# Wormald, Nicholas C.

## Highly arc transitive two ended digraphs ★★

Author(s): Cameron; Praeger; Wormald

\begin{conjecture} If $G$ is a highly arc transitive digraph with two ends, then every tile of $G$ is a disjoint union of complete bipartite graphs. \end{conjecture}

Keywords: arc transitive; digraph; infinite graph

## Universal highly arc transitive digraphs ★★★

Author(s): Cameron; Praeger; Wormald

An \emph{alternating} walk in a digraph is a walk $v_0,e_1,v_1,\ldots,v_m$ so that the vertex $v_i$ is either the head of both $e_i$ and $e_{i+1}$ or the tail of both $e_i$ and $e_{i+1}$ for every $1 \le i \le m-1$. A digraph is \emph{universal} if for every pair of edges $e,f$, there is an alternating walk containing both $e$ and $f$

\begin{question} Does there exist a locally finite highly arc transitive digraph which is universal? \end{question}

Keywords: arc transitive; digraph