  A parallel $d$-stable trace is a closed walk which traverses every edge of a
graph exactly twice in the same direction and for every vertex $v$, there is no
subset $X \subseteq N(v)$ with $1 \leq |N| \leq d$ such that every time the
walk enters $v$ from $X$, it also exits to a vertex in $X$. In the past,
$d$-stable traces were investigated as a mathematical model for an innovative
biotechnological procedure -- self-assembling of polypeptide structures. Among
other, it was proven that graphs that admit parallel $d$-stable traces are
precisely Eulerian graphs with minimum degree strictly larger than $d$. In the
present paper we give an alternative, purely combinatorial proof of this
result.
