A lower bound for the complex flow number of a graph: A geometric approach

Let (Formula presented.) be a real number. A complex nowhere-zero (Formula presented.) -flow on a graph (Formula presented.) is an orientation of (Formula presented.) together with an assignment (Formula presented.) such that, for all (Formula presented.), the Euclidean norm of the complex number (Formula presented.) lies in the interval (Formula presented.) and, for every vertex, the incoming flow is equal to the outgoing flow. The complex flow number of a bridgeless graph (Formula presented.), denoted by (Formula presented.), is the minimum of the real numbers (Formula presented.) such that (Formula presented.) admits a complex nowhere-zero (Formula presented.) -flow. The exact computation of (Formula presented.) seems to be a hard task even for very small and symmetric graphs. In particular, the exact value of (Formula presented.) is known only for families of graphs where a lower bound can be trivially proved. Here, we use geometric and combinatorial arguments to give a nontrivial lower bound for (Formula presented.) in terms of the odd-girth of a cubic graph (Formula presented.) (i.e., the length of a shortest odd cycle) and we show that this lower bound is tight. This result relies on the exact computation of the complex flow number of the wheel graph (Formula presented.). In particular, we show that for every odd (Formula presented.), the value of (Formula presented.) arises from one of three suitable configurations of points in the complex plane according to the congruence of (Formula presented.) modulo 6.