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!