didericis
10a53e9de1
- Paper: Conjecture 5.5 restated with the hypothesis that every face
of H has length ≥ 5 (= the cubic plane analogue of "no triangles
or quadrilaterals as faces"). This kills K_4 and the n=8 trivial
counterexamples (girth 3) and the ad-hoc n=40 counterexample
(which has 2 triangles and 4 quadrilaterals). A new remark catalogues
these excluded counterexamples and the smallest cubic plane graphs
satisfying the hypothesis (dodecahedron at |V| = 20).
- search_smaller_counterexample.py: --min-face=N option to filter
cubic planar graphs by minimum face length.
- search_min_face5_counterexample.py: enumerates triangulations T
with min degree ≥ 5 via graphs.triangulations(n, minimum_degree=5),
takes planar dual (= cubic plane with all faces ≥ 5), and runs the
Heawood-constancy check.
- Result: smallest counterexample at triangulation order n_T = 16,
whose dual is a 28-vertex cubic plane graph (graph6
[kG[A?_A?_?_?K?D?@_CO?o?@_??A??@C??O??AG?C????`???a???W???A_???F).
Faces: 12 pentagons + 4 hexagons (a C28 fullerene). Both
K_{red, blue} and K_{red, green} are 12-cycles sharing the
colour-red edge (0, 1) and both have h_φ ≡ -1. 8 of 28 vertices
lie outside V(K_0) ∪ V(K_1).
- verify_28_vertex_counterexample.py: reproduces the counterexample,
verifies all properties, and renders figures/min-face-5-counterexample.png.
Note on the boundary: face-length ≥ 6 is impossible for cubic plane
graphs by Euler (6F = 6(V/2 + 2) > 3V = sum face lengths for V > 4).
So face-length ≥ 5 is the strongest face-length restriction admitting
any cubic plane graphs at all.
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