we-solve (beta)

Math-2090 · mathematics · Combinatorics

"Seymour's Second Neighborhood Conjecture" Any oriented graph has a vertex whose outdegree is at most its second outdegr

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

An oriented graph is a finite directed graph with no loops, no parallel arcs, and no pair of opposite arcs. For a vertex $v$, put \[ N_1^+(v)=\{u:v\to u\}, \] and let \[ N_2^+(v)=\{w\notin N_1^+(v)\cup\{v\}: v\to u\to w\text{ for some }u\}. \] The latter is the set of vertices at directed distance exactly two from $v$. Multiple paths to the same vertex count only once.

Seymour's second neighborhood conjecture asks whether every nonempty oriented graph has a vertex $v$ satisfying \[ |N_2^+(v)|\ge |N_1^+(v)|. \] A sink meets this inequality because both sets are empty, so a potential counterexample necessarily has positive outdegree at every vertex. No assumption of strong connectivity is made.

Excluding first neighbors from the second neighborhood matters: having a two-step walk to an existing first neighbor does not create a new vertex at distance two. Nor does the question compare the number of walks of lengths one and two. The orientation hypothesis rules out directed digons, for which the assertion in this form can fail. A proof must find at least one suitable vertex; the inequality is not asserted at every vertex of the graph.

Short English statement

Must a nonempty oriented graph contain a vertex with at least as many distinct second outneighbors as first outneighbors?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2090-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2090

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.

"Seymour's Second Neighborhood Conjecture" Any oriented graph has a vertex whose outdegree is at most its second outdegr

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

Go to contribution →