TY - CHAP

T1 - An algorithmic answer to the ore-type version of Dirac’s question on disjoint cycles

AU - Kierstead, Henry

AU - Kostochka, A. V.

AU - Molla, T.

AU - Yager, D.

N1 - Funding Information:
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.

PY - 2018

Y1 - 2018

N2 - 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 G̲ satisfies the following Ore-type condition: dG̲(v)+dG̲(u)≥4k−3 for all nonadjacent u, v ∈ V (G).

AB - 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 G̲ satisfies the following Ore-type condition: dG̲(v)+dG̲(u)≥4k−3 for all nonadjacent u, v ∈ V (G).

UR - http://www.scopus.com/inward/record.url?scp=85054178766&partnerID=8YFLogxK

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U2 - 10.1007/978-3-319-94830-0_8

DO - 10.1007/978-3-319-94830-0_8

M3 - Chapter

AN - SCOPUS:85054178766

T3 - Springer Optimization and Its Applications

SP - 149

EP - 168

BT - Springer Optimization and Its Applications

PB - Springer International Publishing

ER -