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 orientationD of a graph G, if for any b : V (G) → Awith  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 w f (e) −  e=uv is directed from u to v f (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. © 2011 Elsevier B.V. All rights reserved.