Weighted Modulo Orientations of Graphs and Signed Graphs

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 Esperet et al. (2018), 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.