Related papers: A Lower Bound for Kruskal's Weak Tree Function tre…

In a complete graph $K_n$ with edge weights drawn independently from a uniform distribution $U(0,1)$ (or alternatively an exponential distribution $\operatorname{Exp}(1)$), let $T_1$ be the MST (the spanning tree of minimum weight) and let…

We study the multi-level Steiner tree problem: a generalization of the Steiner tree problem in graphs where terminals $T$ require varying priority, level, or quality of service. In this problem, we seek to find a minimum cost tree…

We study rooted spiral trees in 2,3 and 4 dimensions on a hyper cubical lattice using exact enumeration and Monte-Carlo techniques. On the square lattice, we also obtain exact lower bound of 1.93565 on the growth constant $\lambda$. Series…

The fractal dimension of minimal spanning trees on percolation clusters is estimated for dimensions $d$ up to $d=5$. A robust analysis technique is developed for correlated data, as seen in such trees. This should be a robust method…

In this paper, we estimate the weak saturation numbers of trees. As a case study, we examine caterpillars and obtain several tight estimates. In particular, this implies that for any $\alpha\in [1,2]$, there exist caterpillars with $k$…

We study graph estimation and density estimation in high dimensions, using a family of density estimators based on forest structured undirected graphical models. For density estimation, we do not assume the true distribution corresponds to…

A normal (phylogenetic) network with $k$ reticulations displays $2^k$ phylogenetic trees. In this paper, we establish an analogous result for tree-child (phylogenetic) networks with no underlying $3$-cycles. In particular, we show that a…

In the laminar-constrained spanning tree problem, the goal is to find a minimum-cost spanning tree which respects upper bounds on the number of times each cut in a given laminar family is crossed. This generalizes the well-studied…

In the first part of this paper, we consider 3 x 3 x 3 arrays with complex entries, and provide a complete self-contained proof of Kruskal's theorem that the maximum rank is 5. In the second part, we provide a complete classification of the…

We consider the randomized decision tree complexity of the recursive 3-majority function. We prove a lower bound of $(1/2-\delta) \cdot 2.57143^h$ for the two-sided-error randomized decision tree complexity of evaluating height $h$ formulae…

Given a set $X$ and a sufficiently large integer $t$, let $\mathcal{F}$ be a family of $k$-subsets of $X$. The Kruskal-Katona theorem states that if $|\mathcal{F}|\geq \binom{t}{k}$, then $|\partial_{k-1}\mathcal{F}|\geq\binom{t}{k-1}$. The…

A set N is called a "weak epsilon-net" (with respect to convex sets) for a finite set X in R^d if N intersects every convex set that contains at least epsilon*|X| points of X. For every fixed d>=2 and every r>=1 we construct sets X in R^d…

In 1952, J.H.Braun claimed to have established a formula giving a lower bound for certain partitions of sets of integers into weakly sum-free classes. However, no proof or supporting construction was published at that time. In today's…

We study the query complexity of finding a Tarski fixed point over the $k$-dimensional grid $\{1,\ldots,n\}^k$. Improving on the previous best upper bound of $\smash{O(\log^{\lceil 2k/3\rceil} n)}$ [FPS20], we give a new algorithm with…

This report discusses the improved bound of the cluster expansion, recently proposed by Procacci and Yuhjtman (Lett. Math. Phys. 107, 31, 2017). Brydges and Helmuth noticed the relevance of Kruskal's algorithm, which allows to streamline…

Let $X_n(k)$ be the number of vertices at level $k$ in a random recursive tree with $n+1$ vertices. We are interested in the asymptotic behavior of $X_n(k)$ for intermediate levels $k=k_n$ satisfying $k_n\to\infty$ and $k_n=o(\log n)$ as…

We prove some properties of the Kruskal-Katona function, and apply to the following variation of cross-intersecting antichains. Let $n\ge 4$ be an even integer and $\mathscr{A}$ and $\mathscr{B}$ be two cross-intersecting antichains of…

The threshold-$k$ metric dimension ($\mathrm{Tmd}_k$) of a graph is the minimum number of sensors -- a subset of the vertex set -- needed to uniquely identify any vertex in the graph, solely based on its distances from the sensors, when the…

In this paper we study termination of term graph rewriting, where we restrict our attention to acyclic term graphs. Motivated by earlier work by Plump we aim at a definition of the notion of simplification order for acyclic term graphs. For…

This paper tightens the best known analysis of Hein's 1989 algorithm to infer the topology of a weighted tree based on the lengths of paths between its leaves. It shows that the number of length queries required for a degree-$k$ tree of $n$…