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Assume a game on a Graph $G$ with two players called Alice and Bob. They alternate their moves and Alice always begins.

In the beginning Alice puts a coin on arbitrarily vertex of the Graph. In all following moves, the coin must be moved along an edge from $G$ to an unvisited vertex of $G$ (i.e. a vertex on which the coin has never been placed in the history of the game).

The first player who can no longer make a move (because all neighboring vertices have already been visited) loses the game.

For all matching $M$ of the graph $G$ there exists a Strategie called $S_M$. If the coin is on the vertex $v$ before the start of the move, make the following move:

  1. If $v$ is in a edge $\{u,v\} \in M$ and $u$ is unvisited, than go along the edge $\{u,v\}$ to $u$.

  2. Otherwise: Give up (the other player wins).

(i) Assume for this $M$ isn't a perfect matching and Alice puts in here first move the coin on a vertex which isn't in $M$ and than she plays according to $S_M$. How can I proof that if Alice lost the game, then $M$ is not a maximum matching?

My thought: When I try to prove this with contraposition: $M$ is a maximum matching $\implies$ Alice wins the game. But i have found a counterexample. Assume a Graph with 3 vertices. Two are connected with one edge, and one vertex is isolated and $M = \{\}$, hence $M$ is not perfect. Alice puts the coin on the isolated vertex. She wins the game, but $M$ is not a maximum matching - how could that be?

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Statement 1: If Alice loses then $M$ is not a maximum matching.

Statement 2: If Alice wins then $M$ is a maximum matching.

These statements do not say the same thing. The task asks you to prove Statement 1. Your counterexample is a counterexample for statement 2.

Maybe this is easier to understand if I add another statement:

Statement 3: If $M$ is a maximum matching then Alice wins.

Now statement 3 is equivalent to statement 1. But statement 3 is different from statement 2. In your post you try to prove statement 1 by proving statement 3 (which is also called an indirect proof and this works perfectly fine). But your counterexample is a counterexample for statement 2 and thus does not invalidate statements 1 and 3.

And another idea which might help: Assume that we try to use your 3-vertex graph as counterexample for statement 1. In order to do that we'd have to find a maximum matching on that graph where Alice loses. But that is not possible since a maximum matching on your graph contains the only edge and thus Alice will put the coin on the isolated vertex and win.

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