A graph is called H-free if it has no induced subgraph isomorphic to H. A graph is called Ni\documentclass[12pt]{minimal}
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\begin{document}$$N^i$$\end{document}-locally connected if G[{x∈V(G):1≤dG(w,x)≤i}]\documentclass[12pt]{minimal}
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\begin{document}$$G[\{ x\in V(G): 1\le d_G(w, x)\le i\}]$$\end{document} is connected and N2\documentclass[12pt]{minimal}
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\begin{document}$$N_2$$\end{document}-locally connected if G[{uv:{uw,vw}⊆E(G)}]\documentclass[12pt]{minimal}
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\begin{document}$$G[\{uv: \{uw, vw\}\subseteq E(G)\}]$$\end{document} is connected for every vertex w of G, respectively. In this paper, we prove the following.Every 2-connected P7\documentclass[12pt]{minimal}
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\begin{document}$$P_7$$\end{document}-free graph of minimum degree at least three other than the Petersen graph has a spanning Eulerian subgraph. This implies that every H-free 3-connected graph (or connected N4\documentclass[12pt]{minimal}
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\begin{document}$$N^4$$\end{document}-locally connected graph of minimum degree at least three) other than the Petersen graph is supereulerian if and only if H is an induced subgraph of P7\documentclass[12pt]{minimal}
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\begin{document}$$P_7$$\end{document}, where Pi\documentclass[12pt]{minimal}
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\begin{document}$$P_i$$\end{document} is a path of i vertices.Every 2-edge-connected H-free graph other than {K2,2k+1:kis a positive integer}\documentclass[12pt]{minimal}
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\begin{document}$$\{K_{2, 2k+1}:k ~\text {is a positive integer}\}$$\end{document} is supereulerian if and only if H is an induced subgraph of P4\documentclass[12pt]{minimal}
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\begin{document}$$P_4$$\end{document}.If every connected H-free N3\documentclass[12pt]{minimal}
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\begin{document}$$N^3$$\end{document}-locally connected graph other than the Petersen graph of minimum degree at least three is supereulerian, then H is an induced subgraph of P7\documentclass[12pt]{minimal}
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\begin{document}$$P_7$$\end{document} or T2,2,3\documentclass[12pt]{minimal}
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\begin{document}$$T_{2, 2, 3}$$\end{document}, i.e., the graph obtained by identifying exactly one end vertex of P3,P3,P4\documentclass[12pt]{minimal}
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\begin{document}$$P_3, P_3, P_4$$\end{document}, respectively.If every 3-connected H-free N2\documentclass[12pt]{minimal}
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\begin{document}$$N_2$$\end{document}-locally connected graph is hamiltonian, then H is an induced subgraph of K1,4\documentclass[12pt]{minimal}
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\begin{document}$$K_{1,4}$$\end{document}. We present an algorithm to find a collapsible subgraph of a graph with girth 4 whose idea is used to prove our first conclusion above. Finally, we propose that the reverse of the last two items would be true.
