The exact interpolation among second-neighborhood inequalities

Definitions and mathematical statement

The exact permanent record consists only of the fragment

"Conjecture 6.1 (specifically 6.4), we might be led to ask if the following holds:"

The supplied verification identifies it as the second half of a transition sentence split across extracted records. It contains no conclusion after "the following," so no standalone conjecture can be recovered from this sentence alone. The aid classifies it as context_only and not_a_problem, despite registry open metadata.

The supplied surrounding-context analysis concerns a nonempty finite oriented graph: no loops or antiparallel arc pairs. For a vertex $v$, let $d^+(v)$ and $d^-(v)$ be outdegree and indegree, and let $d^{++}(v)$ count vertices at directed distance exactly two, excluding immediate outneighbors and $v$. In the aid's terminology the four pointwise predicates are \[ \begin{array}{ll} \text{Seymour:}&d^{++}\ge d^+,\\ \text{Sullivan--1 (AIM Compromise):}&d^{++}\ge d^-,\\ \text{Sullivan--2 (AIM 6.4):}&d^{++}+d^+\ge2d^-,\\ \text{Sullivan--3:}&d^{++}\ge\min\{d^+,2d^- -d^+\}. \end{array} \] These formulas are contextual interpretations supplied by the aid, not content printed in the fragment.

The reported Sullivan--3 slack is \[ s_3=d^{++}-d^-+|d^+-d^-|. \] When $d^+\le d^-$, it equals Seymour's slack $d^{++}-d^+$; when $d^-\le d^+$, it equals Sullivan--2's slack $d^{++}+d^+-2d^-$. At equal degrees these agree. Thus, at one fixed vertex, Sullivan--3 is exactly the disjunction of Seymour and Sullivan--2, and the reported implication structure is \[ (\text{Seymour and Sullivan--2}) \ \Longrightarrow\ \text{Sullivan--1} \ \Longrightarrow\ \text{Sullivan--3} = (\text{Seymour or Sullivan--2}). \] The aid says four small oriented examples separate all invalid additional pointwise implications, but their arc sets are not supplied. Pointwise implications must be distinguished from claims that a graph has some qualifying vertex, or universal conjectures that every oriented graph has one.

For an $n$-vertex, $m$-arc oriented graph with no Sullivan--3 vertex, the aid reports the necessary integer defect conditions \[ d^-(v)\ge d^{++}(v)+|d^+(v)-d^-(v)|+1\quad(\forall v), \] \[ \sum_v d^{++}(v)+\sum_v|d^+(v)-d^-(v)|\le m-n. \] These are obstructions to a putative counterexample, not a proof that one cannot exist and not a reconstructed missing theorem. The exact intended continuation of the source sentence still requires its adjacent source context.

Short English statement

This entry is an incomplete transition, not an independent conjecture. The supplied aid analyzes how several second-neighborhood inequalities relate and gives necessary conditions for a counterexample to the weakest one, without recovering the missing continuation or proving the general conjectures.

Sources

• Local UnsolvedMath registry, permanent ID $20001045$, source code catalogue entry; aim-combinatorics-notes.json, notes[169], tag conjecture.
• AIM workshop notes, "The Caccetta--Häggkvist conjecture," fragment referring to Conjectures 6.1 and 6.4: https://aimath.org/WWN/caccetta/caccetta.pdf. Statement status exact; the supplied verification identifies a split transition sentence. No PDF was accessed for this draft.
• Seymour, Sullivan--1/AIM Compromise, Sullivan--2/AIM 6.4, and Sullivan--3 are contextual labels supplied by the aid. No full authoritative external citations, original remarks, or literature field are supplied.
• Research attempt 1: context_only, problem status at run not_a_problem, rigor label proved, novelty confidence low; registry status open, classification OPEN-TRIAGE, suggested difficulty L2. Created and updated 14 August 2026. This machine-generated aid is not peer reviewed; independently verify before citation.
• Ulam AI / UnsolvedMath Contributors; CC BY 4.0.