# Is the maximum matching problem trivial when the graph is complete?

I have a quick question.

The maximum matching problem is an easy problem but not a trivial one. I was wondering that if the bipartite graph was complete, is it a trivial problem?

I think we can just select an arbitrary edge $\{u,v\}$ and remove the vertices $u$ and $v$ from the graph and continue until there is no vertices.

• By a complete bipartite graph, do you mean that there exist two disjoint sets of vertices where no two vertices in the same set have an edge between them but any two vertices from different sets do have an edge between them? There are no complete bipartite graphs with more than two vertices. – Fricative Melon Jan 18 '16 at 17:02
• As you said, I mean this: there exist two disjoint sets of vertices where no two vertices in the same set have an edge between them but any two vertices from different sets do have an edge between them. – Brika Jan 18 '16 at 17:17
• @user1657355 The term "complete bipartite graph" is completely standard, with the meaning you give. – David Richerby Jan 18 '16 at 17:51
• What have you tried? Have you tried working through some examples to see if you see a pattern? If yes, have you tried making a guess and then tried proving it? – D.W. Jan 18 '16 at 20:04
• What I tired is in the question. – Brika Jan 18 '16 at 20:31