Neighborhood Unions and Z 3-Connectivity in Graphs

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.