In this note, we study bounds on the maximum cardinality of a b-matching, where a b-matching is an edge set Mb ⊆ E in a graph G = (V, E) with the constraint that dMb (v) ≤ b for every vertex v. Here dMb (v) is the degree of v in the subgraph induced by Mb and b ∈ N is a constant. If b = 1, then b-matchings are ordinary matchings. For the maximum cardinality of an ordinary matching, we derive ν (G) ≥ frac(2 m, 3 k - 1) for k ≥ 3, where ν (G) denotes the maximum cardinality of a matching in G, m is the number of edges, and k is the maximum degree of G. This answers an open question proposed by Biedl, Demaine, Duncan, Fleischer and Kobourov [T. Biedl, E. Demaine, C. Duncan, R. Fleischer, S. Kobourov, Tight bounds on maximal and maximum matchings, Discrete Math. 285 (1-3) (2004) 7-15]. For the maximum cardinality of a b-matching, we derive frac(νb (G), νb - 1 (G)) ≤ 1 + frac(4 b - 2, 3 b2 - 5 b + 2) for b ≥ 3, where νb (G) denotes the maximum cardinality of a b-matching in G. © 2008 Elsevier B.V. All rights reserved.
Feng, W. (2009). Bounds on maximum b-matchings. Discrete Mathematics, 309(12), 4162–4165. https://doi.org/10.1016/j.disc.2008.10.018