Received: 27 August 2015 Revised: 1 August 2017 Accepted: 20 August 2017
One-way inﬁnite 2-walks in planar graphs
Daniel P. Biebighauser
M. N. Ellingham
Department of Mathematics, Concordia
College, Moorhead, MN 56562
Department of Mathematics, Vanderbilt
University, Nashville, TN 37240
M. N. Ellingham, Department of Math-
ematics, Vanderbilt University, 1326
Stevenson Center, Nashville, TN 37240.
Supported by a sabbatical leave from Concordia
Grant sponsor: National Security Agency;
Contract grant numbers: H98230-04-1-0110
and H98230-13-1-0233; Grant sponsor: Simons
Foundation; Contract grant numbers: 245715
We prove that every 3-connected 2-indivisible inﬁnite pla-
nar 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
inﬁnite spanning walk, planar graph, 3-connected
For terms not deﬁned in this article, see . All graphs are simple (having no loops or multiple edges)
and may be inﬁnite, unless we explicitly state otherwise.
A cutset in a graph is a set ⊆() such that − is disconnected. A -cut is a cutset with
|| = . A graph is -connected if it has at least +1vertices and no cutset with || <.The
connectivity of a graph is the smallest for which it is -connected.
The ﬁrst major result on the existence of Hamilton cycles in graphs embedded in surfaces was by
Whitney  in 1931, who proved that every 4-connected ﬁnite planar triangulation is Hamiltonian.
Tutte extended this to all 4-connected ﬁnite planar graphs in 1956 , and gave another proof in
1977 . Tutte actually proved a more general result, using subgraphs which have since been called
“Tutte subgraphs” (deﬁned in Section 3).
To extend these results to inﬁnite graphs, one can look for inﬁnite spanning paths. We say that
⋯ is a 1-way inﬁnite path, and ⋯
⋯ is a 2-way inﬁnite path, if each
distinct vertex and consecutive vertices are adjacent.
If deleting ﬁnitely many vertices in an inﬁnite graph leaves more than one (two) inﬁnite compo-
nent(s), then the graph has no 1-way (2-way) inﬁnite spanning path. Nash-Williams  deﬁned a
graph to be -indivisible, for a positive integer , if, for any ﬁnite ⊆(), − has at most
110 © 2017 Wiley Periodicals, Inc. wileyonlinelibrary.com/journal/jgt J Graph Theory. 2018;88:110–130.