Locally self-avoiding eulerian tours

Abstract

It was independently conjectured by H\"aggkvist in 1989 and Kriesell in 2011 that given a positive integer $\ell$, every simple eulerian graph with high minimum degree (depending on $\ell$) admits an eulerian tour such that every segment of length at most $\ell$ of the tour is a path. Bensmail, Harutyunyan, Le and Thomass\'e recently verified the conjecture for 4-edge-connected eulerian graphs. Building on that proof, we prove here the full statement of the conjecture. This implies a variant of the path case of Bar\'at-Thomassen conjecture that any simple eulerian graph with high minimum degree can be decomposed into paths of fixed length and possibly an additional shorter path.