Avoidable Paths In Graphs

We prove a recent conjecture of Beisegel et al. that for every positive integer k, every graph containing an induced P-k also contains an avoidable P-k. Avoidability generalises the notion of simpliciality best known in the context of chordal graphs. The conjecture was only established for k is an element of {1, 2} (Ohtsuki et al. 1976, and Beisegel et al. 2019, respectively). Our result also implies a result of Chvatal et al. 2002, which assumed cycle restrictions. We provide a constructive and elementary proof, relying on a single trick regarding the induction hypothesis. In the line of previous works, we discuss conditions for multiple avoidable paths to exist.