A note on nowhere-zero 3-flows and Zsub3-connectivity - Chen, Fuyuan (Ars mathematica contemporanea, 2016)
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 J. Comb. Theory, Ser. B 56, No. 2, 165--182 (1992) conjecture that every 5-edge-connected graph is ?$Z_3$?-connected and Kochol's J. Comb. Theory, Ser. B 83, No. 2, 258--261 (2001) 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). C. Thomassen J. Comb. Theory, Ser. B 102, No. 2, 521--529 (2012) proved that every 8-edge-connected graph is ?$Z_3$?-connected and therefore admits a nowhere-zero 3-flow. Furthermore, Lovász et al. J. Comb. Theory, Ser. B 103, No. 5, 587--598 (2013) 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.
Keywords: celoštevilski tok; nikjer-ničeln 3-tok; orientacija po modulu 3; povezavni rezi