Locally self-avoiding Eulerian tours

摘要

It was independently conjectured by Haggkvist in 1989 and Kriesell in 2011 that given a positive integer l, 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 Thomasse 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 Barest-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. (C) 2018 Elsevier Inc. All rights reserved.