  It was conjectured by Mkrtchyan, Petrosyan, and Vardanyan that every graph
$G$ with $\Delta(G)-\delta(G) \le 1$ has a maximum matching $M$ such that any
two $M$-unsaturated vertices do not share a neighbor. In this note, we confirm
the conjecture for all $k$-regular simple graphs and also $k$-regular
multigraphs with $k\le 4$.
