Flows and parity subgraphs of graphs with large odd-edge-connectivity

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 least 4⌈log2|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.