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

Abstract For a graph S , a vertex with degree one in S is an end-vertex of S and an edge is a pendant edge if one of its ends is an end-vertex. A net N is a graph obtained from a K 3 by adding a pendant edge at each vertex of the K 3 . A P 6 is a path on 6 vertices. In this paper, we solve two conjectures and prove that for 2-connected claw-free graphs H and for a fixed graph S ∈ { 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 (1993); the case for S = P 6 was conjectured by Cada et al. (2016).

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