I am wondering if anyone can point me in the right direction for the answer to the following question. It has come up in my research and I am not sure where to look or who to ask.
I am wondering under what conditions a graph has a matching that touches all maximum degree vertices. Obviously this is not always possible (complete graphs of odd order). Sometimes it is possible though (complete graphs of even order). Does anyone know of a known theorem that describes this?
I am also wondering how close to such a matching one can get, if such a matching cannot be found. For example, in $K_5$ we can choose three edges that almost make a matching, but two of them share a vertex (i.e., a single edge and a $P_2$). For an arbitrary graph, can you always cover the maximum degree vertices if you allow for a single $P_2$ (the rest of the edges form a matching)? How many $P_2$'s will you need to allow in general to cover all the maximum degree vertices?
Thank you!
Edit: I forgot to mention---I know that bipartite graphs have matchings that cover the maximum degree vertices.