Transitive graphs in counterexamples to Karp's conjecture

Highly arc-transitive digraphs -- counterexamples and structure

We resolve two problems of [Cameron, Praeger, and Wormald -- Infinite highly arc transitive digraphs and universal covering digraphs, Combinatorica 1993]. First, we construct a locally finite highly arc-transitive digraph with universal reachability relation. Second, we provide constructions of 2-ended highly arc transitive digraphs where each `building block' is a finite bipartite graph that is not a disjoint union of complete bipartite graphs. This was conjectured impossible in the above paper. We also describe the structure of 2-ended highly arc transitive digraphs in more generality, altho…

关于多循环猜想的一个无穷反例族

An infinite family of counterexamples to the Polycirculant Conjecture

反驳多循环猜想，证明存在无半正则自同构的顶点传递图。采用Chen等人方法构造度为16464的难以捉摸群\(7^6.\mathrm{PSU}_3(3)\)，其是2 - 闭的，此例扩展出多循环猜想无穷反例及相关顶点传递图。

On the structure of asymptotic expanders

In this paper, we use geometric tools to study the structure of asymptotic expanders and show that a sequence of asymptotic expanders always admits a "uniform exhaustion by expanders". It follows that asymptotic expanders cannot be coarsely embedded into any $L^p$-space, and that asymptotic expanders can be characterised in terms of their uniform Roe algebra. Moreover, we provide uncountably many new counterexamples to the coarse Baum--Connes conjecture. These appear to be the first counterexamples that are not directly constructed by means of spectral gaps. Finally, we show that vertex-transi…

对称图中将路径闭合为环

Closing paths to cycles in symmetric graphs

本文研究对称图中路径闭合为环的问题，完全解决了顶点传递图、边传递图及非星形边传递图中路径长度阈值，并给出反例族。

可数无限图的难以捉摸性质

Elusive properties of countably infinite graphs

本文研究可数无限图在允许算法经无限多步终止时的难以捉摸性，证明连通性仍难以捉摸，并给出强化版反例否定 Aanderaa–Karp–Rosenberg 猜想的无限版本。

正则图的临界阈值

Critical threshold for regular graphs

本文刻画拟传递 d-正则图的临界渗流阈值，证明其达到下界 1/(d−1) 当且仅当图为树，并用反例说明拟传递条件不可缺少。