Thomasse's path-girth conjecture: refutation and exact cyclic blow-ups

Definitions and mathematical statement

For a finite digraph $D$, let $\ell(D)$ be the maximum number of arcs in a simple directed path, let $\delta^+(D)$ be minimum outdegree, and let $g(D)$ be directed girth in arcs. In the finite-girth setting the repaired source Conjecture 6.34, attributed to Thomassé, asks \[ \ell(D)\ge\delta^+(D)(g(D)-1). \] The detached degree notation is a product, not an exponent. The original notes that the $g=3$ case is open and that the conjecture implies Caccetta--Häggkvist. For acyclic graphs the source does not specify a convention for $g=\infty$ and the product with possible zero minimum outdegree.

The attached aid reports the conjecture refuted for every $g\ge4$, while retaining the $g=3$ oriented-graph case as open. In particular it claims a simple oriented family $X_d$, for every integer $d\ge3$, with \[ g(X_d)=4,\quad\delta^+(X_d)=d,\quad\ell(X_d)=2d+2<3d. \] Its vertex and arc construction is not reproduced in this summary. The refutation of this stronger path conjecture does not refute Caccetta--Häggkvist itself.

The aid separately gives an exact positive class. Let $m\ge3$ and let $B(a_0,\ldots,a_{m-1})$ have nonempty independent layers $L_i$ of sizes $a_i$, with every arc from $L_i$ to $L_{i+1\bmod m}$ and no other arcs. Put $s=\min_i a_i$ and let $R$ be the largest cyclic consecutive run of layers with size at least $s+1$, using $R=0$ if all sizes equal $s$. Because some layer has minimum size, $0\le R\le m-1$. The aid's formula is \[ \ell(B)=sm+R-1,\qquad g(B)=m,\qquad\delta^+(B)=s. \] Consequently $\ell(B)-\delta^+(B)(g(B)-1)=s+R-1\ge0$, with equality exactly when all $a_i=1$, the bare directed cycle. Thus nonuniform cyclic blow-ups satisfy the original target and are not the refuting family.

Short English statement

The supplied aid reports Thomassé's path-girth conjecture false from girth four onward while preserving the open girth-three case. It also gives an exact longest-path formula for cyclic blow-ups, a family that satisfies the target rather than providing counterexamples.

Sources

• Local UnsolvedMath registry, permanent ID $20001079$, source code catalogue entry; canonical aim-combinatorics-notes.json, notes[203].
• American Institute of Mathematics, "The Caccetta--Häggkvist conjecture" workshop record, Conjecture 6.34 attributed to Thomassé: https://aimath.org/WWN/caccetta/caccetta.pdf. The supplied verification identifies the corrected statement on page 11 after damaged PDF extraction; no PDF inspection is claimed here.
• Attached research attempt 1: counterexample, recorded run status refuted, rigor proved, novelty confidence low; registry status solved, classification SOLVED-IN-LITERATURE. It supplies no full citation for the girth-at-least-four refutations or construction of $X_d$. The machine-generated aid is not peer reviewed; independently verify before citation.