Full Orientability of the Square of a Cycle

Abstract

Let \(D\) be an acyclic orientation of a simple graph \(G\). An arc of \(D\) is called dependent if its reversal creates a directed cycle. Let \(d(D)\) denote the number of dependent arcs in \(D\). Define \(d_{\min}(G)\) (\(d_{\max}(G)\)) to be the minimum (maximum) number of \(d(D)\) over all acyclic orientations \(D\) of \(G\). We call \(G\) fully orientable if \(G\) has an acyclic orientation with exactly \(k\) dependent arcs for every \(k\) satisfying \(d_{\min}(G) \leq k \leq d_{\max}(G)\). In this paper, we prove that the square of a cycle \(C_n\) is fully orientable except for \(n = 6\).