Doubly transitive 2‐factorizations

Abstract

Let F be a 2-factorization of the complete graph K v admitting an automorphism group G acting doubly transitively on the set of vertices. The vertex-set V (K v ) can then be identified with the point-set of AG(n, p) and each 2-factor of F is the union of p-cycles which are obtained from a parallel class of lines of AG(n, p) in a suitable manner, the group G being a subgroup of AGL(n, p) in this case. The proof relies on the classification of 2-(v, k, 1) designs admitting a doubly transitive automorphism group. The same conclusion holds even if G is only assumed to act doubly homogeneously.

Some Results on 1‐Rotational Hamiltonian Cycle Systems

“…Theorem Let be a 1‐rotational HCS under G . Then unless is the unique doubly transitive HCS( p ) with p prime. Proof As a consequence of the main result in , the only HCSs admitting an automorphism group acting doubly transitively on the vertices are those of prime order p which, up to isomorphism, can be described as follows: the vertex‐set is ; the cycles are with . The full automorphism group of this HCS( p ) is , that is the group of all affine linear transformations with , .…”

On perfect Γ‐decompositions of the complete graph

“…This, considering that W 8 has diameter 2, assures that B is a perfect (F q , W 8 , 1)-DF and hence the assertion follows from Theorem 2.1. We report the primes p = 28n + 1 < 5,000 for which our Theorem 3.1 succeeds in finding a cyclic Steiner W 8 …”

On the Existence of Elementary Abelian Cycle Systems

“…In what follows we will deal with the existence problem of cycle systems for the elementary abelian case. We recall that an elementary abelian group is a finite group where every non-zero element has order a prime p. The only results we know about this case are the following: Theorem 1.2 (Bonisoli, Buratti, Mazzuoccolo [7]). There exists a 2-transitive elementary abelian Kirkman k-cycle system of order v if and only if (k, v) = ( p, p n ) for some odd prime p and some positive integer n. Theorem 1.3 (Granville, Moisiadis, Rees [13]).…”

“…cycle systems and, in particular, about Steiner cycle systems can be found in [1], [3], [7], [12], [13], [15], [16], [17]. Here we only recall that a necessary condition for the existence of a Sk S(v) is that v and k be odd integers.…”