Extendability of Contractible Configurations for Nowhere-Zero Flows and Modulo Orientations |
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Authors: | Yanting Liang Hong-Jian Lai Rong Luo Rui Xu |
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Institution: | 1.Department of Mathematics,University of Wisconsin-Fond du Lac,Fond Du Lac,USA;2.Department of Mathematics,West Virginia University,Morgantown,USA;3.Department of Mathematics,University of West Georgia,Carrollton,USA |
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Abstract: | Let H be a connected graph and G be a supergraph of H. It is trivial that for any k-flow (D, f) of G, the restriction of (D, f) on the edge subset E(G / H) is a k-flow of the contracted graph G / H. However, the other direction of the question is neither trivial nor straightforward at all: for any k-flow \((D',f')\) of the contracted graph G / H, whether or not the supergraph G admits a k-flow (D, f) that is consistent with \((D',f')\) in the edge subset E(G / H). In this paper, we will investigate contractible configurations and their extendability for integer flows, group flows, and modulo orientations. We show that no integer flow contractible graphs are extension consistent while some group flow contractible graphs are also extension consistent. We also show that every modulo \((2k+1)\)-orientation contractible configuration is also extension consistent and there are no modulo (2k)-orientation contractible graphs. |
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