Highly arc transitive two ended digraphs

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

An end of an undirected graph is an equivalence class of one-way infinite paths, where two paths are equivalent when their tails lie in the same component after deletion of every finite vertex set. For a digraph, ends refer to the underlying undirected graph. A digraph is highly arc-transitive if its automorphisms are transitive on every set of non-backtracking directed $s$-arcs, for all $s\geq0$. In the two-ended setting of the source, choose its automorphism-invariant partition $(X_i)_{i\in\mathbb Z}$ into finite blocks, with each arc directed from $X_i$ to $X_{i+1}$. The tile at $i$ is the bipartite digraph induced between these two consecutive blocks.

Statement recorded in the supplied source.

The source conjecture asks whether every tile of a two-ended highly arc-transitive digraph is a disjoint union of complete bipartite digraphs, each with all arcs oriented consistently from $X_i$ to $X_{i+1}$. The cited primary paper constructs counterexamples to this historical assertion. [S2] [S3]

Short English statement

Must each finite layer-to-layer piece of a two-ended highly symmetric digraph split into complete bipartite components? Counterexamples to this historical statement are known.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3305 (OPG-691). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Highly arc transitive two ended digraphs, node 691, original problem page. Original question and full discussion. https://www.openproblemgarden.org/op/highly_arc_transitive_two_ended_digraphs.
• S3 Matt DeVos, Bojan Mohar and Robert Šámal, Highly arc-transitive digraphs -- counterexamples and structure (2011), abstract. https://arxiv.org/abs/1110.2945.