2,162 research outputs found

    Resolution of the Oberwolfach problem

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    The Oberwolfach problem, posed by Ringel in 1967, asks for a decomposition of K2n+1K_{2n+1} into edge-disjoint copies of a given 22-factor. We show that this can be achieved for all large nn. We actually prove a significantly more general result, which allows for decompositions into more general types of factors. In particular, this also resolves the Hamilton-Waterloo problem for large nn.Comment: 28 page

    A Generalization of the Hamilton-Waterloo Problem on Complete Equipartite Graphs

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    The Hamilton-Waterloo problem asks for which ss and rr the complete graph KnK_n can be decomposed into ss copies of a given 2-factor F1F_1 and rr copies of a given 2-factor F2F_2 (and one copy of a 1-factor if nn is even). In this paper we generalize the problem to complete equipartite graphs K(n:m)K_{(n:m)} and show that K(xyzw:m)K_{(xyzw:m)} can be decomposed into ss copies of a 2-factor consisting of cycles of length xzmxzm; and rr copies of a 2-factor consisting of cycles of length yzmyzm, whenever mm is odd, s,r1s,r\neq 1, gcd(x,z)=gcd(y,z)=1\gcd(x,z)=\gcd(y,z)=1 and xyz0(mod4)xyz\neq 0 \pmod 4. We also give some more general constructions where the cycles in a given two factor may have different lengths. We use these constructions to find solutions to the Hamilton-Waterloo problem for complete graphs
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