Amalgamated Factorizations of Complete Graphs

Dugdale, J. K.; Hilton, A. J. W.

doi:10.1017/s0963548300001127

Cited by 4 publications

(3 citation statements)

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“…Theorem 6 is an adaptation of the outline/amalgamation result obtained for Hamiltonian decompositions of complete graphs in [8]. It also generalizes results in [5,9]. It concerns a K m edge-coloured with t colours, c 1 , .…”

Section: Extending Edge-colouringsmentioning

confidence: 64%

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Some Results on the Oberwolfach Problem

Hilton¹,

Johnson²

2001

J. Lond. Math. Soc.

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The well-known Oberwolfach problem is to show that it is possible to 2-factorize K n (n odd) or K n less a 1-factor (n even) into predetermined 2-factors, all isomorphic to each other; a few exceptional cases where it is not possible are known. A completely new technique is introduced that enables it to be shown that there is a solution when each 2-factor consists of k r-cycles and one (n − kr)-cycle for all n > 6kr − 1. Solutions are also given (with three exceptions) for all possible values of n when there is one r-cycle, 3 6 r 6 9, and one (n − r)-cycle, or when there are two r-cycles, 3 6 r 6 4, and one (n − 2r)-cycle.

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Section: Extending Edge-colouringsmentioning

confidence: 64%

“…Let 3 6 r 6 9, n > r + 3 be integers. Then OP(r, n − r) has a solution except for the cases OP(3, 3) and OP (4,5). Let 3 6 r 6 4, n > 2r + 3 be integers.…”

Section: Introductionmentioning

confidence: 99%

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Some Results on the Oberwolfach Problem

Hilton¹,

Johnson²

2001

J. Lond. Math. Soc.

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A Fast Algorithm for Computing a Nearly Equitable Edge Coloring with Balanced Conditions

Shioura

Yagiura

2009

Lecture Notes in Computer Science

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We discuss the nearly equitable edge coloring problem on a multigraph and propose an efficient algorithm for solving the problem, which has a better time complexity than the previous algorithms. The coloring computed by our algorithm satisfies additional balanced conditions on the number of edges used in each color class, where conditions are imposed on the balance among all edges in the multigraph as well as the balance among parallel edges between each vertex pair. None of the previous algorithms are guaranteed to satisfy these balanced conditions simultaneously. To achieve these improvements, we propose a new recoloring procedure, which is based on a set of edge-disjoint alternating walks, while the existing algorithms are based on an Eulerian circuit or a single alternating walk. This new recoloring procedure makes it possible to reduce the time complexity of the algorithm.

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