Optimally Balanced Orientation of Graphs

Every graph has orientation $\delta$ with the property that the indegree and outdegree of each vertex differ by no more than a unity. For a subset A of vertices of a digraph D the indegree of A is the number of arcs pointing into A and the outdegree of A is the number of arcs pointing out of A. The flux at A is the difference of the two (‘in’ minus ‘out’.) For a fixed graph G consider the set $\triangle$ of all orientations of G. We calculate “worstcase” flux as the “min-max” flux: the maximum flux over all subsets of vertices and the minimum over all orientations. The min-max flux over A with respect to orientation $\delta$ is the “flux” of the graph $\phi_{\delta}(A)$ where\begin{equation*}\min_{\delta\in\delta A}\max_{\subset V}\phi(A;\delta). \tag{1}\end{equation*}An orientation $\delta$ of G achieving the min-max is said to be optimally-balanced. In this paper we characterize optimally-balanced graphs.