The Flow Index of Regular Class I Graphs

For integers k and d with k ≥ 2 d > 0, a circular k/d -flow of a graph G is an orientation together with a mapping from E ( G ) to {± d, ± ( d + 1) , . . . , ± ( k − d ) } such that, for each vertex of G , the sum of images on outgoing edges is equal to the sum of images on incoming edges. Related to the Four Color Problem, a classical result of Tutte shows that a cubic graph admits a circular 4 / 1-flow if and only if it is Class I (i.e., 3-edge-colorable). Tutte’s 3-flow conjecture implies that every 5-regular Class I graph admits a nowhere-zero 3-flow (equivalently, a circular 6 / 2-flow) as a special case. Steffen in 2015 conjectured that every (2 t + 1)-regular Class I graph admits a circular (2 t + 2) /t -flow. He also proposed a more general conjecture that every (2 t + 1)-odd-edge-connected (2 t + 1)-regular graph admits a circular (2 t + 2) /t -flow for any integer t ≥ 2, which includes the Circular Flow Conjecture of Jaeger(1981) stating that every 2 t -edge-connected graph admits a circular (2 t + 2) /t -flow for any even t ≥ 2. Jaeger’s conjecture was disproved in 2018 for all even t ≥ 6, and based on these results, Mattiolo and Steffen recently constructed counterexamples to Steffen’s conjecture for Class I graphs when t = 4 k + 2 for any integer k ≥ 1. -edge-connected (2 t +1)-regular Class I graphs without circular (2 t +2) /t -flows for any integer t ∈ { 6 , 8 , 10 } or t ≥ 12. Our result provides more general counterexamples to Steffen’s two conjectures for both even and odd t , and simultaneously generalizes the counterexamples of Jaeger’s Circular Flow Conjecture to regular Class I graphs.