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THE IDEA, MADE VISIBLE

Give every edge its own difference.

Change a labeling and test the graceful condition directly on a small tree.

Method & partial progressRooted Weyl Alternants for the Graceful Tree Conjecture

Calculated illustration · change the inputs to inspect the mechanism.

01 / 04Guided chapter

Pick a tree

A tree has no cycles. With n vertices, it has n−1 edges. A graceful labeling uses every vertex label from zero through n−1.

Tree vertices

Tree

Labeling

Vertex labels: 6
Distinct differences: 5
Graceful?: Yes

Differences: 5, 4, 3, 2, 1. This particular labeling passes.

Edge uv receives |label(u) − label(v)|

TRY THIS

On a path, compare graceful and sequential labels. Sequential labels repeat difference 1 on every edge.

What this experiment represents. Exact finite labeling check. The tree drawings are teaching instances, not a rerun of the manuscript’s 7,812-type audit or a universal graceful-tree proof.

Source manuscript & release ↗Read the full explanation ↓

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1. Pick a tree

A tree has no cycles. With n vertices, it has n−1 edges. A graceful labeling uses every vertex label from zero through n−1.

2. Take edge differences

For each edge, subtract its endpoint labels and take the absolute value. The target differences are exactly 1 through n−1.

3. Find collisions

Red edges repeat a difference. Distinct vertex labels alone are not enough: the edge differences must also cover the required range.

4. Understand the research method

The paper encodes labelings using signed alternants, leaf recurrences and symmetry. This direct finite checker explains the target condition; it does not resolve the universal Graceful Tree Conjecture.

Mathematics · Preprint

Rooted Weyl Alternants for the Graceful Tree Conjecture

Global alternants, extremal-edge flows, and exact pruning tests

Method & partial progressMF-PRISM-MATH-2026-10 / v1.2

An exact signed-enumeration framework for graceful tree labelings, with global alternants, directed edge classes, and two leaf recurrences. The reconstructed finite audit reproduces 7,812 rooted types and the pruning results through thirteen vertices.

Read the paper Release & files

Cover of Rooted Weyl Alternants for the Graceful Tree Conjecture — Current public editionv1.2 · 2026-09-12

Identifier: MF-PRISM-MATH-2026-10
Series: Mathematics
Edition: Version 1.2
Length: 9 pages
Reserved DOI: 10.5281/zenodo.22728328 (record reserved)

Plain-language guide

3 minute explanation

What this paper is really saying.

The paper turns graceful tree labeling into an exact signed-enumeration problem that can certify and prune finite cases, but it does not prove that every tree is graceful.

graph labelinggraceful treesalternantsexact computation

A graceful labeling assigns distinct numbers to the vertices of a tree so that the absolute differences across edges are also all distinct. The famous conjecture says every tree has such a labeling.

Instead of searching through labels one by one, the paper encodes candidate labelings in algebraic alternants. Coefficients can then act as exact witnesses: a nonzero coefficient signals surviving labelings, while leaf recurrences and symmetry reductions remove large blocks of redundant work.

How it works

The argument, without the notation.

The paper’s technical details matter, but the basic route can be understood in three moves.

Encode labelings algebraically

Use global and rooted Weyl-style alternants to represent signed assignments and edge-difference constraints.

Exploit tree structure

Direct edge classes, remove leaves recursively, and group symmetric sibling configurations when the coefficient identities permit it.

Audit finite cases

Reconstruct 7,812 rooted types and reproduce the published pruning results through thirteen vertices.

What it adds

The useful takeaways.

The manuscript supplies an exact algebraic framework for graceful-labeling searches.
Two leaf recurrences and symmetry reductions make finite certification substantially more compact.
The finite audit is partial progress only; universal nonvanishing for all trees remains open.

Scope & limits

What this does—and does not—establish.

Current status

Method and partial progress; the universal Graceful Tree Conjecture is not resolved.

This manuscript develops a method and finite evidence, but it does not resolve the full open problem. The unresolved boundary is part of the published result.

No universal proof of the Graceful Tree Conjecture is claimed.
Finite nonvanishing certificates do not imply nonvanishing for every tree size.
The validity of every symmetry reduction must be checked coefficient by coefficient.

Why it matters

Why anyone should care.

The framework turns parts of the graceful-labeling search into auditable algebraic certificates and exact finite pruning tests, while keeping the universal conjecture explicitly open.

Key terms

The vocabulary, decoded.

These are the terms needed to understand the claim. The full paper uses them more precisely.

Graceful labeling

A vertex labeling whose edge differences are all distinct and use the required range.

Tree

A connected graph with no cycles.

Alternant

A signed algebraic sum whose cancellations encode symmetry and distinctness.

Certificate

A compact object that can be checked independently to verify a finite claim.

Independent review

Where scrutiny should concentrate.

The strongest review is not a general reaction. It tests the steps most capable of changing the conclusion.

01
Orientation and inherited ordering conventions in the global and rooted identities.
02
Validity of sibling-orbit reductions for every coefficient, not just endpoint counts.
03
The distinction between finite nonvanishing certificates and a universal graceful-tree proof.

Publication record

Go from explanation to evidence.

This page is a reading guide, not a substitute for the manuscript. The public record links the explanation to the paper, source package, review materials, and persistent identifier.

GitHub release Source folder

Document: Preprint
Review status: Open for independent review
Published: 2026-09-12
DOI status: Reserved · 10.5281/zenodo.22728328
Canonical record: MF-PRISM-MATH-2026-10

Metriq PRISM Laboratory, Rooted Weyl Alternants for the Graceful Tree Conjecture, Metriq PRISM Laboratory Mathematics Research Paper MF-PRISM-MATH-2026-10, Version 1.2, 2026. Corresponding contributor: Daniel H. Jeffery, ORCID 0009-0001-1200-6042. DOI: 10.5281/zenodo.22728328.