Degree sum condition for -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 ) → A with ∑ 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 x y ∈ E ( G ) , then G is not Z 3 -connected if and only if G is either one of 15 specific graphs or one of K 2 , n − 2 , K 3 , n − 3 , K 2 , n − 2 + or K 3 , n − 3 + for n ≥ 6 , where K r , s + denotes the graph obtained from K r , 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.