Multiple weak 2-linkage and its applications on integer flows of signed graphs

Lu, You; Luo, Rong; Zhang, Cun-Quan

doi:10.1016/j.ejc.2017.09.002

Cited by 12 publications

(7 citation statements)

References 16 publications

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“…In the context of flows on signed graphs partial results for hard conjectures are obtained for the aforementioned classes of signed graphs [5,6,10,12].…”

Section: Definitions and Basic Resultsmentioning

confidence: 99%

“…Reed [9] proved that for every s ≥ 2 there exists an s-frustrated signed graph (G s , Σ s ) which does not contain any edge-disjoint negative circuits. By Proposition Signed graphs which do not contain two edge-disjoint negative circuits are characterized by Lu et al [6]. Let H be a contraction of a graph G and let x ∈ V (G).…”

Section: Non-decomposable Critical Signed Graphsmentioning

confidence: 99%

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Frustration-critical signed graphs

Cappello,

Steffen

2021

Preprint

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We study decomposition and subdivision of critical signed graphs. We characterize critical signed graphs and completely determine the set of t-critical graphs, for t ≤ 2 and the set S * of non-decomposable k-critical signed graphs which do not contain a decomposable t-critical subgraph for every t ≤ k. We prove that S * consists of cyclically 4-edge-connected, projective-planar cubic graphs. We further construct k-critical graphs of S * for every k ≥ 1.

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“…In the context of flows on signed graphs partial results for hard conjectures are obtained for the aforementioned classes of signed graphs [5,6,10,12].…”

Section: Definitions and Basic Resultsmentioning

confidence: 99%

Section: Non-decomposable Critical Signed Graphsmentioning

confidence: 99%

Frustration-critical signed graphs

Cappello,

Steffen

2021

Preprint

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“…By applying this lemma, Lu, Luo, and Zhang extended it to integer flows in the following lemma. Lemma 3.3 (Lu, Luo, and Zhang [8]). Let k be a positive integer, and let G be a graph with an orientation τ and admitting a k-NZF.…”

Section: Preliminariesmentioning

confidence: 99%

Six‐flows on almost balanced signed graphs

Wang

Zhang

et al. 2019

Journal of Graph Theory

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In 1983, Bouchet conjectured that every flow‐admissible signed graph admits a nowhere‐zero 6‐flow. By Seymour's 6‐flow theorem, Bouchet's conjecture holds for signed graphs with all edges positive. Recently, Rollová et al proved that every flow‐admissible signed cubic graph with two negative edges admits a nowhere‐zero 7‐flow, and admits a nowhere‐zero 6‐flow if its underlying graph either contains a bridge, or is 3‐edge‐colorable, or is critical. In this paper, we improve and extend these results, and confirm Bouchet's conjecture for signed graphs with frustration number at most two, where the frustration number of a signed graph is the smallest number of vertices whose deletion leaves a balanced signed graph.

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“…Lemma 3.2. (Lu, Luo and Zhang [8]) Let G be an unsigned graph with an orientation τ and assume that G admits a nowhere-zero k-flow. If a vertex u of G has degree at most 3 and γ :…”

Section: Integer Flows and Modulo Flows 31 Integer Flowsmentioning

confidence: 99%

“…Lu et al [8] also showed that every flow-admissible cubic signed graph without long barbells admits a nowhere-zero 6-flow. In Section 3 we will verify Bouchet's 6-flow conjecture for the family of signed graphs without long barbells.…”

Section: Introductionmentioning

confidence: 99%

Flows on signed graphs without long barbells

Luo

Schubert

et al. 2019

Preprint

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Many basic properties in Tutte's flow theory for unsigned graphs do not have their counterparts for signed graphs. However, signed graphs without long barbells in many ways behave like unsigned graphs from the point view of flows. In this paper, we study whether some basic properties in Tutte's flow theory remain valid for this family of signed graphs. Specifically let (G, σ) be a flow-admissible signed graph without long barbells. We show that it admits a nowhere-zero 6-flow and that it admits a nowhere-zero modulo k-flow if and only if it admits a nowhere-zero integer k-flow for each integer k ≥ 3 and k = 4. We also show that each nowhere-zero positive integer k-flow of (G, σ) can be expressed as the sum of some 2-flows. For general graphs, we show that every nowhere-zero p q -flow can be normalized in such a way, that each flow value is a multiple of 1 2q . As a consequence we prove the equality of the integer flow number and the ceiling of the circular flow number for flow-admissible signed graphs without long barbells.

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Multiple weak 2-linkage and its applications on integer flows of signed graphs

Cited by 12 publications

References 16 publications

Frustration-critical signed graphs

Frustration-critical signed graphs

Six‐flows on almost balanced signed graphs

Flows on signed graphs without long barbells

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