Group Connectivity of 3-Edge-Connected Chordal Graphs

Abstract.

 Let A be a finite abelian group and G be a digraph. The boundary of a function f: E(G)↦A is a function ∂f: V(G)↦A given by ∂f(v)=∑_{e leaving v} f(e)−∑_{e entering v} f(e). The graph G is A-connected if for every b: V(G)↦A with ∑_{v∈ V(G)} b(v)=0, there is a function f: E(G)↦A{0} such that ∂f=b. In [J. Combinatorial Theory, Ser. B 56 (1992) 165–182], Jaeger et al showed that every 3-edge-connected graph is A-connected, for every abelian group A with |A|≥6. It is conjectured that every 3-edge-connected graph is A-connected, for every abelian group A with |A|≥5; and that every 5-edge-connected graph is A-connected, for every abelian group A with |A|≥3.¶ In this note, we investigate the group connectivity of 3-edge-connected chordal graphs and characterize 3-edge-connected chordal graphs that are A-connected for every finite abelian group A with |A|≥3.