In the flow network, integer max flow sends at most unit through each vertex (by capacity constraints). A flow of units means edge-disjoint paths from to , each passing through one vertex in and one in .
This set of paths corresponds exactly to a matching of size . Since max flow finds the largest such flow, it finds the maximum matching. Algorithms like Dinic or Ford-Fulkerson solve this in or faster.
Space complexity is for the data structures used.