A　maximum　matching　of　a　graph　G　is　a　matching　of　G　with　the　largest　number　of　edges.　The　matching　number　of　a　graph　G,　denoted　by　alpha＇(G),　is　the　number　of　edges　in　a　maximum　matching　of　G.　In　1966,　Gallai　conjectured　that　all　the　longest　paths　of　a　connected　graph　have　a　common　vertex.　Although　this　conjecture　has　been　disproved,　finding　some　nice　classes　of　graphs　that　support　this　conjecture　is　still　very　meaningful　and　interesting.　In　this　short　note,　we　prove　that　Gallai＇s　conjecture　is　true　for　every　connected　graph　G　with　alpha＇(G)　a　(c)　1/2　3.