Leírás
In this talk, I would like to present some new results on the recently introduced notion: the \emph{Frank number} of an undirected $3$-edge-connected graph.
In an orientation $O$ of the undirected graph $G$, an arc $e$ is called \emph{deletable} if and only if $O-e$ is strongly connected.
For a $3$-edge-connected graph $G$, the \emph{Frank number} is the minimum $k$ for which $G$ admits $k$ strongly connected orientations such that for every edge $e$ of $G$ the corresponding arc is deletable in at least one of the $k$ orientations. Hörsch and Szigeti conjectured that the Frank number is at most $3$ for every $3$-edge-connected graph $G$.
We present some examples and an infinite family of graphs with Frank number equal to 3. With respect to the upper bound, I would like to give a summary of improvements from the last two years. This is a joint work with János Barát.