Abstract
Corrádi and Hajnal in 1963 proved the following theorem on the NP-complete problem on the existence of k disjoint cycles in an n-vertex graph G: For all k ≥ 1 and n ≥ 3k, every (simple) n-vertex graph G with minimum degree δ(G) ≥ 2k contains k disjoint cycles. The same year, Dirac described the 3-connected multigraphs not containing two disjoint cycles and asked the more general question: Which (2k − 1)-connected multigraphs do not contain k disjoint cycles? Recently, Kierstead, Kostochka, and Yeager resolved this question. In this paper, we sharpen this result by presenting a description that can be checked in polynomial time of all multigraphs G with no k disjoint cycles for which the underlying simple graph \( \underline {G}\) satisfies the following Ore-type condition: \(d_{ \underline {G}}(v)+d_{ \underline {G}}(u)\geq 4k-3\) for all nonadjacent u, v ∈ V (G).
Dedicated to Gregory Gutin on the occasion of his 60th birthday
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Notes
- 1.
Dirac used the word graphs, but in [3] this appears to mean multigraphs.
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Acknowledgements
We thank a referee for a number of helpful comments. Research of A. Kostochka is supported in part by NSF grant DMS-1600592 and by grants 18-01-00353A and 16-01-00499 of the Russian Foundation for Basic Research. Research of T. Molla is supported in part by NSF grant DMS-1500121. Research of D. Yager is supported by the Campus Research Board of the University of Illinois.
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Kierstead, H.A., Kostochka, A.V., Molla, T., Yager, D. (2018). An Algorithmic Answer to the Ore-Type Version of Dirac’s Question on Disjoint Cycles. In: Goldengorin, B. (eds) Optimization Problems in Graph Theory. Springer Optimization and Its Applications, vol 139. Springer, Cham. https://doi.org/10.1007/978-3-319-94830-0_8
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