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Math-1566 · mathematics · Combinatorics

Frontier Hall obstructions for exact distance shells

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

For a finite simple digraph $D$, define the outgoing distance $d_D(v,w)$ to be the minimum length of a directed path from $v$ to $w$, taking it to be infinity when no path exists and setting $d_D(v,v)=0$. For each positive integer $j$, its $j$th outgoing layer is \[ N_j^+(v)=\{w:d_D(v,w)=j\}. \] This is an exact-distance layer, so its members cannot have a shorter path from $v$ even if they also occur at the end of a $j$-edge walk. Different paths ending at the same vertex are not counted separately.

Fix an integer $k\ge2$. Suppose $D$ is nonempty and has no directed cycle of length at most $k$; in particular, it is loopless and has no digons. The proposed higher-neighbourhood conjecture asserts that some vertex $v$ satisfies \[ |N_k^+(v)|\ge|N_{k-1}^+(v)|. \] The assertion is required for every $k\ge2$ and every such finite digraph, with the vertex allowed to depend on both $D$ and $k$. Layers may be empty, and no minimum-degree assumption is made. For $k=2$ this is the second-neighbourhood conjecture. Larger $k$ compare two successive layers rather than a layer to the entire ball preceding it. The range $k\ge2$ also keeps both indices within the positive-layer notation used by the source.

Short English statement

In a digraph with no cycle of length at most k, must some vertex have a kth outgoing layer at least as large as its preceding layer?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-1566-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-1566

Seymour second neighborhoods: The theorem proves the exact-distance-shell assertion only for k=2; the record asks for every integer k>=2.

This page records partial support. Complete resolution of the catalogue question is not established by this record.

Frontier Hall obstructions for exact distance shells

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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