Nowhere-zero 3-flows and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" display="inline" overflow="scroll"><mml:msub><mml:mrow><mml:mi>Z</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub></mml:math>-connectivity of a family of graphs

Jin, Yan

doi:10.1016/j.disc.2011.05.011

Discrete Mathematics

2011

DOI: 10.1016/j.disc.2011.05.011

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Nowhere-zero 3-flows and Z3-connectivity of a family of graphs

Yan Jin

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Cited by 6 publications

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

2018

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Degree condition and Z3-connectivity

Lai

Shao

2012

Discrete Mathematics

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Degree sum of 3 independent vertices andZ3-connectivity

Yang

2013

Discrete Mathematics

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Nowhere-zero 3-flows and Z3-connectivity of a family of graphs

Cited by 6 publications

References 10 publications

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

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

Degree condition and Z3-connectivity

Degree sum of 3 independent vertices andZ3-connectivity

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