Hamiltonicity and Degrees of Adjacent Vertices in Claw‐Free Graphs
Zhi‐Hong Chen · Journal of Graph Theory · 2017
Abstract For a graph H, let for every edge . For and , let be a set of k‐edge‐connected K3‐free graphs of order at most r and without spanning closed trails. We show that for given and ε, if H is a k‐connected claw‐free graph of order n with and , and if n is sufficiently large, then either H is Hamiltonian or the Ryjác̆ek's closure where G is an essentially k‐edge‐connected K3‐free graph that can be contracted to a graph in . As applications, we prove: For , if and if and n is sufficiently large, then H is Hamiltonian. For , if and n is sufficiently large, then H is Hamiltonian. These bounds are sharp. Furthermore, since the graphs in are fixed for given p and can be determined in a constant time, any improvement to (i) or (ii) by increasing the value of p and so enlarging the number of exceptions can be obtained computationally.