Dλ-cycles in λ-claw-free graphs Original Research Article

For a given integer λ ⩾ 1, we define a λ-claw as a graph H which is connected and contains a vertex w such that H-w is the union of three pairwise remote connected subgraphs with exactly λ vertices each. We define a graph to be λ-claw-free if it contains no induced subgraph isomorphic to a λ-claw. We show that if G is a k-connected (k integer, k ⩾ 2), λ-claw-free graph such that if for every two remote connected subgraphs S1, S2 of G of order λ we have d(S1) + d(S2) > n − (k + 1)λ − k, or if for every three pairwise remote, connected subgraphs S1, S2, S3 of G of order λ we have d(S1) + d(S2) + d(S3) > n − (k + 1)λ + 1, then G contains a cycle C such that every component of G − V(C) has at most λ − 1 vertices.