The Oberwolfach problem and factors of uniform odd length cycles

The Hamilton—Waterloo problem: The case of triangle‐factors and one Hamilton cycle

“…Each resolution class of the KTS gives n − 2 parallel classes of triples for a total of v−1 2 · (n − 2) parallel classes of this type. The HW(n)'s together contribute n−3 2 parallel classes of triples and there are v − 1 parallel classes formed by the classes in the 3-RGDD(2 v ) combined with the partial parallel classes from the TD(3, n)-TD (3,2). The total number of parallel classes of triples is vn−3 2 as required.…”

“…Generalizing to higher k, a resolvable k−cycle system of order n is a 2-factorization of K n in which each 2-factor consists exclusively of k−cycles. In 1989, Alspach, Schellenberg, Stinson and Wagner [2] proved that the necessary conditions are sufficient for the existence of a resolvable k−cycle system of order n, namely that n is odd and that k ≡ n mod(2n).…”

Bipartite 2‐Factorizations of Complete Multipartite Graphs

“…The problem of finding a 2-factorisation of K n in which the 2-factors are isomorphic to a given 2-factor F is the Oberwolfach Problem. The Oberwolfach Problem has been completely settled for infinitely many values of n [8], when F consists of cycles of uniform length [2], and in many other special cases. The known results on the Oberwolfach Problem up to 2007 can be found in the survey [7], and several new results appearing after [7] was published are cited in the introduction of [6].…”

“…Häggkvist [10] observed that for any bipartite 2-regular graph F on 2m vertices, there is a 2-factorisation of C (2) m into two copies of F . The following very useful result, on which many of our constructions depend, is an immediate consequence of Häggkvist's observation and the fact that F (2) is a factorisation of K (2) when F is a factorisation of K. If F is a Hamilton decomposition of K, then we obtain a 4-factorisation of K (2) into copies of C (2) m (where m is the number of vertices in K), and we then obtain the required 2-factorisation of K (2) by factorising each copy of C (2) m into two copies of the required bipartite 2-regular graph.…”

“…The complete multipartite graph K n r , r ≥ 2, has a 2-factorisation into 2-factors composed of k-cycles if and only if k divides rn, (r − 1)n is even, k is even when r = 2, and (k, r, n) is none of (3, 3, 2), (3, 6, 2), (3,3,6), (6,2,6).…”