The Chvátal–Erdös Condition for Group Connectivity in Graphs

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 ?(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.