Flow-critical graphs

TL;DR

The paper tackles the problem of flow-critical graphs by extending Lovász et al.'s nowhere-zero $3$-flow result to configurations where a distinguished vertex $z$ may have unbounded degree, provided there exists a second high-degree vertex $x$ with no small cut separating $x$ from $z$. It introduces a canvas framework for $ ext{Z}_3$-bordered graphs, along with the notions of tameness and flow-criticality, and develops a robust inductive approach built on a carefully designed partial order to rule out minimal counterexamples. The authors prove tall tame easels are not critical and formulate generation theorems for flow-critical canvases and $(k,r)$-tall easels, yielding an algorithmic scheme to enumerate all relevant structures with bounded parameters in polynomial time (excluding the time of feasibility tests). They derive density bounds for flow-critical graphs, notably a bound of $|E(G)| leq 3|V(G)|-5$ under certain degree configurations, advancing toward Li et al.'s conjecture on the density of flow-critical graphs and contributing to the broader pursuit of the $3$-flow conjecture. Overall, the work provides a structural and algorithmic framework that connects flow-criticality, decade-old conjectures, and density considerations, with potential implications for understanding when three-edge decompositions guarantee nowhere-zero $3$-flows.

Abstract

Lovász et al. proved that every $6$-edge-connected graph has a nowhere-zero $3$-flow. In fact, they proved a more technical statement which says that there exists a nowhere zero $3$-flow that extends the flow prescribed on the incident edges of a single vertex $z$ with bounded degree. We extend this theorem of Lovász et al. to allow $z$ to have arbitrary degree, but with the additional assumption that there is another vertex $x$ with large degree and no small cut separating $x$ and $z$. Using this theorem, we prove two results regarding the generation of minimal graphs with the property that prescribing the edges incident to a vertex with specific flow does not extend to a nowhere-zero $3$-flow. We use this to further strengthen the theorem of Lovász et al., as well as make progress on a conjecture of Li et al.

Figures (11)

• Figure 1: On the left, we have a flow-critical graph $G$ where $z$ has boundary zero and the vertices where $z$ has in arcs to have boundary $2$, and the remaining have boundary $1$. On the right, we have the three tip-respecting contractions and a nowhere-zero $3$-flow that does not extend to $G$.
• Figure 2: A flow-critical graph $(G,\beta)$ where $G-z$ is not connected. Here $\beta(v) =0$ for all $v \in V(G)$. Two tip-respecting contractions which do not extend to $G$ are shown.
• Figure 3: The four possible cases in Lemma \ref{['lemma-sepx']} for how we choose the edges $e_{1}$ and $e_{2}$. Note that in cases $1$ and $2$, we must have all of the edges from $z$ to $C$ oriented in the same direction, whereas in cases $3$ and $4$, either there is no edge from $z$ to $A$, or not all the edges go the same direction. Hence, these figures are merely examples of the four possible cases.
• Figure 4: The three cases in Lemma \ref{['lemma-no4']}. Note in case (i), the edge $e_{2}$ may be incident to $z$, and in case (iii) there may be parallel edges.
• Figure 5: The situation at the start of Step 4. All vertices $v \in \mathfrak{G} - \{z,x\}$ have $\tau(v) >0$, the vertex $z$ has degree at most $\deg(x) +1$, and moreover $z$ does not have an arc to $x$. The final reduction is to take any arc incident to $z$, reverse it, and observe that the resulting easel contradicts our choice of counterexample.

Theorems & Definitions (150)

• Conjecture 1.1: $3$-flow-conjecture, tutte1954contribution
• Theorem 1.2: Kochol KOCHOL
• Theorem 1.3: L. M. Lovász et al. ltwz
• Conjecture 1.3: Li et al. li20223
• Conjecture 1.4: Li et al. li20223
• Theorem 1.5
• Definition 1.6
• Definition 1.7
• Definition 1.8
• Definition 1.9