Circular flow numbers of $r$-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 $X$. An $s$-graph is an $s$-regular graph satisfying $|\delta_G(X)|\ge s$ whenever $|X|$ is odd.

Statement recorded in the supplied source.

For every integer $t>1$ and every $(2t+1)$-graph $G$, is \[F_c(G)\le2+\frac2t?\] [S2]

Short English statement

Does the odd-cut condition force the proposed circular-flow bound for every odd regular graph?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3136 (OPG-59997). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Circular flow numbers of $r$-graphs, original node 59997, statement and mathematical discussion. https://www.openproblemgarden.org/op/circular_flow_numbers_of_r_graphs.