Antiparallel d-stable traces and a stronger version of ore problem.

In 2013 a novel self-assembly strategy for polypeptide nanostructure design which could lead to significant developments in biotechnology was presented in Gradišar et al. (Nat Chem Bio 9:362-366, 2013). It was since observed that a polyhedron P can be realized by interlocking pairs of polypeptide chains if its corresponding graph G(P) admits a strong trace. It was since also demonstrated that a similar strategy can also be expanded to self-assembly of designed DNA (Kočar, Nat commun 7:1-8, 2016). In this direction, in the present paper we characterize graphs which admit closed walk which traverses every edge exactly once in each direction and for every vertex v, there is no subset N of its neighbors, with [Formula: see text], such that every time the walk enters v from N, it also exits to a vertex in N. This extends Thomassen's characterization (Thomassen, J Combin Theory Ser B 50:198-207, 1990) for the case [Formula: see text].