30 research outputs found

    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

    On Hamilton decompositions of infinite circulant graphs

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    The natural infinite analogue of a (finite) Hamilton cycle is a two-way-infinite Hamilton path (connected spanning 2-valent subgraph). Although it is known that every connected 2k-valent infinite circulant graph has a two-way-infinite Hamilton path, there exist many such graphs that do not have a decomposition into k edge-disjoint two-way-infinite Hamilton paths. This contrasts with the finite case where it is conjectured that every 2k-valent connected circulant graph has a decomposition into k edge-disjoint Hamilton cycles. We settle the problem of decomposing 2k-valent infinite circulant graphs into k edge-disjoint two-way-infinite Hamilton paths for k=2, in many cases when k=3, and in many other cases including where the connection set is ±{1,2,...,k} or ±{1,2,...,k - 1, 1,2,...,k + 1}

    Perfect 1-factorisations of circulants with small degree

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    A 1-factorisation of a graph G is a decomposition of G into edge-disjoint 1-factors (perfect matchings), and a perfect 1-factorisation is a 1-factorisation in which the union of any two of the 1-factors is a Hamilton cycle. We consider the problem of the existence of perfect 1-factorisations of even order circulant graphs with small degree. In particular, we characterise the 3-regular circulant graphs that admit a perfect 1-factorisation and we solve the existence problem for a large family of 4-regular circulants. Results of computer searches for perfect 1-factorisations of 4-regular circulant graphs of orders up to 30 are provided and some problems are posed

    An Infinite Family of Connected 1-Factorisations of Complete 3-Uniform Hypergraphs

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    A connected 1-factorisation is a 1-factorisation of a hypergraph for which the union of each pair of distinct 1-factors is a connected hypergraph. A uniform 1-factorisation is a 1-factorisation of a hypergraph for which the union of each pair of distinct 1-factors is isomorphic to the same subhypergraph, and a uniform-connected 1-factorisation is a uniform 1-factorisation in which that subhypergraph is connected. Chen and Lu [Journal of Algebraic Combinatorics, 46(2) 475--497, 2017] describe a family of 1-factorisations of the complete 3-uniform hypergraph on q+1q+1 vertices, where q≡2(mod3)q\equiv 2\pmod 3 is a prime power. In this paper, we show that their construction yields a connected 1-factorisation only when q=2,5,11q=2,5,11 or q=2pq=2^p for some odd prime pp, and a uniform 1-factorisation only for q=2,5,8q=2,5,8 (each of these is a uniform-connected 1-factorisation).Comment: 11 page

    On factorisations of complete graphs into circulant graphs and the Oberwolfach problem

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    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 2p where p ≡ 5 (mod 8) is prime

    Nonextendible Latin Cuboids

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    We show that for all integers m >= 4 there exists a 2m x 2m x m latin cuboid that cannot be completed to a 2mx2mx2m latin cube. We also show that for all even m > 2 there exists a (2m-1) x (2m-1) x (m-1) latin cuboid that cannot be extended to any (2m-1) x (2m-1) x m latin cuboid

    The Molecular Identification of Organic Compounds in the Atmosphere: State of the Art and Challenges

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    Common multiples of complete graphs

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    A graph H is said to divide a graph G if there exists a set S of subgraphs of G, all isomorphic to H, such that the edge set of G is partitioned by the edge sets of the subgraphs in S. Thus, a graph G is a common multiple of two graphs if each of the two graphs divides G