Group connectivity of graphs with diameter at most 2

• Authors: Hong-Jian Lai;Xiangjuan Yao
• Venue: European Journal of Combinatorics
• Year: 2006

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 D(G) such that for every function b : V(G) ↦A satisfying Σv ∈ 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 Λ g(G) = min{n: G is A-connected for every abelian group A with |A| ≥ n}, and show that any such graph G satisfies Λ g (G) ≤ 6. Furthermore, we show that if G is such a 2-edge-connected diameter 2 graph, then Λ g(G) = 6 if and only if G is the 5-cycle; and when G is not the 5-cycle, then Λ g (G) = 5 if and only if G is the Petersen graph or G belongs to two infinite families of well characterized graphs.