Infinite paths in planar graphs II, structures and ladder nets
Xingxing Yu · Journal of Graph Theory · 2005
Abstract A graph is k‐indivisible , where k is a positive integer, if the deletion of any finite set of vertices results in at most k – 1 infinite components. In 1971, Nash‐Williams conjectured that a 4‐connected infinite planar graph contains a spanning 2‐way infinite path if and only if it is 3‐indivisible. In this paper, we prove a structural result for 2‐indivisible infinite planar graphs. This structural result is then used to prove Nash‐Williams conjecture for all 4‐connected 2‐indivisible infinite planar graphs. © 2005 Wiley Periodicals, Inc. J Graph Theory 48: 247–266, 2005