I have been having a little difficulty with the question below.
Given the following graph:
Give the maximum matching i.e a matching with the most edges.
My problem is that is it possible for there to be multiple possibilities for the answer? As you could have {(v1, v4),(v3, v6)} or {(v1, v6), (v2, v4)} etc. each of which have 4 edges.
There are precisely two maximum matchings in that graph. Since there are matchings consisting of three edges (and this is the largest possible since there are six vertices), $v_1$ must be matched to either $v_2$ or $v_3$. If $v_1 v_2$ is in the matching, then $v_3 v_5$ and $v_4 v_6$ must be in the matching. If $v_1 v_3$ is in the matching, then $v_2 v_4$ and $v_5 v_6$ are in the matching. Thus there are exactly two maximum matchings.
One must be careful when asking and answering questions like these to use indefinite articles like "a" and "an" instead of "the", since "the" connotes uniqueness.