1,413 research outputs found

    A complete solution to the infinite Oberwolfach problem

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    Let FF be a 22-regular graph of order vv. The Oberwolfach problem, OP(F)OP(F), asks for a 22-factorization of the complete graph on vv vertices in which each 22-factor is isomorphic to FF. In this paper, we give a complete solution to the Oberwolfach problem over infinite complete graphs, proving the existence of solutions that are regular under the action of a given involution free group GG. We will also consider the same problem in the more general contest of graphs FF that are spanning subgraphs of an infinite complete graph K\mathbb{K} and we provide a solution when FF is locally finite. Moreover, we characterize the infinite subgraphs LL of FF such that there exists a solution to OP(F)OP(F) containing a solution to OP(L)OP(L)

    The Hamilton-Waterloo Problem with even cycle lengths

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    The Hamilton-Waterloo Problem HWP(v;m,n;α,β)(v;m,n;\alpha,\beta) asks for a 2-factorization of the complete graph KvK_v or KvIK_v-I, the complete graph with the edges of a 1-factor removed, into α\alpha CmC_m-factors and β\beta CnC_n-factors, where 3m<n3 \leq m < n. In the case that mm and nn are both even, the problem has been solved except possibly when 1{α,β}1 \in \{\alpha,\beta\} or when α\alpha and β\beta are both odd, in which case necessarily v2(mod4)v \equiv 2 \pmod{4}. In this paper, we develop a new construction that creates factorizations with larger cycles from existing factorizations under certain conditions. This construction enables us to show that there is a solution to HWP(v;2m,2n;α,β)(v;2m,2n;\alpha,\beta) for odd α\alpha and β\beta whenever the obvious necessary conditions hold, except possibly if β=1\beta=1; β=3\beta=3 and gcd(m,n)=1\gcd(m,n)=1; α=1\alpha=1; or v=2mn/gcd(m,n)v=2mn/\gcd(m,n). This result almost completely settles the existence problem for even cycles, other than the possible exceptions noted above

    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

    Infinitely many cyclic solutions to the Hamilton-Waterloo problem with odd length cycles

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    It is conjectured that for every pair (,m)(\ell,m) of odd integers greater than 2 with m1  (mod)m \equiv 1\; \pmod{\ell}, there exists a cyclic two-factorization of KmK_{\ell m} having exactly (m1)/2(m-1)/2 factors of type m\ell^m and all the others of type mm^{\ell}. The authors prove the conjecture in the affirmative when 1  (mod4)\ell \equiv 1\; \pmod{4} and m2+1m \geq \ell^2 -\ell + 1.Comment: 31 page
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