Hamilton circles in infinite planar graphs

  • Authors:

  • Qing Cui;Jian Wang;Xingxing Yu

  • Affiliations:

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

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2009

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Abstract

A circle in a graph G is a homeomorphic image of the unit circle in the Freudenthal compactification of G, a topological space formed from G and the ends of G. Bruhn conjectured that every locally finite 4-connected planar graph G admits a Hamilton circle, a circle containing all points in the Freudenthal compactification of G that are vertices and ends of G. We prove this conjecture for graphs with no dividing cycles. In a plane graph, a cycle C is said to be dividing if each closed region of the plane bounded by C contains infinitely many vertices.