Modulo Flows and Integer Flows in Signed Graphs

ABSTRACT Modulo flow is a powerful tool in the study of flows in both ordinary graphs and signed graphs. For ordinary graphs, Tutte showed that a graph admits a nowhere‐zero ‐flow if and only if it admits a nowhere‐zero ‐flow. However, such equivalence does not hold any more for signed graphs. Mačajova and Škoviera [ SIAM Journal of Discrete Mathematics 31 (2017) 1937–1952] proved that every flow‐admissible signed graph with a nowhere‐zero ‐flow admits a nowhere‐zero 4‐flow. DeVos et al. [ Journal of Combinatorial Theory Series B 149 (2021) 198–221] proved that every flow‐admissible signed graph admits a nowhere‐zero 11‐flow by converting certain special nowhere‐zero ‐flows into integer flows, and as a key step, they showed that every signed graph with a nowhere‐zero ‐flow admits a nowhere‐zero 5‐flow. In this paper, we study how to convert ‐flows and ‐flows into integer flows by proving the following two results: (1) every flow‐admissible signed graph with a nowhere‐zero ‐flow admits a nowhere‐zero 8‐flow; (2) every bridgeless signed graph with a nowhere‐zero ‐flow admits a nowhere‐zero 7‐flow. Combining known results, it follows that every flow‐admissible signed graph with a nowhere‐zero ‐flow admits a nowhere‐zero ‐flow for each integer .