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In 2010, Mkrtchyan, Petrosyan and Vardanyan proved that every graph $G$ with$2\leq \delta(G)\leq \Delta(G)\leq 3$ contains a maximum matching whoseunsaturated vertices do not have a common neighbor, where $\Delta(G)$ and$\delta(G)$ denote the maximum and minimum degrees of vertices in $G$,respectively. In the same paper they suggested the following conjecture: everygraph $G$ with $\Delta(G)-\delta(G)\leq 1$ contains a maximum matching whoseunsaturated vertices do not have a common neighbor. Recently, Picouleaudisproved this conjecture by constructing a bipartite counterexample $G$ with$\Delta(G)=5$ and $\delta(G)=4$. In this note we show that the conjecture isfalse for graphs $G$ with $\Delta(G)-\delta(G)=1$ and $\Delta(G)\geq 4$, andfor $r$-regular graphs when $r\geq 7$.

On maximum matchings in almost regular graphs