# The Maximum Path of a Graph Contained In Maximum Matching

I just started learning some basic graph theory stuff and I was wondering if the following claim/proof is valid and whether it has any implications.

I want to show that the Maximum Path of any graph must be one of the M-alternating paths formed by the maximum matching.

My reasoning for this is that if there exists a maximum matching in a graph, then that graph does not have any M-augmenting paths, therefore, the longest path in such a graph would have to be M-alternating.

Is this correct? Does this have any consequences? Any small pointers are helpful!! Thank you!

Consider this graph on 6 vertices, $$\{0, 1, 2, 3, 4, 5\}$$, and the following edges: $$(0, 1)$$, $$(2, 3)$$, $$(4, 5)$$ - which form the maximum matching - and also $$(1, 3)$$ and $$(3, 5)$$.
Then your maximal (acyclic) path is $$0-1-3-5-4$$, which is not an alternating path.