Abstract
Let G be a graph and A an abelian group with the identity element 0 and \({|A| \geq 4}\) . Let D be an orientation of G. The boundary of a function \({f: E(G) \rightarrow A}\) is the function \({\partial f: V(G) \rightarrow A}\) given by \({\partial f(v) = \sum_{e \in E^+(v)}f(e) - \sum_{e \in E^-(v)}f(e)}\) , where \({v \in V(G), E^+(v)}\) is the set of edges with tail at v and \({E^-(v)}\) is the set of edges with head at v. A graph G is A-connected if for every b: V(G) → A with \({\sum_{v \in V(G)} b(v) = 0}\) , there is a function \({f: E(G) \mapsto A-\{0\}}\) such that \({\partial f = b}\) . A graph G is A-reduced to G′ if G′ can be obtained from G by contracting A-connected subgraphs until no such subgraph left. Denote by \({\kappa^{\prime}(G)}\) and α(G) the edge connectivity and the independent number of G, respectively. In this paper, we prove that for a 2-edge-connected simple graph G, if \({\kappa^{\prime}(G) \geq \alpha(G)-1}\) , then G is A-connected or G can be A-reduced to one of the five specified graphs or G is one of the 13 specified graphs.
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X. Li is supported by the Natural Science Foundation of China (11171129).
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Zhang, X., Li, X. The Chvátal–Erdös Condition for Group Connectivity in Graphs. Graphs and Combinatorics 30, 769–781 (2014). https://doi.org/10.1007/s00373-013-1288-x
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DOI: https://doi.org/10.1007/s00373-013-1288-x