Eulerian Orientations and Circulations

Abstract

Let $G = ( V,E )$ be an (undirected) eulerian graph. An eulerian orientation$\vec G$ of G is a directed graph obtained by giving each edge of G an orientation in such a way that $\vec G$ contains a directed eulerian tour. If each edge $[ u,v ]$ of G has real costs $c( u,v)$ and $c ( v,u )$ associated with it, depending on which way it is oriented, then the cost of an eulerian orientation $\vec G$ is the sum of the costs of its arcs. We show how the problem of finding a minimum cost eulerian orientation can be transformed into a minimum cost circulation problem. We also describe direct algorithms for the minimum cost eulerian orientation problem which can be viewed as specializations of general network flow algorithms for the transformed problem. The orientation graph $\theta ( G )$ has a node for every eulerian orientation, and the nodes corresponding to $\vec G$ and $\tilde G$ are adjacent in $\theta ( G )$ if and only if $\vec G$ and $\tilde G$ differ only for the edges belonging to a simple cycle C of G. (Necessarily, the corresponding arcs will comprise directed cycles in $\vec G$ and $\tilde G$.) We show that $\theta ( G )$ is either isomorphic to a d-dimensional hypercube for some d or else is hamilton connected (i.e. each pair of nodes is joined by a hamiltonian path of $\theta ( G )$).