didericis
88c74efd28
Introduce Conjecture 1.31 (tire inner-boundary three-colour) as a decomposition-native weakening of the level-cycle conjecture 1.29: every maximal planar graph admits a vertex source and proper 4-colouring under which each tire inner boundary omits a colour. Remark 1.32 shows inner boundaries are single-level cycles, so the vertex-source form of 1.29 implies it on 2-connected boundaries. Extend check_level_cycle_three_color.py with --restriction inner-boundary (reconstructs the tire-tree decomposition from the embedding; inner boundary = level-(d+1) vertices of each depth-d dual component) and a --min-connectivity flag for the 5-connected slice. Verified: full census 4<=n<=13 (57716 triangulations) and 5-connected slice 14<=n<=24 (9732 graphs) all admit witnesses; no counterexample. Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
math-research
Personal mathematics research repository by Eric Bauerfeld. Papers are written in AMS-LaTeX using the amsart document class and live under papers/.
Papers
All papers are at papers/<name>/paper.tex. The current set:
| Directory | Title |
|---|---|
colored_edge_flip_classes |
Colored Edge Flip Classes |
colored_pentagon_contractions |
Colored Pentagon Reductions |
coloring_nested_tire_dual_graphs |
Coloring Nested Tire Dual Graphs |
coloring_nested_tire_graphs |
Coloring Nested Tire Graphs |
even_level_graph_generators |
Even Level Graph Generators: a constructive conjecture stronger than the Four Color Theorem |
face_monochromatic_pairs |
Face-Monochromatic Pairs and the Four Colour Theorem |
iterated_reduction_in_reduced_dual |
An Iterated Reduction in the Reduced Dual |
level_resolutions_of_maximal_planar_graphs |
Level Resolutions of Maximal Planar Graphs |
level_switching |
Level Switching |
plane_depth |
Plane Depth |
plane_depth_sequencing |
Plane Depth Sequencing |
plane_diamond_coloring |
Plane Diamond Coloring |
The papers form a connected programme around plane triangulations, BFS-level structure, and the Four Colour Theorem. plane_depth introduces the level / dual-depth framework that downstream papers build on; coloring_nested_tire_graphs develops the tire-tread tree-of-treads decomposition.
Creating a New Paper
Use run.sh to scaffold a new paper from the AMS-LaTeX template:
./run.sh init_paper "Your Paper Title"
This creates papers/<name>/ (the name is derived from the title, lower-cased, spaces → underscores) containing a paper.tex pre-filled with the title and author.
Setup
The Python library code in lib/ requires SageMath. Run setup once per machine:
./run.sh setup <sage_python_path> <sage_site_packages> [system_name]
sage_python_path— path to the SageMath Python interpreter (e.g./opt/sage/local/bin/python3)sage_site_packages— path to SageMath's site-packages directorysystem_name— optional label for this machine (defaults tohostname -s); used to store per-machine env files as.env.<system_name>
On subsequent runs the paths default to whatever was saved in .env, so ./run.sh setup alone re-runs setup with the existing configuration.
Setup also compiles the plantri submodule via make.
Running Sage
To run a Sage script with plantri available on PATH:
./run.sh sage <script.py> [args...]
Or to open an interactive Sage session:
./run.sh sage
Linting
./run.sh lint
Runs pyright and pylint on lib/ using the SageMath Python interpreter.
Shell Completion
To enable tab-completion for run.sh in zsh, add this to your .zshrc:
eval "$(path/to/run.sh completion)"
Or source it once in the current shell session:
eval "$(./run.sh completion)"
Building
Papers are compiled with LaTeX. From within a paper directory:
latexmk -pdf paper.tex