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

Let $r\ge 2$ be a real number. A complex nowhere-zero $r$-flow on a graph $G$ is an orientation of $G$ together with an assignment $\phi :E\left(G\right)\to C$ such that, for all $e\in E\left(G\right)$, the Euclidean norm of the complex number $\phi \left(e\right)$ lies in the interval $\left[1,r-1\right]$ and, for every vertex, the incoming flow is equal to the outgoing flow. The complex flow number of a bridgeless graph $G$, denoted by ${\varphi }_C\left(G\right)$, is the minimum of the real numbers $r$ such that $G$ admits a complex nowhere-zero $r$-flow. The exact computation of ${\varphi }_C$ seems to be a hard task even for very small and symmetric graphs. In particular, the exact value of ${\varphi }_C$ 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 ${\varphi }_C\left(G\right)$ in terms of the odd-girth of a cubic graph $G$ (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 $W_n$. In particular, we show that for every odd $n$, the value of ${\varphi }_C\left(W_n\right)$ arises from one of three suitable configurations of points in the complex plane according to the congruence of $n$ modulo 6.