This is part of a bigger proof I'm trying to solve, which eventually came down to one thing:
Let $G=(V,E)$ an un-directed, complete, metric graph (maintains the triangle inequality) with an even number of vertices. Let $PM$ denote a minimum-weight-perfect matching on $E$ and let $MST$ denote a minimum-spanning-tree of $G$.
Is it true that $w(PM) \leq w(MST)$ ?
I'm not sure if the fact that the graph is metric is useful, but I have pointed that out any way.
I know $PM$ and $MST$ do not necessarily share all edges, but maybe some of them? I was trying and couldn't find a counter-example to disprove this.
My gut says it's true, but I cannot find a definitive way to prove it.