an:06065690 Zbl 1244.05108 Shu, Jinlong; Zhang, Cun-Quan; Zhang, Taoye Flows and parity subgraphs of graphs with large odd-edge-connectivity. EN J. Comb. Theory, Ser. B 102, No. 4, 839-851 (2012). ISSN 0095-8956 2012

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*05C21 05C40 integer flows; parity subgraph; odd-edge-connectivity Summary: 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 [{\it F. Jaeger}, ``Flows and generalized coloring theorems in graphs,'' J. Comb. Theory, Ser. B 26, 205--216 (1979; Zbl 0422.05028)] proved that every 4-edge-connected graph admits a nowhere-zero 4-flow. Galluccio and Goddyn [{\it A. Galluccio} and {\it L. A. Goddyn}, ``The circular flow number of a 6-edge-connected graph is less than four,'' Combinatorica 22, No. 3, 455--459 (2002; Zbl 1006.05049)] 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$\lceil \log_{2}|V(G)|\rceil +1$, 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 {\it H.-J. 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.