How to find a triangle inside a graph?

Question

Here is an exercise in the Algorithm Design Manual.

Consider the problem of determining whether a given undirected graph G = (V, E) contains a triangle

Answer 1

A possible O(|V||E|) solution, is the same idea of the brute-force in (a):

for each edge (u, v):
  for each vertex w:
     if (v, w) is an edge and (w, u) is an edge:
          return true
return false

check all edges, and not all vertices pairs - with another vertex that forms a triangle - it is enough information to determine if the edge and vertex form a feasible solution.

Counter example to BFS solution:

       A
     / | \
    /  |  \
   B   C   D
   |   |   |
   |   |   |
   F---G---H
   |       |
   ---------
    (F, H) is also an edge

Note that father[F] != father[G] != father[H], thus the algorithm will return false - but nevertheless, (F, G, H) is a feasible solution!