arXiv subjects
Publications and source records attributed to Felix Joos.
Menger's theorem on $A$--$B$-paths and Gallai's theorem on $A$-paths are among the most useful results in structural graph theory. Many variants and extensions are known. We add to this line of research and prove results that relate the maximal number of vertex-disjoint paths between vertex sets and a wall to the minimum number of vertices meeting all these paths. We also include types of paths that start and end in the wall.
Suppose $G$ is a $k$-uniform hypergraph on $n$ vertices such that every $(k-1)$-subset $S$ of $V(G)$ belongs to at least $\delta n$ edges, where $\delta> 1/2$. Let $\Psi(G)$ denote the number of tight Hamilton cycles in $G$, that is, cyclic orderings of $V(G)$ in which every $k$ consecutive vertices form an edge. We prove that $\log\Psi(G)\ge kh(G)-n\log{n\choose k-1}+n\log n-n\log e-o(n)$, where $h(G)$ is the hypergraph entropy of $G$, defined via perfect fractional matchings. This bound is tight, for example, for all (nearly) regular hypergraphs, in particular for the binomial random hypergraph. It also implies a conjecture by Ferber, Hardiman and Mond, stating that $\Psi(G)\ge (\delta-o(1))^n n!$.
The Prague dimension of a graph $G$ is defined as the minimum number of complete graphs whose direct product contains $G$ as an induced subgraph. Introduced in the 1970s by Ne\v{s}et\v{r}il, Pultr, and R\"odl -- and motivated by the work of Dushnik and Miller, as well as by the induced Ramsey theorem -- determining the Prague dimension of a graph is a notoriously hard problem. In this paper, we show that for all $\varepsilon > 0$ and $p$ such that $ n^{-1+\varepsilon} \le p \le n^{-\varepsilon}$, with high probability the Prague dimension of $G_{n,p}$ is $\Theta_{\varepsilon}(pn)$, which improves upon a recent result by Molnar, R\"odl, Sales and Schacht. Inspired by the work of Bennett and Bohman, our approach centres on analysing a random greedy process that builds an independent set of size $\Omega(p^{-1}\log pn)$ by iteratively selecting vertices uniformly at random from the common non-neighbourhood of those already chosen. Using the differential equation method, we show that every non-edge is essentially equally likely to be covered by this process, which is key to establishing our bound.
Let $k\geq 2$ and fix a $k$-uniform hypergraph $\mathcal{F}$. Consider the random process that, starting from a $k$-uniform hypergraph $\mathcal{H}$ on $n$ vertices, repeatedly deletes the edges of a copy of $\mathcal{F}$ chosen uniformly at random and terminates when no copies of $\mathcal{F}$ remain. Let $R(\mathcal{H},\mathcal{F})$ denote the number of edges that are left after termination. We show that $R(\mathcal{H},\mathcal{F})=n^{k-1/ρ\pm o(1)}$, where $ρ:=(\lvert E(\mathcal{F})\rvert-1)/(\lvert V(\mathcal{F})\rvert -k)$, holds with high probability provided that $\mathcal{F}$ is strictly $k$-balanced and $\mathcal{H}$ is sufficiently dense with pseudorandom properties. Since we may in particular choose $\mathcal{F}$ and $\mathcal{H}$ to be complete graphs, this confirms the major folklore conjecture in the area in a very strong form.
We consider real tensors of order $D$, that is $D$-dimensional arrays of real numbers $T_{a^1a^2 \dots a^D}$, where each index $a^c$ can take $N$ values. The tensor entries $T_{a^1a^2 \dots a^D}$ have no symmetry properties under permutations of the indices. The invariant polynomials built out of the tensor entries are called trace invariants. We prove that for a Gaussian random tensor with $D\ge 3$ indices (that is such that the entries $T_{a^1a^2 \dots a^D}$ are independent identically distributed Gaussian random variables) the cumulant, or connected expectation, of a product of trace invariants is not always suppressed in scaling in $N$ with respect to the product of the expectations of the individual invariants. Said otherwise, not all the multi-trace expectations factor at large $N$ in terms of the single-trace ones and the Gaussian scaling is not subadditive on the connected components. This is in stark contrast to the $D=2$ case of random matrices in which the multi-trace expectations always factor at large $N$. The best one can do for $D\ge 3$ is to identify restricted families of invariants for which the large $N$ factorization holds and we check that this indeed happens when restricting to the family of melonic observables, the dominant family in the large $N$ limit.
