Weighted modulo orientations of graphs and signed graphs (Q2112587)

Summary: Given a graph \(G\) and an odd prime \(p\), for a mapping \(f: E(G) \to \mathbb{Z}_p\setminus\{0\}\) and a \(\mathbb{Z}_p\)-boundary \(b\) of \(G\), an orientation \(\tau\) is called an \((f, b; p)\)-orientation if the net out \(f\)-flow is the same as \(b(v)\) in \(\mathbb{Z}_p\) at each vertex \(v\in V(G)\) under orientation \(D\). This concept was introduced by \textit{L. Esperet} et al. [SIAM J. Discrete Math. 32, No. 1, 534--542 (2018; Zbl 1380.05092)], generalizing mod \(p\)-orientations and closely related to Tutte's nowhere zero 3-flow conjecture. They proved that \((6p^2 - 14p + 8)\)-edge-connected graphs have all possible \((f, b; p)\)-orientations. In this paper, the framework of such orientations is extended to signed graph through additive bases. We also study the \((f, b; p)\)-orientation problem for some (signed) graphs families including complete graphs, chordal graphs, series-parallel graphs and bipartite graphs, indicating that much lower edge-connectivity bound still guarantees the existence of such orientations for those graph families.