# Edmonds-Karp bound doesn't seem to be tight

The proof that the Edmonds-Karp algorithm will require at most $$O(|V||E|)$$ uses the fact that when an augmenting path has critical edge $$(u, v)$$, $$\delta_f(u)$$ strictly increases. Therefore, that particular edge can only be critical at most $$|V|$$ times and the total number of augmenting paths until the max flow is at most $$|E| * O(|V|) = O(|V||E|)$$.

But consider that vertex $$u$$ and all its outgoing edges. The sum of the times those outgoing edges are critical must be $$O(|V|)$$ by a similar reasoning. Couldn't we then take this fact and say the number of augmenting paths will be at most $$|V| * O(|V|)$$, by summing over the vertices (instead of summing over the edges like the textbook proof)? This would give $$O(|V|^2)$$ as the bound on the number of augmenting paths.

Put another way, consider the sum $$\sum_{v \in v} \delta_f(v)$$. This sum is at most $$|V|^2$$, but is strictly increased after every augmentation, so $$O(|V|^2)$$ augmentations will be required.

Is there any flaw in this reasoning? We know that $$O(|V|^2|E|)$$ is possible using Dinic's, but if Edmond's-Karp requires at most $$O(|V|^2)$$ augmentations we would get this same bound.

I tried searching for this and found this post, but I do not understand the response or comment discussion. Maybe someone can rephrase it?