Infinite paths in planar graphs III, 1-way infinite paths

  • Authors:

  • Xingxing Yu

  • Affiliations:

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

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

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Abstract

An infinite graph is 2-indivisible if the deletion of any finite set of vertices from the graph results in exactly one infinite component. Let G be a 4-connected, 2-indivisible, infinite, plane graph. It is known that G contains a spanning 1-way infinite path. In this paper, we prove a stronger result by showing that, for any vertex x and any edge e on a facial cycle of G, there is a spanning 1-way infinite path in G from x and through e. Results will be used in two forthcoming papers to establish a conjecture of Nash-Williams. © 2005 Wiley Periodicals, Inc. J Graph Theory