On sharply vertex transitive 2-factorizations of the complete graph

Abstract

We introduce the concept of a 2-starter in a group G of odd order. We prove that any 2-factorization of the complete graph admitting G as a sharply vertex transitive automorphism group is equivalent to a suitable 2-starter in G. Some classes of 2-starters are studied, with special attention given to those leading to solutions of some Oberwolfach or Hamilton-Waterloo problems

Uniformly resolvable decompositions of Kv into paths on two, three and four vertices

“…The case (0, 9,8) corresponds to a (P 2 , P 3 , P 4 ) − URGDD(0, 9, 8) of type 12 3 which is known to exist [15], while the case (1, 6, 10) is given by Lemma 4.5.…”

“…The Oberwolfach problem has been solved for a single cycle size [3] and recently for the case where the factors contain exactly two cycles [35]; for solutions to the analogous Oberwolfach Problem for complete multipartite graphs see [6,7,23,24], and [28]. As regards the Hamilton-Waterloo problem, several cases have been solved in [2,11,12], and [18], where the authors have focused on the case where both H and W are uniform, i.e., the case where all cycles of H have an assigned length h and all cycles of W have an assigned length w; in particular, the case where h = 3 and w = 4 is solved in [11], with some exceptions which are solved in [5]; the case where h = 3 and w = v is studied in [12] and [18]; finally, many other solutions to the Oberwolfach problem and to the Hamilton-Waterloo problem can be found in the literature (see, for instance, [8] and [9]). The existence problem for H-factorizations of K v have been studied in the case where: H = {K k } with k = 3, 4, 5 (for k = 5 there are only four undecided values of v), see [1]; H = {P k } for any k ≥ 2 [4,17,19]); H is a set of two complete graphs of order at most five [13,29,31,33,32,34]; H is a set of two paths on two, three or four vertices [15,16]; H = {P 3 , K 3 +e} [14];…”

Doubly transitive 2‐factorizations

“…These results were presented by the first author at the conference "Giornate di Geometria" held at the Università "La Sapienza" in Rome, December [4][5][6]2003.…”

“…In his lecture at the International Symposium on Graphs, Designs, and Applications, held in Messina (Italy) from Sept. 30, 2003 to Oct. 4,2003, Alexander Rosa remarked, among other things, that doubly transitive 1-factorizations of complete graphs are classified by the work of Cameron and Korchmàros [5], while no such classification for doubly transitive 2-factorizations of complete graphs was available yet, see [11,Section 6]. Here "doubly transitive" means there exists an automorphism group of the 2-factorization acting doubly transitively on the vertices of the underlying complete graph.…”

On the Hamilton‐Waterloo Problem for Bipartite 2‐Factors

“…There are numerous other results on the Oberwolfach Problem, dealing with various special families of 2-regular graphs, see [9,13,14,25,28,31,33,40,42,43,45,47] and see [11] for a survey of results up to…”