A Note on Mod \((2p + 1)\)-Orientable Graphs

Fix a positive integer \(k\). A mod \(k\)-orientation of a graph \(G\) is an assignment of edge directions to \(E(G)\) such that at each vertex \(v \in V(G)\), the number of edges directed in is congruent to the number of edges directed out
modulo \(k\). The main purpose of this note is to correct an error in [JCMCC, 9 (1991), 201-207] by showing that a connected graph \(G\) has a mod \((2p + 1)\)-orientation for any \(p \geq 1\) if and only if \(G\) is Eulerian.