We provide a combinatorial characterization of all testable properties of $k$-uniform hypergraphs ($k$-graphs for short). Here, a $k$-graph property $P$ is testable if there is a randomized algorithm which makes a bounded number of edge queries and distinguishes with probability $2/3$ between $k$-graphs that satisfy $P$ and those that are far from satisfying $P$. For the $2$-graph case, such a combinatorial characterization was obtained by Alon, Fischer, Newman and Shapira. Our results for the $k$-graph setting are in contrast to those of Austin and Tao, who showed that for the somewhat stronger concept of local repairability, the testability results for graphs do not extend to the $3$-graph setting. Our proof relies on a random subhypergraph sampling result proved in a companion paper.
We study conditions under which a given hypergraph is randomly robust Hamiltonian, which means that a random sparsification of the host graph contains a Hamilton cycle with high probability. Our main contribution provides nearly optimal results whenever the host graph is Hamilton connected in a locally robust sense, which translates to a typical induced subgraph of constant order containing Hamilton paths between any pair of suitable ends. The proofs are based on the recent breakthrough on Talagrand's conjecture, which reduces the problem to specifying a distribution on the desired guest structure in the (deterministic) host structure. We find such a distribution via a new argument that reduces the problem to the case of perfect matchings in a higher uniformity. As applications, we obtain asymptotically optimal results for perfect tilings in graphs and hypergraphs both in the minimum degree and uniformly dense setting. We also prove random robustness for powers of cycles under asymptotically optimal minimum degrees and degree sequences. We solve the problem for loose and tight Hamilton cycles in hypergraphs under a range of asymptotic minimum degree conditions. This includes in particular $k$-uniform tight Hamilton cycles under minimum $d$-degree conditions for $1\leq k-d \leq 3$. In all cases, our bounds on the sparseness are essentially best-possible.
The number of independent sets in regular bipartite expander graphs can be efficiently approximated by expressing it as the partition function of a suitable polymer model and truncating its cluster expansion. While this approach has been extensively used for graphs, surprisingly little is known about analogous questions in the context of hypergraphs. In this work, we apply this method to asymptotically determine the number of independent sets in regular $k$-partite $k$-uniform hypergraphs which satisfy natural expansion properties. The resulting formula depends only on the local structure of the hypergraph, making it computationally efficient. In particular, we provide a simple closed-form expression for linear hypergraphs.
We develop a method to study sufficient conditions for perfect mixed tilings. Our framework allows the embedding of bounded degree graphs $H$ with components of sublinear order. As a corollary, we recover and extend the work of Kühn and Osthus regarding sufficient minimum degree conditions for perfect $F$-tilings (for an arbitrary fixed graph $F$) by replacing the $F$-tiling with the aforementioned graphs $H$. Moreover, we obtain analogous results for degree sequences and in the setting of uniformly dense graphs. Finally, we asymptotically resolve a conjecture of Komlós in a strong sense.
Recently, researchers have extended the concept of matchings to the more general problem of finding $b$-matchings in hypergraphs broadening the scope of potential applications and challenges. The concept of $b$-matchings, where $b$ is a function that assigns positive integers to the vertices of the graph, is a natural extension of matchings in graphs, where each vertex $v$ is allowed to be matched to up to $b(v)$ edges, rather than just one. The weighted $b$-matching problem then seeks to select a subset of the hyperedges that fulfills the constraint and maximizes the weight. In this work, we engineer novel algorithms for this generalized problem. More precisely, we introduce exact data reductions for the problem as well as a novel greedy initial solution and local search algorithms. These data reductions allow us to significantly shrink the input size. This is done by either determining if a hyperedge is guaranteed to be in an optimum $b$-matching and thus can be added to our solution or if it can be safely ignored. Our iterated local search algorithm provides a framework for finding suitable improvement swaps of edges. Experiments on a wide range of real-world hypergraphs show that our new set of data reductions are highly practical, and our initial solutions are competitive for graphs and hypergraphs as well.
Recent work showing the existence of conflict-free almost-perfect hypergraph matchings has found many applications. We show that, assuming certain simple degree and codegree conditions on the hypergraph $ \mathcal{H} $ and the conflicts to be avoided, a conflict-free almost-perfect matching can be extended to one covering all of the vertices in a particular subset of $ V(\mathcal{H}) $, by using an additional set of edges; in particular, we ensure that our matching avoids all of a further set of conflicts, which may consist of both old and new edges. This setup is useful for various applications, and our main theorem provides a black box which encapsulates many long and tedious calculations, massively simplifying the proofs of results in generalised Ramsey theory.
