Strongly Spanning Trailable Graphs with Small Circumference and Hamilton-Connected Claw-Free Graphs

A graph G is strongly spanning trailable if for any e₁= u₁v₁, e₂= u₂v₂∈ E(G) (possibly e₁= e₂), G(e₁, e₂) , which is obtained from G by replacing e₁ by a path u1ve1v1 and by replacing e₂ by a path u2ve2v2, has a spanning (ve1,ve2)-trail. A graph G is Hamilton-connected if there is a spanning path between any two vertices of V(G). In this paper, we first show that every 2-connected 3-edge-connected graph with circumference at most 8 is strongly spanning trailable with an exception of order 8. As applications, we prove that every 3-connected { K_{1 , 3}, N_{1 , 2 , 4}} -free graph is Hamilton-connected and every 3-connected { K_{1 , 3}, P₁₀} -free graph is Hamilton-connected with a well-defined exception. The last two results extend the results in Hu and Zhang (Graphs Comb 32: 685–705, 2016) and Bian et al. (Graphs Comb 30: 1099–1122, 2014) respectively.