Papers › The Frank number and nowhere-zero flows on graphs
The Frank number and nowhere-zero flows on graphs
Jan Goedgebeur, Edita Máčajová, Jarne Renders
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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 $2$.
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