Symmetric road interchanges

Abstract

A road interchange where $n$ roads meet and in which the drivers are not allowed to change lanes can be modelled as an embedding of a 2-coloured (hence bipartite) multigraph $G$ with equal-sized colour classes into an orientable surface such that there is a face bounded by a Hamiltonian cycle (Kurauskas, 2017). The case of $G$ a complete bipartite graph $K_{n,n}$ corresponds to a complete $n$-way interchange where drivers approaching from each of $n$ directions can exit to any other direction. The genus of the underlying surface can be interpreted as the number of bridges in the interchange. In this paper we study the minimum genus, or the minimum number of bridges, of a complete interchange with a restriction that it is symmetric under the cyclic permutation of its roads. We consider both (a) abstract combinatorial/topological symmetry, and (b) symmetry in the 3-dimensional Euclidean space $\mathbb{R}^3$. The proof of (a) is based on the classic voltage and transition graph constructions. For (b) we use, among other techniques, a simple new combinatorial lower bound.