Related papers: Parking on a Random Tree
Generating trees are a useful technique in the enumeration of various combinatorial objects, particularly restricted permutations. Quite often the generating tree for the set of permutations avoiding a set of patterns requires infinitely…
We derive a sufficient condition for a sparse graph G on n vertices to contain a copy of a tree T of maximum degree at most d on (1-\epsilon)n vertices, in terms of the expansion properties of G. As a result we show that for fixed d\geq 2…
We consider the quantity $P(G)$ associated with a graph $G$ that is defined as the probability that a randomly chosen subtree of $G$ is spanning. Motivated by conjectures due to Chin, Gordon, MacPhee and Vincent on the behaviour of this…
Designing fare systems for public transportation networks is a challenging task. A popular approach is to partition the network into fare zones (``zoning'') and fix journey prices depending on the number of traversed zones (``pricing''). In…
Let $\mathcal{B}$ be the set of rooted trees containing an infinite binary subtree starting at the root. This set satisfies the metaproperty that a tree belongs to it if and only if its root has children $u$ and $v$ such that the subtrees…
We consider the stochastic geometry model where the location of each node is a random point in a given metric space, or the existence of each node is uncertain. We study the problems of computing the expected lengths of several…
Within the framework of a simple model of car traffic on a one-lane highway, we study the probability for car accidents to occur when drivers do not respect the safety distance between cars, and, as a result of the blockage during the time…
We study a new optimal stopping problem: Let $G$ be a fixed graph with $n$ vertices which become active on-line in time, one by another, in a random order. The active part of $G$ is the subgraph induced by the active vertices. Find a…
Working with tree graphs is always easier than with loopy ones and spanning trees are the closest tree-like structures to a given graph. We find a correspondence between the solutions of random K-satisfiability problem and those of spanning…
Recently, the authors extended the notion of parking functions to parking sequences, which include cars of different sizes, and proved a product formula for the number of such sequences. We here give a refinement of that result involving…
It has been known for a few years that the occupation measure of several models of embedded trees converges, after a suitable normalization, to the random measure called ISE (Integrated SuperBrownian Excursion). Here, we prove a local…
Kemeny's constant $\kappa(G)$ of a connected graph $G$ is a measure of the expected transit time for the random walk associated with $G$. In the current work, we consider the case when $G$ is a tree, and, in this setting, we provide lower…
We show that for every $\Delta\in\mathbb N$, there exists a constant $C$ such that if $G$ is an $(n,d,\lambda)$-graph with $d/\lambda\ge C$ and $d$ is large enough, then $G^2$ contains every $n$-vertex tree with maximum degree bounded by…
Finding interactions between variables in large and high-dimensional datasets is often a serious computational challenge. Most approaches build up interaction sets incrementally, adding variables in a greedy fashion. The drawback is that…
The study of "random segments" is a classic issue in geometrical probability, whose complexity depends on how it is defined. But in apparently simple models, the random behavior is not immediate. In the present manuscript the following…
The properties of randomly evolving special trees having defined and analyzed already in two earlier papers (arXiv:cond-mat/0205650 and arXiv:cond-mat/0211092) have been investigated in the case when the continuous time parameter converges…
In this article, we study a simple random walk on a decorated Galton-Watson tree, obtained from a Galton-Watson tree by replacing each vertex of degree $n$ with an independent copy of a graph $G_n$ and gluing the inserted graphs along the…
Using the methods originally developed for Random Matrix Theory we derive an exact mathematical formula for number variance (introduced in [4]) describing a rigidity of particle ensembles with power-law repulsion. The resulting relation is…
We study the local convergence of critical Galton-Watson trees and Levy trees under various conditionings. Assuming a very general monotonicity property on the functional of random trees, we show that random trees conditioned to have large…
Packing graphs is a combinatorial problem where several given graphs are being mapped into a common host graph such that every edge is used at most once. In the planar tree packing problem we are given two trees T1 and T2 on n vertices and…