Geodesic
A note on nowhere-zero 3-flows and Z_3-connectivity
Ars Mathematica Contemporanea, Tome 10 (2016) no. 1, pp. 91-98.

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There are many major open problems in integer flow theory, such as Tutte's 3-flow conjecture that every 4-edge-connected graph admits a nowhere-zero 3-flow, Jaeger et al.'s conjecture that every 5-edge-connected graph is Z_3-connected and Kochol's conjecture that every bridgeless graph with at most three 3-edge-cuts admits a nowhere-zero 3-flow (an equivalent version of 3-flow conjecture). Thomassen proved that every 8-edge-connected graph is Z_3-connected and therefore admits a nowhere-zero 3-flow. Furthermore, Lovasz, Thomassen, Wu and Zhang improved Thomassen's result to 6-edge-connected graphs. In this paper, we prove that: (1) Every 4-edge-connected graph with at most seven 5-edge-cuts admits a nowhere-zero 3-flow. (2) Every bridgeless graph containing no 5-edge-cuts but at most three 3-edge-cuts admits a nowhere-zero 3-flow. (3) Every 5-edge-connected graph with at most five 5-edge-cuts is Z_3-connected. Our main theorems are partial results to Tutte's 3-flow conjecture, Kochol's conjecture and Jaeger et al.'s conjecture, respectively.
DOI : 10.26493/1855-3974.676.1a9
Keywords: Integer flow, nowhere-zero 3-flow, Z_3-connected, modulo 3-orientation, edge-cuts. 
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Fuyuan Chen; Bo Ning. A note on nowhere-zero 3-flows and Z_3-connectivity. Ars Mathematica Contemporanea, Tome 10 (2016) no. 1, pp. 91-98. doi : 10.26493/1855-3974.676.1a9. http://geodesic.mathdoc.fr/articles/10.26493/1855-3974.676.1a9/

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