One-way infinite 2-walks in planar graphs

TL;DR

This paper proves that certain infinite planar graphs have a 1-way infinite 2-walk, extending previous results by removing the local finiteness assumption and using Tutte subgraphs.

Contribution

It establishes the existence of 1-way infinite 2-walks in 3-connected 2-indivisible infinite planar graphs without assuming local finiteness.

Findings

Every 3-connected 2-indivisible infinite planar graph has a 1-way infinite 2-walk.

Results extend to bipartite graphs and infinite triangulations.

The prism over such graphs contains a spanning 1-way infinite path.

Abstract

We prove that every 3-connected 2-indivisible infinite planar graph has a 1-way infinite 2-walk. (A graph is 2-indivisible if deleting finitely many vertices leaves at most one infinite component, and a 2-walk is a spanning walk using every vertex at most twice.) This improves a result of Timar, which assumed local finiteness. Our proofs use Tutte subgraphs, and allow us to also provide other results when the graph is bipartite or an infinite analog of a triangulation: then the prism over the graph has a spanning 1-way infinite path.