Let $T$ be an oriented tree on $n$ vertices with maximum degree at most $e^{o(\sqrt{\log n})}$. If $G$ is a digraph on $n$ vertices with minimum semidegree $δ^0(G)\geq(\frac12+o(1))n$, then $G$ contains $T$ as a spanning tree, as recently shown by Kathapurkar and Montgomery (in fact, they only require maximum degree $o(n/\log n)$). This generalizes the corresponding result by Komlós, Sárközy and Szemerédi for graphs. We investigate the natural question how many copies of $T$ the digraph $G$ contains. Our main result states that every such $G$ contains at least $|Aut(T)|^{-1}(\frac12-o(1))^nn!$ copies of $T$, which is optimal. This implies the analogous result in the undirected case.
Fix $k\geq 2$, choose $\frac{\log n}{n^{(k-1)/k}}\leq p\leq 1-Ω(\frac{\log^4 n}{n})$, and consider $G\sim G(n,p)$. For any pair of vertices $v,w\in V(G)$, we give a simple and precise formula for the expected number of steps that a random walk on $G$ starting at $w$ needs to first arrive at $v$. The formula only depends on basic structural properties of $G$. This improves and extends recent results of Ottolini and Steinerberger, as well as Ottolini, who considered this problem for constant as well as for mildly vanishing $p$.
Increasing the connectivity of a graph is a pivotal challenge in robust network design. The weighted connectivity augmentation problem is a common version of the problem that takes link costs into consideration. The problem is then to find a minimum cost subset of a given set of weighted links that increases the connectivity of a graph by one when the links are added to the edge set of the input instance. In this work, we give a first implementation of recently discovered better-than-2 approximations. Furthermore, we propose three new heuristic and one exact approach. These include a greedy algorithm considering link costs and the number of unique cuts covered, an approach based on minimum spanning trees and a local search algorithm that may improve a given solution by swapping links of paths. Our exact approach uses an ILP formulation with efficient cut enumeration as well as a fast initialization routine. We then perform an extensive experimental evaluation which shows that our algorithms are faster and yield the best solutions compared to the current state-of-the-art as well as the recently discovered better-than-2 approximation algorithms. Our novel local search algorithm can improve solution quality even further.
It is a well known result due to Korshunov and Sapozhenko that the hypercube in $n$ dimensions has $(1 + o(1)) \cdot 2 \sqrt e \cdot 2^{2^{n-1}}$ independent sets. Jenssen and Keevash investigated in depth Cartesian powers of cycles of fixed even lengths far beyond counting independent sets. They wonder to which extent their results extend to cycles of odd length, where not even the easiest case, counting independent sets in Cartesian powers of the triangle, is known. In this paper, we make progress on their question by providing a lower bound, which we believe to be tight. We also obtain a less precise lower bound for the number of independent sets in Cartesian powers of arbitrary odd cycles and show how to approach this question both with the cluster expansion method as well as more directly with isoperimetric inequalities.
For a collection $\mathbf{G}=\{G_1,\dots, G_s\}$ of not necessarily distinct graphs on the same vertex set $V$, a graph $H$ with vertices in $V$ is a $\mathbf{G}$-transversal if there exists a bijection $ϕ:E(H)\rightarrow [s]$ such that $e\in E(G_{ϕ(e)})$ for all $e\in E(H)$. We prove that for $|V|=s\geq 3$ and $δ(G_i)\geq s/2$ for each $i\in [s]$, there exists a $\mathbf{G}$-transversal that is a Hamilton cycle. This confirms a conjecture of Aharoni. We also prove an analogous result for perfect matchings.
We prove that in all regular robust expanders $G$ every edge is asymptotically equally likely contained in a uniformly chosen perfect matching $M$. We also show that given any fixed matching or spanning regular graph $N$ in $G$, the random variable $|M\cap E(N)|$ is approximately Poisson distributed. This in particular confirms a conjecture and a question due to Spiro and Surya, and complements results due to Kahn and Kim who proved that in a regular graph every vertex is asymptotically equally likely contained in a uniformly chosen matching. Our proofs rely on the switching method and the fact that simple random walks mix rapidly in robust expanders.
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.