Group connectivity of graphs: a nonhomogeneous analogue of nowhere-zero flow properties
Journal of Combinatorial Theory Series B
Discrete Mathematics
Nowhere-zero 3-flows of highly connected graphs
Discrete Mathematics
Circular chromatic number: a survey
Discrete Mathematics
An equivalent version of the 3-flow conjecture
Journal of Combinatorial Theory Series B
Proceedings of the Fifth Colloquium on Automata, Languages and Programming
Combinatorica
Clustering, community partition and disjoint spanning trees
ACM Transactions on Algorithms (TALG)
On (k, d)-colorings and fractional nowhere-zero flows
Journal of Graph Theory
Circular flow numbers of regular multigraphs
Journal of Graph Theory
Circular flows of nearly Eulerian graphs and vertex-splitting
Journal of Graph Theory
Construction of graphs with given circular flow numbers
Journal of Graph Theory
Claw-decompositions and tutte-orientations
Journal of Graph Theory
Graph Theory
Nowhere-zero 3-flows and modulo k-orientations
Journal of Combinatorial Theory Series B
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The odd-edge-connectivity of a graph G is the size of the smallest odd edge cut of G. Tutte conjectured that every odd-5-edge-connected graph admits a nowhere-zero 3-flow. As a weak version of this famous conjecture, Jaeger conjectured that there is an integer k such that every k-edge-connected graph admits a nowhere-zero 3-flow. Jaeger [F. Jaeger, Flows and generalized coloring theorems in graphs, J. Combin. Theory Ser. B 26 (1979) 205-216] proved that every 4-edge-connected graph admits a nowhere-zero 4-flow. Galluccio and Goddyn [A. Galluccio, L.A. Goddyn, The circular flow number of a 6-edge-connected graph is less than four, Combinatorica 22 (2002) 455-459] proved that the flow index of every 6-edge-connected graph is strictly less than 4. This result is further strengthened in this paper that the flow index of every odd-7-edge-connected graph is strictly less than 4. The second main result in this paper solves an open problem that every odd-(2k+1)-edge-connected graph contains k edge-disjoint parity subgraphs. The third main theorem of this paper proves that if the odd-edge-connectivity of a graph G is at least4@?log"2|V(G)|@?+1, then G admits a nowhere-zero 3-flow. This result is a partial result to the weak 3-flow conjecture by Jaeger and improves an earlier result by Lai et al. The fourth main result of this paper proves that every odd-(4t+1)-edge-connected graph G has a circular(2t+1)even subgraph double cover. This result generalizes an earlier result of Jaeger.