Tutte's 4-Flow Conjecture for Petersen-Minor-Free Graphs

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Graphs are finite and undirected, with loops and parallel edges permitted. An edge is a bridge if deleting it increases the number of connected components. A graph minor is obtained by deleting vertices or edges and contracting non-loop edges. Define the Petersen graph $P$ using vertices $a_i,b_i$ for $i\in\mathbb Z/5\mathbb Z$ and edges $a_ia_{i+1}$, $a_ib_i$, and $b_ib_{i+2}$. A graph is Petersen-minor-free if it has no minor isomorphic to $P$. A nowhere-zero $4$-flow consists of an orientation of the edges and an integer assignment $f:E(G)\to\{-3,-2,-1,1,2,3\}$ satisfying, at every vertex $v$, \[ \sum_{e\text{ leaving }v}f(e)=\sum_{e\text{ entering }v}f(e). \] A loop contributes once to each sum; the empty assignment is a flow on an edgeless graph.

Statement recorded in the supplied source.

Does every finite bridgeless Petersen-minor-free graph admit a nowhere-zero $4$-flow? No restriction to cubic graphs, connected graphs, or exclusion of a proper minor of the Petersen graph is imposed. [S2]

Short English statement

Can every graph with no bridge and no Petersen graph minor carry a nonzero integer flow of absolute value less than four on every edge?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1444 (GRAPH-057). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 József Pintér, Nowhere-Zero 4-Flows in Graphs Excluding a Proper Minor of the Petersen Graph, arXiv:2607.22267v2 (2026), abstract and Sections 1--2. https://arxiv.org/pdf/2607.22267v2.