Seymour's r-graph conjecture

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.

Let $r\ge1$. A finite loopless multigraph $G$ is an $r$-graph if it is $r$-regular and $|\delta_G(X)|\ge r$ for every odd-cardinality vertex subset $X$, where $\delta_G(X)$ consists of edges with one endpoint in $X$. The chromatic index $\chi'(G)$ is the minimum number of edge colors with distinct colors on edges having a common endpoint.

Statement recorded in the supplied source.

Does every $r$-graph satisfy $\chi'(G)\le r+1$? [S2]

Short English statement

Can every regular multigraph satisfying the odd-cut condition be edge-colored with one extra color?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3203 (OPG-2226). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Seymour's r-graph conjecture, original node 2226, statement and mathematical discussion. https://www.openproblemgarden.org/op/seymours_r_graph_conjecture.