The minimum stretch spanning tree problem for typical graphs

Quantitative characteristics of cycles and their relations with stretch and spanning tree congestion

The main goal of this article is to introduce new quantitative characteristics of cycles in finite simple connected graphs and to establish relations of these characteristics with the stretch and spanning tree congestion of graphs. The main new parameter is named the support number. We give a polynomial approximation algorithm for the support number with the aid of yet another characteristic we introduce, named the cycle width of the graph.

Simpler and efficient characterizations of tree t-spanners for graphs with few P4's and (k, l)-graphs

A tree $t$-spanner of a graph $G$ is a spanning tree $T$ in which the distance between any two adjacent vertices of $G$ is at most $t$. The smallest $t$ for which $G$ has a tree $t$-spanner is called tree stretch index. The $t$-admissibility problem aims to decide whether the tree stretch index is at most $t$. Regarding its optimization version, the smallest $t$ for which $G$ is $t$-admissible is the stretch index of $G$, denoted by $σ_T(G)$. Given a graph with $n$ vertices and $m$ edges, the recognition of $2$-admissible graphs can be done $O(n+m)$ time, whereas $t$-admissibility is NP-comple…

The diameter of the uniform spanning tree of dense graphs

We show that the diameter of a uniformly drawn spanning tree of a simple connected graph on $n$ vertices with minimal degree linear in $n$ is typically of order $\sqrt{n}$. A byproduct of our proof, which is of independent interest, is that on such graphs the Cheeger constant and the spectral gap are comparable.

The diameter of uniform spanning trees in high dimensions

We show that the diameter of a uniformly drawn spanning tree of a connected graph on $n$ vertices which satisfies certain high-dimensionality conditions typically grows like $Θ(\sqrt{n})$. In particular this result applies to expanders, finite tori $\mathbb{Z}_m^d$ of dimension $d \geq 5$, the hypercube $\{0,1\}^m$, and small perturbations thereof.

随机扰动有向图也具有有界度生成树

Randomly perturbed digraphs also have bounded-degree spanning trees

本研究针对随机扰动有向图的生成树存在性问题，证明在最小半度为αn的n顶点稠密有向图中添加cn条随机边后，将以高概率包含最大度为Δ的固定定向有界度生成树，推广了无向图的相关结果并解答了已有公开问题。

$G$-Parking Functions, Acyclic Orientations and Spanning Trees

Given an undirected graph $G=(V,E)$, and a designated vertex $q\in V$, the notion of a $G$-parking function (with respect to $q$) was independently developed and studied by various authors, and has recently gained renewed attention. This notion generalizes the classical notion of a parking function associated with the complete graph. In this work, we study properties of {\em maximum} $G$-parking functions and provide a new bijection between them and the set of spanning trees of $G$ with no broken circuit. As a case study, we specialize some of our results to the graph corresponding to the disc…