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We generalize the notion of tight geodesics in the curve complex to tight trees. We then use tight trees to construct model geometries for certain surface bundles over graphs. This extends some aspects of the combinatorial model for doubly…
In this survey we describe a recently-developed technique for bounding the number (and controlling the typical structure) of finite objects with forbidden substructures. This technique exploits a subtle clustering phenomenon exhibited by…
We obtain sufficient conditions for the emergence of spanning and almost-spanning bounded-degree {\sl rainbow} trees in various host graphs, having their edges coloured independently and uniformly at random, using a predetermined palette.…
We investigate the problem of sequentially predicting the binary labels on the nodes of an arbitrary weighted graph. We show that, under a suitable parametrization of the problem, the optimal number of prediction mistakes can be…
This article presents two novel algorithms for generating random increasing trees. The first algorithm efficiently generates strictly increasing binary trees using an ad hoc method. The second algorithm improves the recursive method for…
We consider transport over a strongly connected, directed graph. The scheduling amounts to selecting transition probabilities for a discrete-time Markov evolution which is designed to be consistent with certain initial and final marginals.…
Monadic second order logic can be used to express many classical notions of sets of vertices of a graph as for instance: dominating sets, induced matchings, perfect codes, independent sets or irredundant sets. Bounds on the number of sets…
Kim defined a very general combinatorial abstraction of the diameter of polytopes called subset partition graphs to study how certain combinatorial properties of such graphs may be achieved in lower bound constructions. Using Lov\'asz'…
We study a random recursive tree model featuring complete redirection called the random friend tree and introduced by Saram\"aki and Kaski. Vertices are attached in a sequential manner one by one by selecting an existing target vertex and…
We show that the growth of a unimodular random rooted tree $(T,o)$ of degree bounded by $d$ always exists, assuming its upper growth passes the critical threshold $\sqrt{d-1}$. This complements Timar's work who showed the possible…
We propose a general class of co-evolving tree network models driven by local exploration where new vertices attach to the current network via randomly sampling a vertex and then exploring the graph for a random number of steps in the…
This paper extends the study of fringe trees in random plane trees with a given degree statistic. While previous work established the asymptotic normality of the count of fringe trees isomorphic to a fixed tree, we investigate the case…
The problems of detecting and recovering planted structures/subgraphs in Erd\H{o}s-R\'{e}nyi random graphs, have received significant attention over the past three decades, leading to many exciting results and mathematical techniques.…
We establish sufficient mild conditions for a sequence of multitype Bienaym\'e-Galton-Watson trees, conditioned in some sense to be large, to converge to a limiting compact metric space which we call a \emph{multitype L\'{e}vy tree}. More…
This paper is centered on the random graph generated by a Doeblin-type coupling of discrete time processes on a countable state space whereby when two paths meet, they merge. This random graph is studied through a novel subgraph, called a…
We investigate the number of permutations that occur in random labellings of trees. This is a generalisation of the number of subpermutations occurring in a random permutation. It also generalises some recent results on the number of…
Building upon the theory of graph limits and the Aldous-Hoover representation and inspired by Panchenko's work on asymptotic Gibbs measures (Annals of Probability 2013), we construct continuous embeddings of discrete probability…
Gromov asked what a typical (finitely presented) group looks like, and he suggested a way to make the question precise in terms of limiting density. The typical finitely generated group is known to share some important properties with the…
In this series, we introduce and investigate the concept of connectoids, which captures the connectivity structure of various discrete objects such as undirected graphs, directed graphs, bidirected graphs, hypergraphs and finitary matroids.…
The asymptotic dimension of metric spaces is an important notion in geometric group theory introduced by Gromov. The metric spaces considered in this paper are the ones whose underlying spaces are the vertex-sets of graphs and whose metrics…