The Frank number and nowhere-zero flows on graphs

An edge $e$ of a graph $G$ is called deletable for some orientation $o$ if the restriction of $o$ to $G-e$ is a strong orientation. Inspired by a problem of Frank, in 2021 H\"orsch and Szigeti proposed a new parameter for $3$-edge-connected graphs, called the Frank number, which refines $k$-edge-connectivity. The Frank number is defined as the minimum number of orientations of $G$ for which every edge of $G$ is deletable in at least one of them. They showed that every $3$-edge-connected graph has Frank number at most $7$ and that in case these graphs are also $3$-edge-colourable the parameter is at most $3$. Here we strengthen both results by showing that every $3$-edge-connected graph has Frank number at most $4$ and that every graph which is $3$-edge-connected and $3$-edge-colourable has Frank number $2$. The latter also confirms a conjecture by Bar\'at and Bl\'azsik. Furthermore, we prove two sufficient conditions for cubic graphs to have Frank number $2$ and use them in an algorithm to computationally show that the Petersen graph is the only cyclically $4$-edge-connected cubic graph up to $36$ vertices having Frank number greater than $2$.