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, a graph orientation may be fixed, with 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 (2026a) 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.

A three-flow-critical graph is a connected bridgeless graph without a nowhere-zero integer 3-flow, in which every edge contraction produces a graph with such a flow. Jafari (2026a) proved that a three-flow-critical graph on vertices has at most edges, and at most if . Jafari (2026a) reports ChatGPT assistance in developing these density arguments.

For fixed odd , let be a random regular graph sampled uniformly from all -regular simple graphs on labeled vertices. Jafari (2026b) proved that there exists such that with probability tending to 1 as tends to infinity through even integers. In particular, the strict upper bounds for and are and , respectively.