4)=\left(\frac{7}{36}+o(1)\right)n^2$ as $n\to\infty$. More generally, for all $k\in \{3,4\}$, $r\ge 3$ and $t\in [2,r-1]$, we compute the value of the limit $\lim_{n\to \infty} n^{-t}f^{(r)}(n;k(r-t)+t,k)$, which settles a problem of Shangguan and Tamo.
Finding general conditions which ensure that a graph is Hamiltonian is a central topic in graph theory. An old and well known conjecture in the area states that any $d$-regular $n$-vertex graph $G$ whose second largest eigenvalue in absolute value $λ(G)$ is at most $\frac{d}{C}$, for some universal constant $C>0$, has a Hamilton cycle. In this paper, we obtain two main results which make substantial progress towards this problem. Firstly, we settle this conjecture in full when the degree $d$ is at least a small power of $n$. Secondly, in the general case we show that $λ(G) \leq \frac{d}{C(\log n)^{1/3}}$ implies the existence of a Hamilton cycle, improving the 20-year old bound of $\frac{d}{ \log^{1-o(1)} n}$ of Krivelevich and Sudakov. We use in a novel way a variety of methods, such as a robust Pósa rotation-extension technique, the Friedman-Pippenger tree embedding with rollbacks and the absorbing method, combined with additional tools and ideas. Our results have several interesting applications, giving best bounds on the number of generators which guarantee the Hamiltonicity of random Cayley graphs, which is an important partial case of the well known Hamiltonicity conjecture of Lovász. They can also be used to improve a result of Alon and Bourgain on additive patterns in multiplicative subgroups.
An $n$-queens configuration is a placement of $n$ mutually non-attacking queens on an $n\times n$ chessboard. The $n$-queens completion problem, introduced by Nauck in 1850, is to decide whether a given partial configuration can be completed to an $n$-queens configuration. In this paper, we study an extremal aspect of this question, namely: how small must a partial configuration be so that a completion is always possible? We show that any placement of at most $n/60$ mutually non-attacking queens can be completed. We also provide partial configurations of roughly $n/4$ queens that cannot be completed, and formulate a number of interesting problems. Our proofs connect the queens problem to rainbow matchings in bipartite graphs and use probabilistic arguments together with linear programming duality.
A celebrated theorem of Pippenger, and Frankl and Rödl states that every almost-regular, uniform hypergraph $\mathcal{H}$ with small maximum codegree has an almost-perfect matching. We extend this result by obtaining a ``conflict-free'' matching, where conflicts are encoded via a collection $\mathcal{C}$ of subsets $C\subseteq E(\mathcal{H})$. We say that a matching $\mathcal{M}\subseteq E(\mathcal{H})$ is conflict-free if $\mathcal{M}$ does not contain an element of $\mathcal{C}$ as a subset. Under natural assumptions on $\mathcal{C}$, we prove that $\mathcal{H}$ has a conflict-free, almost-perfect matching. This has many applications, one of which yields new asymptotic results for so-called ``high-girth'' Steiner systems. Our main tool is a random greedy algorithm which we call the ``conflict-free matching process''.
We present a modification of the Depth first search algorithm, suited for finding long induced paths. We use it to give simple proofs of the following results. We show that the induced size-Ramsey number of paths satisfies $\hat{R}_{\mathrm{ind}}(P_n)\leq 5\cdot 10^7n$, thus giving an explicit constant in the linear bound, improving the previous bound with a large constant from a regularity lemma argument by Haxell, Kohayakawa and Łuczak. We also provide a bound for the $k$-color version, showing that $\hat{R}_{\mathrm{ind}}^k(P_n)=O(k^3\log^4k)n$. Finally, we present a new short proof of the fact that the binomial random graph in the supercritical regime, $G(n,\frac{1+\varepsilon}{n})$, contains typically an induced path of length $Θ(\varepsilon^2) n$.
We show that for any $d=d(n)$ with $d_0(ε) \le d =o(n)$, with high probability, the size of a largest induced cycle in the random graph $G(n,d/n)$ is $(2\pm ε)\frac{n}{d}\log d$. This settles a long-standing open problem in random graph theory.
In 1994, Erdős, Gyárfás and Łuczak posed the following problem: given disjoint vertex sets $V_1,\dots,V_n$ of size~$k$, with exactly one edge between any pair $V_i,V_j$, how large can $n$ be such that there will always be an independent transversal? They showed that the maximal $n$ is at most $(1+o(1))k^2$, by providing an explicit construction with these parameters and no independent transversal. They also proved a lower bound which is smaller by a $2e$-factor. In this paper, we solve this problem by showing that their upper bound construction is best possible: if $n\le (1-o(1))k^2$, there will always be an independent transversal. In fact, this result is a very special case of a much more general theorem which concerns independent transversals in arbitrary partite graphs that are `locally sparse', meaning that the maximum degree between each pair of parts is relatively small. In this setting, Loh and Sudakov provided a global \emph{maximum} degree condition for the existence of an independent transversal. We show that this can be relaxed to an \emph{average} degree condition. We can also use our new theorem to establish tight bounds for a more general version of the Erdős--Gyárfás--Łuczak problem and solve a conjecture of Yuster from 1997. This exploits a connection to the Turán numbers of complete bipartite graphs, which might be of independent interest.
We survey recent advances in the theory of graph and hypergraph decompositions, with a focus on extremal results involving minimum degree conditions. We also collect a number of intriguing open problems, and formulate new ones.
The MaxCut problem asks for the size ${\rm mc}(G)$ of a largest cut in a graph $G$. It is well known that ${\rm mc}(G)\ge m/2$ for any $m$-edge graph $G$, and the difference ${\rm mc}(G)-m/2$ is called the surplus of $G$. The study of the surplus of $H$-free graphs was initiated by Erdős and Lovász in the 70s, who in particular asked what happens for triangle-free graphs. This was famously resolved by Alon, who showed that in the triangle-free case the surplus is $Ω(m^{4/5})$, and found constructions matching this bound. We prove several new results in this area. Firstly, we show that for every fixed odd $r\ge 3$, any $C_r$-free graph with $m$ edges has surplus $Ω_r\big(m^{\frac{r+1}{r+2}}\big)$. This is tight, as is shown by a construction of pseudorandom $C_r$-free graphs due to Alon and Kahale. It improves previous results of several researchers, and complements a result of Alon, Krivelevich and Sudakov which is the same bound when $r$ is even. Secondly, generalizing the result of Alon, we allow the graph to have triangles, and show that if the number of triangles is a bit less than in a random graph with the same density, then the graph has large surplus. For regular graphs our bounds on the surplus are sharp. Thirdly, we prove that an $n$-vertex graph with few copies of $K_r$ and average degree $d$ has surplus $Ω_r(d^{r-1}/n^{r-3})$, which is tight when $d$ is close to $n$ provided that a conjectured dense pseudorandom $K_r$-free graph exists. This result is used to improve the best known lower bound (as a function of $m$) on the surplus of $K_r$-free graphs. Our proofs combine techniques from semidefinite programming, probabilistic reasoning, as well as combinatorial and spectral arguments.
We show that for $d\ge d_0(ε)$, with high probability, the random graph $G(n,d/n)$ contains an induced path of length $(3/2-ε)\frac{n}{d}\log d$. This improves a result obtained independently by Luczak and Suen in the early 90s, and answers a question of Fernandez de la Vega. Along the way, we generalize a recent result of Cooley, Draganić, Kang and Sudakov who studied the analogous problem for induced matchings.
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 $n$.