A matching is a maximum matching iff it contains no augmenting path.
Berge's Theorem
See also
MatchingExplore with Wolfram|Alpha
References
Berge, C. "Two Theorems in Graph Theory." Proc. Nat. Acad. Sci. USA 43, 842-844, 1957.Pemmaraju, S. and Skiena, S. Computational Discrete Mathematics: Combinatorics and Graph Theory in Mathematica. Cambridge, England: Cambridge University Press, p. 345, 2003.Skiena, S. Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990.Referenced on Wolfram|Alpha
Berge's TheoremCite this as:
Weisstein, Eric W. "Berge's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BergesTheorem.html