Degree sum condition for Z3Z3-connectivity in graphs

Let GG be a 2-edge-connected simple graph on nn vertices, let AA denote an abelian group with the identity element 0, and let DD be an orientation of GG. The boundary   of a function f:E(G)→Af:E(G)→A is the function ∂f:V(G)→A∂f:V(G)→A given by ∂f(v)=∑e∈E+(v)f(e)−∑e∈E−(v)f(e)∂f(v)=∑e∈E+(v)f(e)−∑e∈E−(v)f(e), where E+(v)E+(v) is the set of edges with tail vv and E−(v)E−(v) is the set of edges with head vv. A graph GG is AA-connected   if for every b:V(G)→Ab:V(G)→A with ∑v∈V(G)b(v)=0∑v∈V(G)b(v)=0, there is a function f:E(G)→A−{0}f:E(G)→A−{0} such that ∂f=b∂f=b. In this paper, we prove that if d(x)+d(y)≥nd(x)+d(y)≥n for each xy∈E(G)xy∈E(G), then GG is not Z3Z3-connected if and only if GG is either one of 1515 specific graphs or one of K2,n−2,K3,n−3,K2,n−2+ or K3,n−3+ for n≥6n≥6, where Kr,s+ denotes the graph obtained from Kr,sKr,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.