Avoidable paths in graphs
Abstract
We prove a recent conjecture of Beisegel et al. that for every positive integer k, every graph containing an induced Pk also contains an avoidable Pk. Avoidability generalises the notion of simpliciality best known in the context of chordal graphs. The conjecture was only established for k in 1,2 (Ohtsuki et al. 1976, and Beisegel et al. 2019, respectively). Our result also implies a result of Chv\'atal 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.
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