In a previous work, we gave construction of (not necessarily realizable) oriented matroids from triangulation product two simplices. this follow-up paper, use variant Viro's patchworking to derive topological representation the matroid directly polyhedral structure triangulation, hence finding combinatorial manifestation besides tropical algebraic geometry. We achieve by rephrasing procedure as...

‎An  oriented perfect path double cover (OPPDC) of a‎ ‎graph $G$ is a collection of directed paths in the symmetric‎ ‎orientation $G_s$ of‎ ‎$G$ such that‎ ‎each arc‎ ‎of $G_s$ lies in exactly one of the paths and each‎ ‎vertex of $G$ appears just once as a beginning and just once as an‎ ‎end of a path‎. ‎Maxov{'a} and Ne{v{s}}et{v{r}}il (Discrete‎ ‎Math‎. ‎276 (2004) 287-294) conjectured that ...