Evolving Local Corrections for Global Constructions in Combinatorics

Gergely Bérczi

Home/Archive/arXiv

arXiv · 2603.06692

Evolving Local Corrections for Global Constructions in Combinatorics

Gergely Bérczi

View original source ↗Read source document ↗Export metadata ↓

Abstract

Many open problems in combinatorics admit reformulations in which a global construction can be achieved by the repeated application of small, finite correcting steps. This paper presents three computational case studies of this principle, carried out using AlphaEvolve as an experimental engine for proposing and iteratively refining such certificates. The problems we studied are: reconstruction of bipartite and planar graphs from vertex-deleted subgraphs; the Alon-Tarsi parity problem for Latin squares, approached via sign-reversing involutions built from local trades; Rota's Basis Conjecture, studied through local exchange policies on collections of bases. In these three problems the correcting steps take the form of a reconstruction rule, a parity-reversing involution, and a transversal family of bases, respectively. For each problem, we describe the experimental setup, the scoring protocols, and the outcomes of the searches, leading to concrete conjectures concerning the existence and structure of the relevant correcting steps. The aim is not to claim proofs, but rather to produce explicit algorithms and to reveal structural patterns that appear amenable to subsequent analysis by traditional mathematical methods.

Explore related subjects

math.GM

Keep this discovery

BibTeX RIS

Gergely Bérczi. 2026-03-04. Evolving Local Corrections for Global Constructions in Combinatorics. https://arxiv.org/abs/2603.06692

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Average Chord Lengths in a Triangle

Let $P$ be a point inside a triangle $T$. We consider the average length of the chords of $T$ through $P$, where the direction of the chord is chosen uniformly. An elementary formula is obtained in terms of the distances from $P$ to the sides and vertices of the triangle. Several classical triangle centers give especially simple specializations. For example, if $I$ is the incenter, then \[ M_T(I)=\frac{2r}{\pi} \log\left(\cot\frac A4\cot\frac B4\cot\frac C4\right). \] Our main result is the sharp inequality \[ M_T(P)\le \frac{p}{\pi\sqrt3}\log(2+\sqrt3), \] valid simultaneously for every triangle of perimeter $p$ and every interior point $P$. Thus, among all such pairs $(T,P)$, the largest possible average chord length occurs only when $T$ is equilateral and $P$ is its center. The proof is an elementary symmetrization argument. We close with brief remarks relating the problem to the radial center of a convex body, the electrostatic potential center of a triangle, and dual quermassintegrals.

math.GM↗

arXiv2026

The decrease in sectional curvature of the Fisher--Rao manifold of beta distributions

In the paper, the author develops a general convexity lemma and applies it to demonstrate the monotonic decrease of a function involving polygamma functions, arising from the sectional curvature of the Fisher--Rao manifold of beta distributions.

math.GM↗

arXiv2026

A Proof of Liu's Conjecture on the Fundamental Triangle Inequality

Let $a,b,c$ be the side lengths of a triangle, and let $R$ and $r$ denote its circumradius and inradius, respectively. We prove a conjecture of Liu stating that \[\sum_{\mathrm{cyc}} \left(\frac{a(b+c-a)}{bc}\right)^k \geq 2+\left(\frac{2r}{R}\right)^k,~~k>1, \] with the reverse inequality for $0<k<1$. The proof reduces the problem to three positive variables with fixed sum and product. We also determine the equality cases.

math.GM↗