Cycles in Graphs and Groups

نویسنده

  • William M. Kantor
چکیده

If a group G of automorphisms of a graph Γ acts transitively on the set of vertices, then Γ is d-valent for some d: each vertex is adjacent to exactly d others. This note concerns cycles in Γ, by which we will mean subgraphs isomorphic to a k-cycle for some k ≥ 3; hence there will be no “initial” vertex. If g ∈ G and C is a cycle in Γ, then g(C) is another cycle in Γ: G acts on the set of all cycles in Γ. There is another meaning of the word “cycle”: each element of G can be written as a product of disjoint cycles of the set of vertices. If C = (1, . . . , k) is such a cycle and g ∈ G, then an elementary calculation gives gCg−1 = (g(1), . . . , g(k)), and this is just g(C) if C is viewed as a cyclically ordered set of vertices. In other words, the conjugation and functional actions behave the same. In a 1971 lecture, J. H. Conway [6] presented the following result, which merges these two meanings of “cycle” when G can move any ordered pair of adjacent vertices to any other such pair:

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عنوان ژورنال:
  • The American Mathematical Monthly

دوره 115  شماره 

صفحات  -

تاریخ انتشار 2008