6,578 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

    Long path and cycle decompositions of even hypercubes

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    We consider edge decompositions of the nn-dimensional hypercube QnQ_n into isomorphic copies of a given graph HH. While a number of results are known about decomposing QnQ_n into graphs from various classes, the simplest cases of paths and cycles of a given length are far from being understood. A conjecture of Erde asserts that if nn is even, â„“<2n\ell < 2^n and â„“\ell divides the number of edges of QnQ_n, then the path of length â„“\ell decomposes QnQ_n. Tapadia et al.\ proved that any path of length 2mn2^mn, where 2m<n2^m<n, satisfying these conditions decomposes QnQ_n. Here, we make progress toward resolving Erde's conjecture by showing that cycles of certain lengths up to 2n+1/n2^{n+1}/n decompose QnQ_n. As a consequence, we show that QnQ_n can be decomposed into copies of any path of length at most 2n/n2^{n}/n dividing the number of edges of QnQ_n, thereby settling Erde's conjecture up to a linear factor

    On Hamilton Decompositions of Line Graphs of Non-Hamiltonian Graphs and Graphs without Separating Transitions

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    In contrast with Kotzig's result that the line graph of a 33-regular graph XX is Hamilton decomposable if and only if XX is Hamiltonian, we show that for each integer k≥4k\geq 4 there exists a simple non-Hamiltonian kk-regular graph whose line graph has a Hamilton decomposition. We also answer a question of Jackson by showing that for each integer k≥3k\geq 3 there exists a simple connected kk-regular graph with no separating transitions whose line graph has no Hamilton decomposition
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