Gallai＇s　conjecture　asserts　that　every　connected　graph　of　order　n　can　be　decomposed　into　[Formula　presented]　paths.　A　graph　G　is　k-degenerated　if　each　subgraph　admits　a　vertex　with　degree　no　more　than　k.　In　this　paper,　we　characterize　the　graphs　that　contain　a　path　through　specified　edges.　As　a　result,　we　prove　that　a　connected　3-degenerated　graph　of　order　n　that　is　not　isomorphic　to　K3　or　K5−　can　be　decomposed　into　[Formula　presented]　paths,　which　extends　three　theorems　of　[2,3,12].　©　2024　Elsevier　B.V.