Let　G　be　a　graph　with　n　vertices.　A　path　decomposition　of　G　is　a　set　of　edge-disjoint　paths　including　all　the　edges　of　G.　Let　p(G)　denote　the　minimum　number　of　paths　in　a　path　decomposition　of　G.　Gallai＇s　Conjecture　asserts　that　if　G　is　connected,　then　[Formula　prsented].　The　E-subgraph　of　G　is　the　subgraph　induced　by　the　vertices　of　even　degree　in　G.　A　well-known　result　of　Lovász　is　that　if　the　E-subgraph　of　G　is　empty　or　isomorphic　to　K1,　then　[Formula　prsented].　In　this　paper,　we　prove　that　if　the　E-subgraph　of　G　is　isomorphic　to　Km　with　m≤15,　then　[Formula　prsented],　which　implies,　under　the　condition,　that　Gallai＇s　Conjecture　holds　when　n　is　odd.　A　simple　graph　G　on　n　vertices　is　called　a　semi-clique　if　[Formula　prsented].　By　the　definition,　if　G　is　a　semi-clique　on　n　vertices,　then　n　must　be　odd　and　[Formula　prsented].　As　a　corollary　of　our　main　result,　we　obtain　that　if　G　is　a　semi-clique　on　n　vertices,　then　[Formula　prsented].　©　2025　Elsevier　B.V.