Circular flow number of regular class 1 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 loopless and may have parallel edges. A nowhere-zero real $r$-flow consists of an orientation and real numbers $\phi(e)$ such that $1\le|\phi(e)|\le r-1$ and, at every vertex, the sum on outgoing edges equals the sum on incoming edges. The circular flow number $F_c(G)$ is the infimum of such real $r\ge2$, with value $+\infty$ if none exist. For $X\subseteq V(G)$, $\delta_G(X)$ is the set of edges with exactly one endpoint in $X$. A graph is class $1$ if its chromatic index equals its maximum degree.

Statement recorded in the supplied source.

For every integer $t\ge1$ and every $(2t+1)$-regular class-$1$ graph $G$, is $F_c(G)\le2+2/t$? [S2]

Short English statement

Does an optimal edge coloring of an odd regular graph force the proposed circular-flow bound?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3224 (OPG-59994). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Circular flow number of regular class 1 graphs, original node 59994, statement and mathematical discussion. https://www.openproblemgarden.org/op/circular_flow_number_of_regular_class_1_graphs.