Degree sum condition for Z3-connectivity in graphs

Let G be a 2-edge-connected simple graph on n vertices, let A denote an abelian group with the identity element 0, and let D be an orientation of G. The boundary of a function f : E(G) → A is the function ∂ f : V (G) → A given by ∂ f (v) = ∑ e∈E+(v) f (e) − ∑ e∈E−(v) f (e), where E(v) is the set of edges with tail v and E(v) is the set of edges with head v. A graph G is A-connected if for every b : V (G) → Awith ∑ v∈V (G) b(v) = 0, there is a function f : E(G) → A − {0} such that ∂ f = b. In this paper, we prove that if d(x) + d(y) ≥ n for each xy ∈ E(G), then G is not Z3-connected if and only if G is either one of 15 specific graphs or one of K2,n−2, K3,n−3, K 2,n−2 or K + 3,n−3 for n ≥ 6, where K + r,s denotes the graph obtained from Kr,s by adding an edge joining two vertices of maximum degree. This result generalizes the result in [G. Fan, C. Zhou, Degree sum and Nowhere-zero 3-flows, Discrete Math. 308 (2008) 6233–6240] by Fan and Zhou. © 2010 Elsevier B.V. All rights reserved.