# Avoidable paths in graphs

## TL;DR

This paper proves a conjecture that in any graph with an induced path of length k, an avoidable path of the same length also exists, extending the concept of simpliciality in graph theory.

## Contribution

It provides a constructive, elementary proof of the conjecture for all positive integers k, generalizing previous results and discussing conditions for multiple avoidable paths.

## Key findings

- Confirmed the conjecture for all k using a new proof technique.
- Extended the concept of simpliciality to avoidable paths.
- Discussed conditions for the existence of multiple avoidable paths.

## Abstract

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 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.