Local average outdegree and short directed cycles

Definitions and mathematical statement

Let $D=(V,E)$ be a finite simple loopless digraph on $n$ vertices, with at least one outgoing arc at every vertex. Opposite arcs are allowed. Write $d^+(v)$ for the number of arcs leaving $v$. The directed girth $g(D)$ is the length of its shortest directed cycle; such a cycle exists because following outgoing arcs in a finite graph must eventually revisit a vertex. For a positive integer $r$, assume the local degree-sum condition \[ d^+(u)+d^+(v)\ge2r \qquad\text{for every arc }u\to v. \] Shen's conjecture asks whether these assumptions imply \[ g(D)\le\left\lceil\frac nr\right\rceil. \] The degree sum is imposed on endpoints of every directed edge, not on all pairs of vertices and not merely on average over the graph. A vertex may individually have outdegree smaller than $r$, provided its incident outgoing-edge endpoint sums meet the requirement. Consequently this is a local average-degree extension of the usual minimum-outdegree short-cycle question.

The assertion is uniform over $n,r$ and all such digraphs; $r$ is not the number of vertices or an average degree computed afterwards. The minimum-outdegree-one assumption is retained independently of the degree-sum condition, excluding sinks that the latter might not constrain. Degrees count neighbours in a simple digraph; parallel arcs would artificially enlarge them without reducing girth.

Short English statement

Does a lower bound on the sum of the endpoint outdegrees of each arc force the same short-cycle bound as a minimum-outdegree condition?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20001071 (AIM-COMBINATORICS-0196), AIM Workshop Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 American Institute of Mathematics, The Caccetta-Haggkvist conjecture, item 6. Original workshop question underlying AIM-COMBINATORICS-0196. https://aimath.org/WWN/caccetta/caccetta.pdf.