Periodica Mathematica Hungarica

, Volume 30, Issue 2, pp 145–154 | Cite as

On translations of double rays in graphs

  • Norbert Polat
  • Mark E. Watkins


Let τ be an infinite graph, let π be a double ray in τ, and letd anddπ denote the distance functions in τ and in π, respectively. One calls π anaxis ifd(x,y)=d π (x,y) and aquasi-axis if lim infd(x,y)/d π (x,y)>0 asx, y range over the vertex set of π andd π (x,y)→∞. The present paper brings together in greater generality results of R. Halin concerning invariance of double rays under the action of translations (i.e., graph automorphisms all of whose vertex-orbits are infinite) and results of M. E. Watkins concerning existence of axes in locally finite graphs. It is shown that if α is a translation whose directionD(α) is a thin end, then there exists an axis inD(α) andD−1) invariant under α r for somer not exceeding the maximum number of disjoint rays inD(α).The thinness ofD(α) is necessary. Further results give necessary conditions and sufficient conditions for a translation to leave invariant a quasi-axis.

Mathematics subject classification numbers, 1991

Primary 05C38 Secondary 05C12 

Key words and phrases

Infinite graph end axis double ray translation geodesic 


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Copyright information

© Akadémiai Kiadó 1995

Authors and Affiliations

  • Norbert Polat
    • 1
  • Mark E. Watkins
    • 2
  1. 1.I. A. EUniversité Jean Moulin (Lyon 3)Lyon Cedex 02France
  2. 2.Mathematics DepartmentSyracuse UniversitySyracuseUSA

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