On translations of double rays in graphs
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(α−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, 1991Primary 05C38 Secondary 05C12
Key words and phrasesInfinite graph end axis double ray translation geodesic
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