Circular Flow Number

The circular flow number of a graph is the infimum of the real numbers for which has a circular -flow. Such a flow consists of a graph orientation and a real value in on each graph edge, with equal incoming and outgoing sums at every graph vertex (Jafari 2026). Equivalently, one may fix an graph orientation and allow signed values whose absolute magnitudes lie in . Set if no such flow exists.

This refines the integer-valued nowhere-zero flow parameter by permitting noninteger values of . Jafari (2026) proved

for every finite -edge-connected multigraph and integer . The connectivity threshold is optimal for this strict inequality. In particular, every six-edge-connected graph has circular flow number strictly below 3.