Locally self-avoiding Eulerian tours.

Abstract It was independently conjectured by Häggkvist in 1989 and Kriesell in 2011 that given a positive integer ℓ , every simple Eulerian graph with high minimum degree (depending on ℓ) admits an Eulerian tour such that every segment of length at most ℓ of the tour is a path. Bensmail, Harutyunyan, Le and Thomassé 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 Barát–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. [ABSTRACT FROM AUTHOR]