Search

Abstract: An orientation of a graph is a -orientation if under this orientation, the net out-degree at every vertex is congruent to zero . If for any function satisfying , always has an orientation such that the net out-degree at every vertex is congruent to , then is strongly -connected. In this paper, we prove that a connected graph has a -orientation if and only if it is a contraction of a -regular bipartite graph. We also proved that every -edge-connected series–parallel graph is strongly -connected, and every simple -connected chordal graph is strongly -connected. [Copyright &y& Elsevier]

Abstract: Tutte introduced the theory of nowhere zero flows and showed that a plane graph has a face -coloring if and only if has a nowhere zero -flow, for any Abelian group with . In 1992, Jaeger et al. extended nowhere zero flows to group connectivity of graphs: given an orientation of a graph , if for any with , there always exists a map , such that at each , in , then is -connected. Let denote the cyclic group of order 3. In , Jaeger et al. (1992) conjectured that every 5-edge-connected graph is -connected. In this paper, we proved the following. [(i)] Every 5-edge-connected graph is -connected if and only if every 5-edge-connected line graph is -connected. [(ii)] Every 6-edge-connected triangular line graph is -connected. [(iii)] Every 7-edge-connected triangular claw-free graph is -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. [Copyright &y& Elsevier]