2-connected Hamiltonian claw-free graphs involving degree sum of adjacent vertices
Tian Tian, Liming Xiong · Discussiones Mathematicae Graph Theory · 2018
For a graph H, define 2 (H) = min{d(u) + d(v)| uv E(H)}. Let H be a 2-connected claw-free simple graph of order n with (H) 3. In [J. Graph Theory 86 (2017) 193-212], Chen proved that if 2 (H) n 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 2 (H) 2n 5 -1, and characterize non-Hamiltonian 2-connected claw-free graphs H of order n sufficiently large with 2 (H) 2n 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].