On weights of induced paths and cycles in claw-free andK1,r-free graphs

Abstract

Let G be a K1,r -free graph (r ≥ 3) on n vertices. We prove that, for any induced path or induced cycle on k vertices in G (k ≥ 2r − 1 or k ≥ 2r, respectively), the degree sum of its vertices is at most (2r − 2)(n − α) where α is the independence number of G. As a corollary we obtain an upper bound on the length of a longest induced path and a longest induced cycle in a K1,r -free graph. Stronger bounds are given in the special case of claw-free graphs (i.e., r = 3). Sharpness examples are also presented. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 131–143, 2001