Weighted Modulo Orientations of Graphs and Signed Graphs

Given a graph $G$ and an odd prime $p$, for mapping $f: E(G) \to {\mathbb Z}_p\setminus\{0\}$ ${\mathbb Z}_p$-boundary $b$ of $G$, orientation $\tau$ is called $(f,b;p)$-orientation if the net out $f$-flow same as $b(v)$ in Z}_p$ at each vertex $v\in V(G)$ under $D$. This concept was introduced by Esperet et al. (2018), generalizing mod $p$-orientations 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, framework such orientations extended signed through additive bases. We also study problem some (signed) families including complete graphs, chordal series-parallel bipartite indicating much lower edge-connectivity bound still guarantees existence those families.