The prime maximal Z15 boundary counterexample

Definitions and mathematical statement

Let $G$ be a finite simple loopless digraph on $n$ vertices. Say it is $3$-free if it has no directed cycle of length at most three, and arc-maximal $3$-free if adjoining any missing ordered arc between distinct vertices creates such a cycle. A directed module is a set $M\subseteq V(G)$ such that every vertex outside $M$ has the same incoming and outgoing adjacency relation to each member of $M$. A nontrivial module has $2\leq|M|\leq n-1$; the source calls these homogeneous sets and excludes substitution constructions.

Thomassé's Conjecture 6.36 as printed states that every arc-maximal $3$-free $G$ with no nontrivial module and \[ \delta^+(G)\geq4n/15 \] is isomorphic, for some integer $k\geq1$, to the directed Cayley graph on ${\mathbb{Z}}/(3k+1){\mathbb{Z}}$ with generators $1,\ldots,k$. Thus every vertex points to the next $k$ vertices in cyclic order.

The source itself notes the boundary graph \[ G_0=\operatorname{Cay}({\mathbb{Z}}/15{\mathbb{Z}},\{1,2,4,8\}),\qquad x\to x+s\quad(s\in\{1,2,4,8\}). \] The attached aid reports a full counterexample certificate: $G_0$ has outdegree and indegree $4$, is $3$-free, is globally arc-maximal $3$-free, and has no nontrivial directed module. Its degree is exactly $4\cdot15/15$, while $15$ is not $3k+1$ for any integer $k$. Hence it refutes the non-strict formulation under those supplied certificate claims. The aid describes sumset identities for maximality and a three-orbit forced-signature closure table for primeness; the table is not included in the record.

The proposed strict repair replaces the hypothesis by $\delta^+(G)>4n/15$ and keeps every other condition and conclusion. It excludes this boundary example. Because outdegrees are integers, the printed and strict hypotheses differ exactly when $15$ divides $n$: \[ \delta^+(G)\geq\lceil4n/15\rceil \quad\text{versus}\quad \delta^+(G)\geq\lfloor4n/15\rfloor+1. \] The aid says the strict version remains open. That is a supplied assessment and not a resolution or independent literature verification of the repaired conjecture.

Short English statement

The printed structural conjecture includes its own degree-boundary counterexample on fifteen vertices, according to the supplied maximality and primeness certificate. Replacing the non-strict degree threshold by a strict one removes that example and gives a different conjecture.

Sources

• Blair D. Sullivan's 2006 AIM report on The Caccetta--Häggkvist conjecture, page 11, Conjecture 6.36 attributed to Thomassé and its ${\mathbb{Z}}/15{\mathbb{Z}}$ note: https://aimath.org/WWN/caccetta/caccetta.pdf.
• Local registry data/unsolvedmath/problems.json, permanent ID 20001081, exact source code catalogue entry; canonical location aim-combinatorics-notes.json notes[205].
• Attached attempt 1, counterexample, problem status at run refuted, rigor proved, low novelty confidence. Its primeness certificate and strict-repair status have no separate complete citation; the aid is machine-generated, not peer reviewed, requiring independent verification before citation.