Abstract
Let M be the number of edges in a maximum matching in graphs with m edges, maximum vertex degree k and shortest simple odd-length cycle length L. We show that
This lower bound is tight.
When no simple odd-length cycle exists it is known previously that \(M\geq \frac{m}{k}\).
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Biedl T, Demaine E, Duncan C, Fleischer R, Kobourov S (2004) Tight bounds on the maximal and maximum matchings. Discrete Math 285(1–3):7–15
Feng W, Qu W, Wang H (2007) Lower bounds on the cardinality of maximum matchings in graphs with bounded degrees. In: Proceedings of the 2007 international conference on foundations of computer science, Las Vegas, pp 110–113
Han Y (2008) Matching for graphs of bounded degree. In: Proceedings of the second international workshop on frontiers in algorithmics (FAW’2008), Changsha, China. Lecture notes in computer science, vol 5059. Springer, Berlin, pp 171–173
König D (1916) Über Graphen und ihre Anwendung auf Determinantentheorie und Mengenlehre. Math Ann 77:453–456
Misra J, Gries D (1992) A constructive proof of Vizing’s theorem. Inf Process Lett 41:131–133
Nishizeki T, Baybars I (1979) Lower bounds on the cardinality of the maximum matchings of planar graphs. Discrete Math 28(3):255–267
Petersen J (1891) Die Theorie der regulären Graphs (The theory of regular graphs). Acta Math 15:193–220
Tutte WT (1947) The factorization of linear graphs. J Lond Math Soc 22:107–111
Vizing VG (1964) On an estimate of chromatic class of a p-graph. Diskretn Anal 3:25–30 (in Russian)
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An erratum to this article is available at http://dx.doi.org/10.1007/s10878-012-9566-8.
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Han, Y. Tight bound for matching. J Comb Optim 23, 322–330 (2012). https://doi.org/10.1007/s10878-010-9299-5
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DOI: https://doi.org/10.1007/s10878-010-9299-5