didericis
b4fece08bc
- Extends Definition 1.5 (tire graph) to allow either B_out or B_in to
be a single vertex (a "degenerate" boundary); the tire is then a
triangulated disk with that vertex as apex.
- Adds Remark 1.6 with vertex/edge/triangle counts for both the
non-degenerate (annulus) and degenerate (disk) cases.
- Adds Lemma 1.7 (tire-component lemma): for a maximal planar graph G
with level source S and depth d ≥ 0, every connected component C' of
the dual subgraph on dual vertices of depth d induces a subgraph C
on G that is a tire graph; its two boundary parts are the level-d
and level-(d+1) induced subgraphs (in either order), and its
triangular faces are exactly the faces of G in F_{C'}.
- Adds Remark 1.8 noting the d=0 degenerate-source case as an example.
- Adds a figure (fig_tire_example.png) illustrating the definition with
m=6 outer cycle, k=4 inner cycle, one chord in O, plus a legend
identifying B_out, B_in, O's chord, and E_ann.
- Adds experiments/tire_def_figure.py to regenerate the figure.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
math-research
Personal mathematics research repository by Eric Bauerfeld. Papers are written in AMS-LaTeX using the amsart document class.
Papers
kempe_style_search_for_smaller_contradiction
Humans Suffice: A Novel Proof of the Four Color Theorem
An in-progress proof of the Four Color Theorem via a minimal counterexample argument. The paper builds on Kempe's 1879 strategy — establishing valid cases for vertices of degree ≤ 4, then extending the argument to the degree-5 case using properties of non-adjacent degree-5 vertices, merged subgraphs, and locked colorings.
plane_depth_labelling
Plane Depth Labelling
Early-stage paper. Title and author information set; content in progress.
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 a new directory (name derived from the title) 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
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eval "$(path/to/run.sh completion)"
Or source it once in the current shell session:
eval "$(./run.sh completion)"
Building
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latexmk -pdf paper.tex