In the last 2 paragraphs of the paper about Hopcroft–Karp algorithm to find the maximum cardinality matching in bipartite graph:
The execution time of a phase is O(m+n), where m is the number of edges in G, and n is the number of vertices. Hence the execution time of the entire algorithm is O((m+n)s), where s is the cardinality of a maximum matching.
If G has n vertices then m <= n^2 / 4 and s < n / 2 so that the execution time is bounded by O(n^(5/2)).
I don't understand given:
m <= n^2 / 4 s <= n / 2
why they concluded:
O((m+n)s) = O(n^(5/2))
Shouldn't it be:
O((m+n)s) = O(n^3)