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 = v w  is directed from  v  to  w f ( e ) − ∑ e = u v  is directed from  u  to  v f ( e ) = b ( v ) in A , then G is A -connected. Let Z 3 denote the cyclic group of order 3. In [9], Jaeger et al. (1992) conjectured that every 5-edge-connected graph is Z 3 -connected. In this paper, we proved the following. (i) 5-edge-connected graph is Z 3 -connected if and only if every 5-edge-connected line graph is Z 3 -connected. 6-edge-connected triangular line graph is Z 3 -connected. 7-edge-connected triangular claw-free graph is Z 3 -connected. rticular, every 6-edge-connected triangular line graph and every 7-edge-connected triangular claw-free graph have a nowhere zero 3-flow.