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.
@article{10_37236_10740,
author = {Jianbing Liu and Miaomiao Han and Hong-Jian Lai},
title = {Weighted modulo orientations of graphs and signed graphs},
journal = {The electronic journal of combinatorics},
year = {2022},
volume = {29},
number = {4},
doi = {10.37236/10740},
zbl = {1505.05058},
url = {http://geodesic.mathdoc.fr/articles/10.37236/10740/}
}
TY - JOUR
AU - Jianbing Liu
AU - Miaomiao Han
AU - Hong-Jian Lai
TI - Weighted modulo orientations of graphs and signed graphs
JO - The electronic journal of combinatorics
PY - 2022
VL - 29
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/10740/
DO - 10.37236/10740
ID - 10_37236_10740
ER -
%0 Journal Article
%A Jianbing Liu
%A Miaomiao Han
%A Hong-Jian Lai
%T Weighted modulo orientations of graphs and signed graphs
%J The electronic journal of combinatorics
%D 2022
%V 29
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/10740/
%R 10.37236/10740
%F 10_37236_10740
Jianbing Liu; Miaomiao Han; Hong-Jian Lai. Weighted modulo orientations of graphs and signed graphs. The electronic journal of combinatorics, Tome 29 (2022) no. 4. doi: 10.37236/10740
