Supereulerian graphs with small circumference and 3-connected hamiltonian claw-free graphs

A graph GG is supereulerian if it has a spanning eulerian subgraph. We prove that every 3-edge-connected graph with the circumference at most 11 has a spanning eulerian subgraph if and only if it is not contractible to the Petersen graph. As applications, we determine collections F1F1, F2F2 and F3F3 of graphs to prove each of the following(i) Every 3-connected {K1,3,Z9}{K1,3,Z9}-free graph is hamiltonian if and only if its closure is not a line graph L(G)L(G) for some G∈F1G∈F1.(ii) Every 3-connected {K1,3,P12}{K1,3,P12}-free graph is hamiltonian if and only if its closure is not a line graph L(G)L(G) for some G∈F2G∈F2.(iii) Every 3-connected {K1,3,P13}{K1,3,P13}-free graph is hamiltonian if and only if its closure is not a line graph L(G)L(G) for some G∈F3G∈F3.