On maximum matchings in almost regular graphs

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

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