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Math-1563 · mathematics · Combinatorics

One-sided lifting and polarized minimal counterexamples for Sullivan-3

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Let $D$ be any nonempty finite simple directed graph without loops or digons, where a digon is a pair $u\to v$, $v\to u$. Write $d^+(v)$ and $d^-(v)$ for the outdegree and indegree of a vertex $v$. Let $N_2^+(v)$ be the set of vertices $w$ for which there is a directed path $v\to u\to w$, but no arc $v\to w$, with $w\ne v$. Equivalently, the shortest directed distance from $v$ to $w$ is exactly two. Several paths with endpoint $w$ contribute only one member to this set.

The question is whether there must exist a vertex $v$ satisfying \[ |N_2^+(v)|+d^+(v)\ge2\min\{d^-(v),d^+(v)\}. \] If $d^-(v)\ge d^+(v)$, this is the inequality $|N_2^+(v)|\ge d^+(v)$. If $d^-(v)\le d^+(v)$, it is instead $|N_2^+(v)|\ge2d^-(v)-d^+(v)$. The graph need only have one vertex satisfying the appropriate inequality for its own degrees. These two cases explain the role of the minimum without imposing a global relation between in- and outdegrees.

No assumptions on the underlying undirected graph, beyond simplicity, are made. A source or sink already satisfies the desired inequality. The statement is Conjecture 6.7 of the source, independently of its motivating implications from stronger rainbow or second-neighbourhood conjectures. Its target is this particular mixed-degree bound.

Short English statement

Is there always a vertex whose first two outgoing layers have size at least twice the smaller of its indegree and outdegree?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-1563-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-1563

Seymour second neighborhoods: At the Seymour vertex |N2+|>=d+, so |N2+|+d+>=2d+>=2 min(d-,d+), proving the Sullivan-3 statement from a stronger local inequality.

The claimed result implies the stated question and also supplies the additional strength described above.

One-sided lifting and polarized minimal counterexamples for Sullivan-3

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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