Let　G　be　a　graph　with　n　vertices.　A　path　decomposition　of　G　is　a　set　of　edge-disjoint　paths　containing　all　the　edges　of　G.　Let　p(G)　denote　the　minimum　number　of　paths　needed　in　a　path　decomposition　of　G.　Gallai　Conjecture　asserts　that　if　G　is　connected,　then　p(G)　＜=　inverted　right　perpendicularn/2inverted　left　perpendicular.　If　G　is　allowed　to　be　disconnected,　then　the　upper　bound　left　perpendicular3/4nright　perpendicular　for　p(G)　was　obtained　by　Donald　[7],　which　was　improved　to　left　perpendicular2/3nright　perpendicular　independently　by　Dean　and　Kouider　[6]　and　Yan　[14].　For　graphs　consisting　of　vertex-disjoint　triangles,　left　perpendicular2/3nright　perpendicular　is　reached　and　so　this　bound　is　tight.　If　triangles　are　forbidden　in　G,　then　p(G)　＜=　left　perpendicularg+1/2g　nright　perpendicular　can　be　derived　from　the　result　of　Harding　and　McGuinness　[11],　where　g　denotes　the　girth　of　G.　In　this　paper,　we　also　focus　on　triangle-free　graphs　and　prove　that　p(G)　＜=　left　perpendicular3n/5right　perpendicular,　which　improves　the　above　result　with　g　=　4.　(C)　2022　Elsevier　B.V.　All　rights　reserved.