TY - JOUR

T1 - Flows and parity subgraphs of graphs with large odd-edge-connectivity

AU - Shu, Jinlong

AU - Zhang, Cun Quan

AU - Zhang, Taoye

N1 - Funding Information:
E-mail address: cqzhang@math.wvu.edu (C.-Q. Zhang). 1 Supported in part by National Natural Science Foundation of China (Nos. 10671074 and 60673048), NSF of Shanghai (No. 05ZR14046) and by Open Research Funding Program of LGISEM. 2 Supported in part by the National Security Agency under Grants H98230-05-1-0080, H98230-12-1-0233 and by a WV-RCG grant.

PY - 2012/7

Y1 - 2012/7

N2 - 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⌈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.

AB - 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⌈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.

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U2 - 10.1016/j.jctb.2012.03.002

DO - 10.1016/j.jctb.2012.03.002

M3 - Article

AN - SCOPUS:84862828777

SN - 0095-8956

VL - 102

SP - 839

EP - 851

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

IS - 4

ER -