Abstract: In 2-connected Hamiltonian claw-free graphs involving degree sum of adjacent vertices [Discuss. Math. Graph Theory 40(2020) 85-106 [24]], Tian and Xiong proved that 2-edge connected triangle-free graph of order at most 7 is either supereulerian or one of G₁. Furthermore, in Edge degree conditions for dominating and spanning closed trails
[Discuss. Math. Graph Theory 44(2024) 363-381 [23]], Tian, Broersma and Xiong proved that 2-edge connected graph of order at most 8 is either supereulerian or one of G₁∪G₂. Motivated by the advance above, we considered 2-edge connected graph G of order at most 9, and show that G is either supereulerian or one of G₁∪G₂∪G₃. Although only one vertex is added, forbidden graphs are more complex and harder to derived.

Key words: Hamilton cycle, Supereulerian graph, Spanning closed trail, Dominating closed trail