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Neighborhood Unions and Z 3-Connectivity in Graphs

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Abstract

Let G be a 2-edge-connected simple graph on n ≥ 14 vertices, and let A be an abelian group with the identity element 0. If a graph G* is obtained by repeatedly contracting nontrivial A-connected subgraphs of G until no such a subgraph left, we say that G can be A-reduced to G*. In this paper, we prove that if for every \({uv\not\in E(G), |N(u) \cup N(v)| \geq \lceil \frac{2n}{3} \rceil}\), then G is not Z 3-connected if and only if G can be Z 3-reduced to one of \({\{C_3,K_4,K_4^-, L\}}\), where L is obtained from K 4 by adding a new vertex which is joined to two vertices of K 4.

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Authors and Affiliations

  1. Department of Mathematics, Huazhong Normal University, Wuhan, 430079, China

    Liangchen Li & Xiangwen Li

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  1. Liangchen Li
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  2. Xiangwen Li
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Correspondence to Liangchen Li.

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Supported by the Natural Science Foundation of China (11171129) and CCNU11A02015.

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Li, L., Li, X. Neighborhood Unions and Z 3-Connectivity in Graphs. Graphs and Combinatorics 29, 1891–1898 (2013). https://doi.org/10.1007/s00373-012-1240-5

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  • DOI: https://doi.org/10.1007/s00373-012-1240-5

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