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Math-1557 · mathematics · Combinatorics

Exact trace certificates and a regular counterexample normal form for Caccetta--Haggkvist

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Let $D=(V,E)$ be a finite simple loopless directed graph on $n$ vertices. There is at most one arc $u\to v$ for an ordered pair of distinct vertices; the two opposite arcs between a pair are allowed. The outdegree $d^+(v)$ counts arcs leaving $v$, and $\delta^+(D)=\min_{v\in V}d^+(v)$. A directed cycle of length $\ell$ is a sequence of distinct vertices $v_0,\ldots,v_{\ell-1}$ with arcs $v_i\to v_{i+1}$, indices modulo $\ell$. In particular, opposite arcs form a cycle of length two.

The Caccetta--Häggkvist conjecture asserts that for every integer $r\ge1$, every such nonempty digraph with $\delta^+(D)\ge r$ has a directed cycle of length at most $\lceil n/r\rceil$. Equivalently, if its shortest directed cycle has length $g$, then $n\ge r(g-1)+1$. Positive minimum outdegree ensures that some cycle exists, by following successive outgoing arcs in a finite graph.

The source also gives a matrix formulation for the digon-free case. Let $A$ be a zero--one $n\times n$ matrix with diagonal entries one and $a_{ij}a_{ji}=0$ for $i\ne j$. If every row sum is at least $r+1$, the proposed conclusion is $\operatorname{tr}(A^{\lceil n/r\rceil})>n$. The diagonal ones allow a closed walk to be padded to that length; the surplus trace detects a nonconstant closed walk and hence a short cycle in the underlying loopless digraph.

Short English statement

Does minimum outdegree at least r force a directed cycle of length at most the ceiling of n divided by r?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-1557-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-1557

Seymour second neighborhoods: The directed-cycle corollary obtains a directed triangle under BOTH minimum indegree and outdegree at least n/3, adding an indegree hypothesis to the recorded outdegree-only assertion.

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

Exact trace certificates and a regular counterexample normal form for Caccetta--Haggkvist

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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