Minimum Number of Vertices of Graphs without Perfect Matching, with Given Edge Connectivity and Minimum and Maximum Degrees

A matching \( M \) in a graph \( G \) is a subset of \( E(G) \) in which no two edges have a vertex in common. A vertex \( V \) is unsaturated by \( M \) if there is no edge of \( M \) incident with \( V \). A matching \( M \) is called a perfect matching if there is no vertex of the graph that is unsaturated by \( M \). Let \( G \) be a \( k \)-edge-connected graph, \( k \geq 1 \), on even \( n \) vertices, with minimum degree \( r \) and maximum degree \( r + e \), \( e \geq 1 \). In this paper, we find a lower bound for \( n \) when \( G \) has no perfect matchings.