Group Connectivity of Certain Graphs

Abstract

Let \(G\) be an undirected graph, \(A\) be an (additive) Abelian group and \(A^* = A – \{0\}\). A graph \(G\) is \(A\)-connected if \(G\) has an orientation such that for every function \(b: V(G) \longmapsto A\) satisfying \(\sum_{v\in V(G)} b(v) = 0\), there is a function \(f: E(G) \longmapsto A^*\) such that at each vertex \(v\in V(G)\) the net flow out of \(v\) equals \(b(v)\). We investigate the group connectivity number \(\Lambda_g(G) = \min\{n; G \text{ is } A\text{-connected for every Abelian group with } |A| \geq n\}\) for complete bipartite graphs, chordal graphs, and biwheels.