Matching Covered Graphs and Subdivisions of $K_4$ and $\overlineC_6$

We give a very simple proof that every non-bipartite matching covered graph contains a nice subgraph that is an odd subdivision of $K_4$ or $\overline{C_6}$. It follows immediately that every brick different from $K_4$ and $\overline{C_6}$ has an edge whose removal preserves the matching covered property. These are classical and very useful results due to Lovász.

1994