Abstract
A 2-factorization of a simple graph \(\Gamma \) is called \(2\)-pyramidal if it admits an automorphism group \(G\) fixing two vertices and acting sharply transitively on the others. Here we show that such a \(2\)-factorization may exist only if \(\Gamma \) is a cocktail party graph, i.e., \(\Gamma = K_{2n}-I\) with \(I\) being a \(1\)-factor. It will be said of the first or second type according to whether the involutions of \(G\) form a unique conjugacy class or not. As far as we are aware, \(2\)-factorizations of the second type are completely new. We will prove, in particular, that \(K_{2n}-I\) admits a 2-pyramidal 2-factorization of the second type if and only if \(n\equiv 1\) (mod 8).
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Work performed under the auspicies of the G.N.S.A.G.A. of the C.N.R. (National Research Council) of Italy and supported by M.I.U.R. project “Disegni combinatorici, grafi e loro applicazioni, PRIN 2008”. The second author is supported by a fellowship of INdAM.
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Buratti, M., Traetta, T. The Structure of \(2\)-Pyramidal \(2\)-Factorizations. Graphs and Combinatorics 31, 523–535 (2015). https://doi.org/10.1007/s00373-014-1408-2
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DOI: https://doi.org/10.1007/s00373-014-1408-2