科研之友
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摘要
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 partial derivative f : V(G) -> A given by partial derivative f(v) =Sigma(e is an element of E+(v)) f(e) - Sigma(e is an element of 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 Sigma(v is an element of V(G)) b(v) = 0, there is a function f : E(G) -> A - {0} such that partial derivative f = b. In this paper, we prove that if d(x) + d(y) >= n for each xy is an element of 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,K-n-2,K- K-3,K-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,K-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.
