Matching and edge-connectivity in graphs with given maximum degree

In a recent paper (2018) [10] the authors determine tight lower bounds on the matching number in a graph with given maximum degree in terms of the number of vertices (order) and number of edges (size). In this paper, we raise the problem to a higher level by proving a tight lower bound on the matching number of a graph with given maximum degree and edge-connectivity in terms of its order and size. For a graph G of order n, size m, matching number α^′(G), edge-connectivity λ(G)≥λ≥2 and maximum degree k≥λ we determine best possible constants a_k,λ, b_k,λ and c_k,λ (depending only on k and λ) such that α^′(G)≥a_k,λ⋅n+b_k,λ⋅m−c_k,λ. We prove that the above bounds are tight for essentially all densities of graphs.