Weighted Modulo Orientations of Graphs and Signed Graphs
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Published:2022-12-16
Issue:4
Volume:29
Page:
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ISSN:1077-8926
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Container-title:The Electronic Journal of Combinatorics
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language:
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Short-container-title:Electron. J. Combin.
Author:
Liu Jianbing,Han Miaomiao,Lai Hong-Jian
Abstract
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.
Publisher
The Electronic Journal of Combinatorics
Subject
Computational Theory and Mathematics,Geometry and Topology,Theoretical Computer Science,Applied Mathematics,Discrete Mathematics and Combinatorics