Infinite paths in planar graphs IV, dividing cycles

  • Authors:

  • Xingxing Yu

  • Affiliations:

  • School of Mathematics, Georgia Institute of Technology, Atlanta, Goeorgia 30332 and Center for Combinatorics, Nankai University, Tianjin, 300071, China

  • Venue:
  • Journal of Graph Theory
  • Year:
  • 2006

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Abstract

Nash-Williams conjectured that a 4-connected infinite planar graph contains a spanning 2-way infinite path if, and only if, the deletion of any finite set of vertices results in at most two infinite components. In this article, we prove this conjecture for graphs with no dividing cycles and for graphs with infinitely many vertex disjoint dividing cycles. A cycle in an infinite plane graph is called dividing if both regions of the plane bounded by this cycle contain infinitely many vertices of the graph. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 173–195, 2006