Hamiltonicity and Degrees of Adjacent Vertices in Claw‐Free Graphs

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 Ryjác?ek'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:

• (i) For , if and if and n is sufficiently large, then H is Hamiltonian.
• (ii) 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.