Reciprocal-weight cycles and a winding-number obstruction

Definitions and mathematical statement

For a finite digraph $G$ with $n\geq1$ vertices and positive arc weights $f(e)$, define reciprocal cycle cost $\operatorname{cost}_f(C)=\sum_{e\in C}1/f(e)$. Zhang's proposed implication in the source is that, for a positive parameter $k$, \[ \sum_{e\in E^+(v)}f(e)\geq n/k\quad\text{for every }v \quad\Longrightarrow\quad \exists\text{ directed cycle }C:\operatorname{cost}_f(C)\leq k. \] The printed weight range includes zero, for which $1/f(e)$ is undefined without a convention. The positive-integer subcase already suffices for the supplied refutation; zeros are not silently assigned a reciprocal.

Charbit's source example has vertex set ${\mathbb{Z}}/8{\mathbb{Z}}$, arcs $x\to x+1$ of weight $2$ and $x\to x+2$ of weight $1$, and $k=3$. The outgoing-weight sum is $2+1=3$, not the printed $4$, and still satisfies $3\geq8/3$. The supplied aid records that every simple directed cycle has reciprocal cost $4$, so the claimed conclusion fails.

More generally the aid gives, for each integer $k\geq2$, the simple digraph on ${\mathbb{Z}}/[2(k+1)]{\mathbb{Z}}$ with the same two generators and weights. Every vertex has outgoing-weight sum $3\geq2(k+1)/k$, while every simple directed cycle has cost $k+1>k$. Its broader winding statement is: if $S\subseteq\{1,\ldots,n-1\}$ and a positive integer $a$ is divisible by every $s\in S$, assign weight $a/s$ to the $s$-arcs of the directed Cayley graph on ${\mathbb{Z}}/n{\mathbb{Z}}$. Every nonempty closed directed walk then has cost $qn/a$, where its positive generator sum is $qn$ for an integer $q\geq1$.

For the two-generator graph on ${\mathbb{Z}}/2t{\mathbb{Z}}$, $t\geq2$, with weights $2$ and $1$, the aid reports every simple directed cycle has cost exactly $t$; the failure interval for a real positive threshold parameter is precisely $2t/3\leq k<t$. This distinguishes simple cycles from repeated closed walks, whose winding cost can be a larger multiple. Finally, it records an affirmative restricted case when every vertex has exactly one outgoing arc: under the displayed outgoing-weight condition that functional digraph contains a cycle of reciprocal cost at most $k$. These are supplied counterexample/generalization claims, not a repaired universal conjecture.

Short English statement

The weighted cycle proposal is false even with positive integer weights. Charbit's cyclic two-generator example has outgoing sum three, correcting a source arithmetic error, but all simple cycles cost four. The aid extends this obstruction to every integer threshold at least two.

Sources

• AIM summary The Caccetta--Häggkvist conjecture, Conjecture 6.24 attributed to Zhang, followed by Charbit's counterexample: https://aimath.org/WWN/caccetta/caccetta.pdf. No separate complete references for these attributions are supplied.
• Local registry data/unsolvedmath/problems.json, permanent ID 20001069, exact source code catalogue entry; canonical location aim-combinatorics-notes.json notes[193].
• Attached attempt 1, counterexample, problem status at run refuted, rigor proved, low novelty confidence. Its winding theorem and functional-digraph case are machine-generated research aid, not peer reviewed, requiring independent verification before citation. The source verification reports a primary-PDF check; this draft did not consult that PDF.