Matching and edge-connectivity in graphs with given maximum degree

A complete description of convex sets associated with matchings and edge‐connectivity in graphs

“…In Sections 5.4 and 5.6, we show that the lower bounds on the matching number presented in Theorem 6, Theorem 7 and Corollary 2 are all tight. As a consequence of these new results and earlier results established in [6], we have the following result where the families , , ,…”

“…≥ which clearly holds under the conditions of i k + 1 ≥ and λ k + 1 ≤ . Therefore by Equation ( 4) and Inequality (6), and by definition of e k λ , and d k λ , , the following holds.…”

“…We note that in this case, the graph G H nm = (, )  to be the family consisting of all graphs H k λ , defined in Definition 2 where X i has order k + 2 for all i [ℓ] ∈ . In all the above cases, k and λ have the same parity and as shown in [6]…”

“…Definition 1 (Henning and Yeo [6]). Let λ 2 ≥ be an arbitrary integer and let k 2 ≥ be an even integer.…”

“…Matchings in graphs are extensively studied in the literature (see, e.g., the classical book on matchings my Lovász and Plummer [8]) with over 8000 publications according to MathSciNet. Results on convex sets and matching can be found, for example, in [3,5], while results on connectivity and matchings can be found, for example, in [6,7,[9][10][11].…”