Häggkvist–Kriesell conjecture on locally self-avoiding Eulerian tours

Let $G$ be an Eulerian graph, and let an $\ell$-step self-avoiding Eulerian tour of $G$ be one in which every segment of length at most $\ell$ is a path. Here, a segment of length $\ell$ is a walk $e_1e_2\ldots e_\ell$ whose consecutive edges occur consecutively in the tour. For every positive integer $\ell$, there is an integer $d_\ell$ such that every Eulerian graph $G$ with minimum degree at least $d_\ell$ admits an $\ell$-step self-avoiding Eulerian tour.

Häggkvist–Kriesell conjecture. For every positive integer $\ell$, there is an integer $d_\ell$ such that every Eulerian graph $G$ with minimum degree at least $d_\ell$ admits an $\ell$-step self-avoiding Eulerian tour.

The conjecture asks for a minimum-degree condition guaranteeing an Eulerian tour with no segment forming a cycle of length at most $\ell$. It is solved in this paper; earlier, the case $\ell=3$ was proved with the sharp bound $d_3=6$, and the result was known for $4$-edge-connected Eulerian graphs.