Nowhere-zero 15-flow in 3-edge-connected bidirected graphs

Wei, Er Ling; Tang, Wen; Ye, Dong

doi:10.1007/s10114-014-2028-8

Cited by 3 publications

(2 citation statements)

References 9 publications

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“…Very recently, the following was shown by Cheng et al [15] for 2-edge-connected signed graphs: The flow number of 3-edge-connected flow-admissible signed graphs was bounded by 25 in [68], which was improved to 15 in [67], and to 9 in [71]. A further improvement, based on the ideas of the proof of the Weak 3-Flow Conjecture [33,60] was obtained by Wu et al [69]: Theorem 8 (Wu, Ye, Zang and Zhang).…”

Section: Higher Edge-connectivitymentioning

confidence: 99%

Nowhere-zero flows in signed graphs: A survey

Kaiser¹,

Rollová²,

Lukoťka³

2016

Preprint

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Section: Higher Edge-connectivitymentioning

confidence: 99%

Nowhere-zero flows in signed graphs: A survey

Kaiser¹,

Rollová²,

Lukoťka³

2016

Preprint

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“…Zero-sum flows are interesting to study, because of their connections to Bouchet's 6-flow conjecture (every bidirected graph that has a nowhere-zero bidirected flow admits a nowhere-zero bidirected 6-flow [7]) and Tutte's 5-flow conjecture (every bridgeless graph has a nowhere-zero 5flow [47]). For more information about these two conjectures, see [28,45,54,53,56,58]. Recently, it was shown that it is NP-complete to determine whether a given graph G has a zero-sum 3-flow [17].…”

Section: Introductionmentioning

confidence: 99%

Not-All-Equal and 1-in-Degree Decompositions: Algorithmic Complexity and Applications

2018

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A Not-All-Equal (NAE) decomposition of a graph G is a decomposition of the vertices of G into two parts such that each vertex in G has at least one neighbor in each part. Also, a 1-in-Degree decomposition of a graph G is a decomposition of the vertices of G into two parts A and B such that each vertex in the graph G has exactly one neighbor in part A. Among our results, we show that for a given graph G, if G does not have any cycle of length congruent to 2 mod 4, then there is a polynomial time algorithm to decide whether G has a 1-in-Degree decomposition. In sharp contrast, we prove that for every r, r ≥ 3, for a given r-regular bipartite graph G determining whether G has a 1-in-Degree decomposition is NP-complete. These complexity results have been especially useful in proving NP-completeness of various graph related problems for restricted classes of graphs. In consequence of these results we show that for a given bipartite 3-regular graph G determining whether there is a vector in the null-space of the 0,1-adjacency matrix of G such that its entries belong to {±1, ±2} is NP-complete. Among other results, we introduce a new version of Planar 1-in-3 SAT and we prove that this version is also NP-complete. In consequence of this result, we show that for a given planar (3, 4)-semiregular graph G determining whether there is a vector in the null-space of the 0,1-incidence matrix of G such that its entries belong to {±1, ±2} is NP-complete.

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Nowhere-Zero Flows on Signed Wheels and Signed Fans

2016

Bull. Malays. Math. Sci. Soc.

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Nowhere-zero 15-flow in 3-edge-connected bidirected graphs

Cited by 3 publications

References 9 publications

Nowhere-zero flows in signed graphs: A survey

Nowhere-zero flows in signed graphs: A survey

Not-All-Equal and 1-in-Degree Decompositions: Algorithmic Complexity and Applications

Nowhere-Zero Flows on Signed Wheels and Signed Fans

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