Gallai’s　conjecture　asserts　that　every　connected　graph　on　n　vertices　can　be　decomposed　into　at　most　n+12　paths.　The　E-subgraph　of　a　graph　G,　denoted　by　Ge,　is　the　subgraph　induced　by　the　vertices　of　even　degree　in　G.　A　triangle　pair　is　a　graph　consisting　of　two　triangles　with　exactly　one　vertex　in　common.　In　this　paper,　it　is　proved　that　Gallai’s　conjecture　is　true　for　graphs　G　in　which　Ge　contains　no　triangle　pair　and　each　block　of　Ge　has　maximum　degree　at　most　3.　©　2022,　Operations　Research　Society　of　China,　Periodicals　Agency　of　Shanghai　University,　Science　Press,　and　Springer-Verlag　GmbH　Germany,　part　of　Springer　Nature.