Group connectivity in line graphs

Tutte introduced the theory of nowhere zero flows and showed that a plane graph G has a face k-coloring if and only if G has a nowhere zero A-flow, for any Abelian group A with |A|≥k. In 1992, Jaeger et al. [9] extended nowhere zero flows to group connectivity of graphs: given an orientation D of a graph G, if for any b:V(G)↦A with ∑v∈V(G)b(v)=0, there always exists a map f:E(G)↦A−{0}, such that at each v∈V(G), ∑e=vw is directed from v to wf(e)−∑e=uv is directed from u to vf(e)=b(v) in A, then G is A-connected. Let Z3 denote the cyclic group of order 3. In [9], Jaeger et al. (1992) conjectured that every 5-edge-connected graph is Z3-connected. In this paper, we proved the following. (i)Every 5-edge-connected graph is Z3-connected if and only if every 5-edge-connected line graph is Z3-connected.(ii)Every 6-edge-connected triangular line graph is Z3-connected.(iii)Every 7-edge-connected triangular claw-free graph is Z3-connected. In particular, every 6-edge-connected triangular line graph and every 7-edge-connected triangular claw-free graph have a nowhere zero 3-flow.