2-connected Hamiltonian claw-free graphs involving degree sum of adjacent vertices

For a graph H, define σ¯2(H)=min{d(u)+d(v)|uv∈E(H)} {{\bar \sigma }_2} ( H ) = \min \left\{ {d ( u ) + d ( v )|uv \in E ( H )} \right\} . Let H be a 2-connected claw-free simple graph of order n with δ(H) ≥ 3. In [J. Graph Theory 86 (2017) 193–212], Chen proved that if σ¯2(H)≥n2−1 {{\bar \sigma }_2} ( H ) \ge {n \over 2} - 1 and n is sufficiently large, then H is Hamiltonian with two families of exceptions. In this paper, we refine the result. We focus on the condition σ¯2(H)≥2n5−1 {{\bar \sigma }_2} ( H ) \ge {{2n} \over 5} - 1 , and characterize non-Hamiltonian 2-connected claw-free graphs H of order n sufficiently large with σ¯2(H)≥2n5−1 {{\bar \sigma }_2} ( H ) \ge {{2n} \over 5} - 1 . As byproducts, we prove that there are exactly six graphs in the family of 2-edge-connected triangle-free graphs of order at most seven that have no spanning closed trail and give an improvement of a result of Veldman in [Discrete Math. 124 (1994) 229–239].