Periodica Mathematica Hungarica

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

On translations of double rays in graphs

  • Norbert Polat
  • Mark E. Watkins
Article

Abstract

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 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    P. C. Bonnington, W. Imrich, andM. E. Watkins, Separating paths in infinite, planar graphs,Discrete Math. (to appear).Google Scholar
  2. [2]
    R. Halin, Über unendliche Wege in Graphen,Math. Ann.,157 (1964), 125–137.Google Scholar
  3. [3]
    R. Halin, Über die Maximalzahl fremder unendlicher Wege in Graphen,Math. Nachr.,30 (1965), 63–85.Google Scholar
  4. [4]
    R. Halin, Automorphisms and endomorphisms of infinite locally finite graphs,Abh. Math. Sem. Univ. Hamburg,39 (1973), 251–283.Google Scholar
  5. [5]
    H. A. Jung, A note on fragments of infinite graphs,Combinatorica,1 (1981), 285–288.Google Scholar
  6. [6]
    H. A. Jung, Some results on ends and automorphisms of graphs,Discrete Math.,95 (1991), 119–133.Google Scholar
  7. [7]
    N. Polat, Topological aspects of infinite graphs, in: (G. Hahn et al. eds.)Cycles and Rays, 1990, Kluwer, pp. 197–220.Google Scholar
  8. [8]
    N. Polat, A Mengerian theorem for infinite graphs with ideal points,J. Combin. Theory, Ser. B,51 (1991), 248–255.Google Scholar
  9. [9]
    C. Thomassen, The Hadwiger number of infinite vertex-transitive graphs,Combinatorica,12 (1992), 481–491.Google Scholar
  10. [10]
    M. E. Watkins, Intinite paths that contain only shortest paths,J. Combin. Theory, Ser. B,41 (1986), 341–355.Google Scholar

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

Personalised recommendations