Infinite paths in planar graphs II, structures and ladder nets

  • Authors:

  • Xingxing Yu

  • Affiliations:

  • School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160 and Center for combinatorics LPMC, Nankai University Tianjin 300071, P. R. China

  • Venue:
  • Journal of Graph Theory
  • Year:
  • 2005

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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