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Degree sum condition for \(Z_{3}\)-connectivity in graphs. (English) Zbl 1210.05028
Summary: 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 \(f: E(G)\to A\) is the function \(\partial f: V(G)\to A\) given by \[ \partial f(v)= \sum_{e\in E^+(v)} f(e)- \sum_{e\in E^-(v)} f(e), \] 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 \(b: V(G)\to A\) with \(\sum_{v\in V(G)} b(v)= 0\), there is a function \(f: E(G)\to A-\{0\}\) such that \(\partial f= b\).
In this paper, we prove that if \(d(x)+ d(y)\geq n\) for each \(xy\in E(G)\), 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 \(n\geq 6\), 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 [Discrete Math. 308, No. 24, 6233–6240 (2008; Zbl 1167.05309)] by G. Fan and C. Zhou.

MSC:
05C07 Vertex degrees
05C40 Connectivity
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