didericis
dfca45e913
Adds a worked example: G = C_8 with three non-crossing chords
{(v_0,v_2), (v_3,v_7), (v_4,v_6)}. Walks through the calculation
of P_e(G, 3) by propagating constraints:
1. Fix chord c_0 = a (3 choices).
2. Forces {c(e_0), c(e_7)} = {b, c} and {c(e_1), c(e_2)} = {b, c}
at v_0 and v_2; cycle constraint at v_1 ties them together.
3. Propagating to chord 3-7 forces c_3 = a and the adjacent
cycle edges to alternate {b, c}.
4. Propagating to chord 4-6 forces c_4 = a and cycle edges
continue the alternation.
Result: cycle edges alternate b, c around C_8 (OK since |C_8| is
even); all 3 chords get the same color a. Total proper 3-edge-
colorings: 3 (choice of a) × 2 (b/c assignment) = 6, verified by
Sage's chromatic-polynomial computation on L(G).
Note that the graph admits a UNIQUE proper 3-edge-coloring modulo
permutation of the 3 colors -- the chord structure forces all
three chords to take the "third" color absent on the polygon cycle.
Adds:
- draw_3chord_example.py
- fig_3chord_example.png
Paper grows from 4 to 5 pages.
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
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