Let $G = (V, E)$ be a graph, and let $M \subseteq E$ be a matching such that there is no augmenting path of length at most 3 for $M$. Prove that $|M| \geq \frac{2}{3} |M^*|,$
where $M^*$ is the maximum matching for $G$.
I've been exploring an idea related to matchings within graphs and would appreciate any insights or feedback. The concept focuses on examining the symmetric difference between a matching $M$ and a maximum matching $M^*$ within a graph $G$.
In this graph formed by the symmetric difference, we observe only even cycles and disjoint paths. Given that M* represents the maximum matching and considering the given condition that there are no augmenting paths of length less than 3 for $M$, it appears that any path starting from M* and ending in $M$ (or vice versa) would have a minimum length of 5. This implies that for every 3 edges in M*, there are at least 2 edges in $M$, due to the alternating nature of the paths in the symmetric difference graph.
This observation suggests the desired ratio, indicating a close relationship between the size of $M$ and $M^*$, under the specific conditions given.
However, I feel this approach might be somewhat speculative or not entirely rigorous. If anyone has any suggestions, directions, or corrections, I would be very grateful to hear them.
Thank you