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Math-1558 · mathematics · Combinatorics

A feedback-set bound for second neighborhoods

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

An oriented graph is a finite simple directed graph with no loops and no pair of opposite arcs. For a vertex $v$, its out-neighbourhood is $N^+(v)=\{u:v\to u\}$, and $d^+(v)=|N^+(v)|$. Its second out-neighbourhood is the set of vertices at directed distance exactly two: \[ N_2^+(v)=\{w\notin N^+(v)\cup\{v\}:\text{there exists }u \text{ with }v\to u\to w\}. \] Distance means the length of a shortest directed path. In particular, a vertex reached by both one and two arcs is excluded from $N_2^+(v)$, and different two-edge paths to the same endpoint count only once.

Seymour's second neighbourhood conjecture asserts that every nonempty oriented graph contains a vertex $v$ satisfying \[ |N_2^+(v)|\ge d^+(v). \] The vertex may depend on the entire graph; the claim does not require the inequality at every vertex or on average. No connectedness, regularity, or tournament hypothesis is imposed. A sink, which has outdegree zero, automatically satisfies the required inequality, so only graphs without sinks can present an obstruction. The condition excluding opposite arcs is essential to the stated problem: a pair joined in both directions can have a nonempty first out-neighbourhood and an empty second one. This formulation preserves the exact-distance convention used in the AIM source throughout.

Short English statement

Must every finite oriented graph have a vertex with at least as many second out-neighbours as immediate out-neighbours?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-1558-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-1558

Seymour second neighborhoods: The paper claims that every finite nonempty oriented graph has a vertex whose second out-neighborhood is at least as large as its first. The second neighborhood excludes vertices already reachable in one step. The parameters, regularity assumptions and normalization in this question are retained in applying that assertion.

With the hypotheses and conventions above, the claimed result addresses the whole mathematical assertion on this page.

A feedback-set bound for second neighborhoods

OpenAI's claim

Seymour second neighborhoods: The paper claims that every finite nonempty oriented graph has a vertex whose second out-neighborhood is at least as large as its first. The second neighborhood excludes vertices already reachable in one step.

Mathematical statement of the claimed result

Mathematical claim 1.
For a nonempty finite oriented graph $D$, with no loops or oppositely directed pair of edges, let $N_1^+(v)$ be the out-neighbors and $N_2^+(v)$ the vertices at directed distance exactly two. The claimed theorem gives \[\exists v\in V(D):\quad|N_1^+(v)|\le|N_2^+(v)|.\] For triangle-free $D$ a corollary yields a vertex with $d^+(v)\le|Z(v)|$ and $n-1\ge2d^+(v)+d^-(v)$, where $Z(v)$ consists of non-neighbors excluding $v$. Hence $\delta^+\ge n/3$ and $\delta^-\ge n/3$ force a directed triangle; a digraph regular in both directions with degree $r$ and girth four has $n\ge3r+1$. These consequences require the specified in-degree conditions and do not give the full out-degree-only Caccetta--Häggkvist assertion.

Outline of the argument

Seymour second neighborhoods: A smallest counterexample is pruned to yield strict neighborhood inequalities for all proper vertex sets. Replacing vertices by transitive tournaments strengthens these to a discrete concavity inequality. Extremal families of ordered pairs and an augmented bipartite matching are then controlled by two factorizations of one matrix, contradicting that inequality.

Formal verification

Family 173 describes Lean coverage for the exact second-out-neighborhood inequality for a suitable vertex of every finite nonempty oriented graph.

Seymour second neighborhoods: The manuscript links SeymourSecondNeighborhood.lean (declaration OAI.SeymourSecondNeighborhood.exists_goodVertex).

The documented selection includes this manuscript’s main assertion; that assertion retains the restrictions stated above.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources

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