Abstract
Various results on factorisations of complete graphs into circulant graphs and on 2-factorisations of these circulant graphs are proved. As a consequence, a number of new results on the Oberwolfach Problem are obtained. For example, a complete solution to the Oberwolfach Problem is given for every 2-regular graph of order 2 p where p ≡ 5 (mod 8) is prime.
Highlights
The Oberwolfach problem was posed by Ringel in the 1960s and is first mentioned in [16]
Let F be an arbitrary 2-regular graph and let n be the order of F
The Oberwolfach Problem OP(F ) asks for a 2-factorisation of Kn into F, and if n is even, OP(F ) asks for a 2-factorisation of Kn − I into F, where Kn − I denotes the graph obtained from Kn by removing the edges of a 1-factor
Summary
The Oberwolfach problem was posed by Ringel in the 1960s and is first mentioned in [16] We show that OP(F ) has a solution for every 2-regular graph of order 2p where p is any prime congruent to 5 (mod 8) (see Theorem 4.2), and we obtain solutions to OP(F ) for broad classes of 2-regular graphs in many other cases (see Theorems 4.3 and 4.4). We obtain results on the generalisation of the Oberwolfach Problem to factorisations of complete multigraphs into isomorphic 2factors (see Theorem 5.4). Our results are obtained by constructing various factorisations of complete graphs into circulant graphs, and showing in Section 3 that these circulant graphs can themselves be factored into isomorphic 2-regular graphs in a wide variety of cases Our results are obtained by constructing various factorisations of complete graphs into circulant graphs in Section 2, and showing in Section 3 that these circulant graphs can themselves be factored into isomorphic 2-regular graphs in a wide variety of cases
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