Group connectivity of graphs with diameter at most 2

AbstractLet 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 D(G) such that for every function b:V(G)↦A satisfying ∑v∈V(G)b(v)=0, there is a function f:E(G)↦A∗ such that at each vertex v∈V(G), the amount of f values on the edges directed out from v minus the amount of f values on the edges directed into v equals b(v). In this paper, we investigate, for a 2-edge-connected graph G with diameter at most 2, the group connectivity number Λg(G)=min{n:G is A-connected for everyabelian group A with |A|≥n}, and show that any such graph G satisfies Λg(G)≤6. Furthermore, we show that if G is such a 2-edge-connected diameter 2 graph, then Λg(G)=6 if and only if G is the 5-cycle; and when G is not the 5-cycle, then Λg(G)=5 if and only if G is the Petersen graph or G belongs to two infinite families of well characterized graphs