Hamiltonicity and restricted degree conditions on induced subgraphs in claw-free graphs, II

A wounded graph W is a graph obtained from a K-3 by adding a pendant edge at one vertex of the K-3 and adding a path of length 2 at another vertex of the K-3 . A net N is a graph obtained from a K-3 by adding a pendant edge at each vertex of the K-3 . A P6 is a path on 6 vertices. C?ada et al. (2016) conjectured that for a 2-connected claw-free graph H and for a fixed graph S is an element of {W, N, P-6}, if the degree at each end-vertex of every induced copy of S is at least (| V (H)| - 2)/3, then H is Hamiltonian. The case for S = N was conjectured by Broersma in [2] (1993). The conjecture for S is an element of {N, P-6} was proved in [9]. In this paper, we prove the conjecture for the last case S = W. A new and shorter proof for the case S = N is also included. (C) 2021 Elsevier B.V. All rights reserved.