2-Connected Hamiltonian Claw-Free Graphs Involving Degree Sum of Adjacent Vertices

For a graph H, define σ̄2(H)=min{d(u)+d(v)|uvâE(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 σ̄2(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 σ̄2(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 σ̄2(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].