Locally self-avoiding eulerian tours

TL;DR

This paper proves a conjecture that high minimum degree simple eulerian graphs have eulerian tours with locally short segments forming paths, extending previous results and impacting graph decomposition theories.

Contribution

It confirms the full conjecture on locally self-avoiding eulerian tours, building on prior partial results and advancing understanding of graph decompositions.

Findings

Proved the conjecture for all high minimum degree simple eulerian graphs.

Established a variant of the Barát-Thomassen conjecture related to path decompositions.

Extended the class of graphs known to admit 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.