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Conformal Decomposition of Integral Flows on Signed Graphs with Outer-Edges

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Abstract

A nonzero integral flow f is conformally decomposable if \(f=f_1+f_2\), where \(f_1,f_2\) are nonzero integral flows, both are nonnegative or both are nonpositive. Conformally indecomposable flows on ordinary graphs are simply graph circuit flows. However, on signed graphs without outer-edges (compact case), conformally indecomposable flows, classified by Chen and Wang (arXiv:1112.0642, 2013) and by Chen et al. (Discret Math 340:1271–1286, 2017), are signed-graph circuit flows plus an extra class of characteristic flows of so-called Eulerian circle-trees. This paper is to classify indecomposable conformal integral flows on signed graphs with outer-edges (non-compact case). A notable feature is that with outer-edges the treatment is natural and results become stronger but proofs are simpler than that without outer-edges.

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Notes

  1. Zaslavsky’s definition [19] of signed graph assumes loops without endpoints which play insignificant role.

  2. This is strongly suggested by Zaslavsky to avoid using cycle which has many other meanings. Tutte [15] used polygon instead. Some use cycle for even graph, i.e., all vertices have even degree.

  3. It is different from the open walk in Chen et al. [9], where outer-edges are not considered and open walk means \(v_0\ne v_n\).

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  1. Department of Mathematics, Hong Kong University of Science and Technology, Kowloon, Hong Kong, China

    Beifang Chen

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  1. Beifang Chen
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Correspondence to Beifang Chen.

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Research is supported by the RGC Competitive Earmarked Research Grants 16300614.

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Chen, B. Conformal Decomposition of Integral Flows on Signed Graphs with Outer-Edges. Graphs and Combinatorics 37, 2207–2225 (2021). https://doi.org/10.1007/s00373-021-02344-3

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  • DOI: https://doi.org/10.1007/s00373-021-02344-3

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