The Frank number and nowhere-zero flows on graphs

TL;DR

This work advances the understanding of the Frank number $fn(G)$ for $3$-edge-connected graphs by proving a sharp general upper bound of $fn(G)\le4$ and showing $fn(G)=2$ for graphs admitting a nowhere-zero $4$-flow (in particular, many $3$-edge-connected, $3$-edge-colourable graphs). Central to the approach is representing nowhere-zero $4$-flows as combinations of two flows, enabling deletability arguments that bound $fn$; the authors also derive two concrete sufficient conditions (Theorems 2OddCycles and 2Odd1Even) that guarantee $fn(G)=2$ for a broad class of cubic graphs. They further establish a reduction to cubic graphs and a local cubic modification technique to transfer lower bounds, and develop heuristic and exact algorithms to detect $fn=2$ in practice. Computational results indicate the Petersen graph is the only cyclically $4$-edge-connected cubic graph up to $36$ vertices with $fn>2$, and all tested strong snarks up to order $38$ have $fn=2$, supporting the conjectured landscape and providing practical tools for verification in snark families.

Abstract

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 Hörsch 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 Barát and Blázsik. 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 $2$.

Figures (3)

• Figure 1: A smooth orientation of the circuits in $F-\{x_1x_2\}\cup M$ and of those in $C$ such that each is consistent on the edges of $N_i$ at distance $1$ from $x_i$, for $i\in\{1,2\}$.
• Figure 2: A part of $G'$ and orientation $(G',o')$ as defined in Lemma \ref{['lemma:oeo']}. The left-hand-side corresponds with the orientation of (a) and the right-hand-side corresponds with the orientation of (b). If the conditions of Lemma \ref{['lemma:oeo']} are met the thick, blue edges will be deletable.
• Figure 3: A smooth orientation of the circuits in $F-\{x_1y_1, y_2x_2\}\cup M$ and of the circuits in $C$ such that each consistent on the edges of $N_i$ at distance $1$ from $x_i$, for $i\in\{1,2\}$, and on the edges of $W$ at distance $1$ from $y_i$ but not incident with $y_{3-i}$, for $i\in\{1,2\}$.

Theorems & Definitions (26)

• definition 1
• Theorem 1
• Theorem 2
• lemma 1
• proof
• proof : Proof of Theorem \ref{['thm:4flow']}
• corollary 1
• lemma 2
• proof
• corollary 2