Abstract
In 1963, Corradi and Hajnal proved that for all k≥1 and n≥3k, every (simple) graph G on n vertices 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, the authors characterized the simple graphs G with minimum degree δ(G)≥2k—1 that do not contain k disjoint cycles. We use this result to answer Dirac's question in full.
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References
K. Corráadi and A. Hajnal: On the maximal number of independent circuits in a graph, Acta Math. Acad. Sci. Hungar. 14 (1963), 423–439.
G. Dirac: Some results concerning the structure of graphs, Canad. Math. Bull. 6 (1963), 183–210.
G. Dirac and P. Erdős: On the maximal number of independent circuits in a graph, Acta Math. Acad. Sci. Hungar. 14 (1963), 79–94.
H. A. Kierstead, A. V. Kostochka and E. C. Yeager: On the Corradi-Hajnal Theorem and a question of Dirac, submitted.
L. Lovasz: On graphs not containing independent circuits, (Hungarian, English summary) Mat. Lapok 16 (1965), 289–299.
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The first two authors thank Institut Mittag-Leffler (Djursholm, Sweden) for the hospitality and creative environment.
Research of this author is supported in part by NSA grant H98230-12-1-0212.
Research of this author is supported in part by NSF grant DMS-1266016 and by Grant NSh.1939.2014.1 of the President of Russia for Leading Scientific Schools.
Research of this author is supported in part by NSF grants DMS 08-38434 and DMS-1266016.
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Kierstead, H.A., Kostochka, A.V. & Yeager, E.C. The (2k-1)-connected multigraphs with at most k-1 disjoint cycles. Combinatorica 37, 77–86 (2017). https://doi.org/10.1007/s00493-015-3291-8
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DOI: https://doi.org/10.1007/s00493-015-3291-8