Z(3)-Connected Graphs With Neighborhood Unions And Minimum Degree Condition

Let G be a 2-edge-connected simple graph on n >= 15 vertices, and let A denote 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 G can be A-reduced to G*. In this paper, we prove that if for every uv is not an element of E(G), vertical bar N(u) boolean OR N (v)vertical bar + delta(G) >= n, 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. Our results extend the early theorem by Li et al. (Graphs and Combin., 29 (2013): 1891-1898).