On maximum matchings in almost regular graphs

In 2010, Mkrtchyan, Petrosyan, and Vardanyan proved that every graph G with 2 ≤ δ ( G ) ≤ Δ ( G ) ≤ 3 contains a maximum matching M such that no two vertices uncovered by M share a 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 ) ≤ 1 contains a maximum matching M such that no two vertices uncovered by M share a 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 ) ≥ 4 , and for r -regular graphs when r ≥ 7 .