didericis
3aec31b3ac
experiments/search_smaller_counterexample.py enumerates 3-connected
cubic planar graphs via graphs.planar_graphs(n, min_deg=3, min_conn=2)
(filtering to cubic), then for each graph tries every proper
3-edge-colouring (backtracking with symmetry-break on first edge),
computes h_φ via the CW rotation from sage's planar embedding, and
checks whether some pair of intersecting Kempe cycles K_{a,b} and
K_{a,c} are both constant-Heawood.
Results (up to n=10 in initial run):
n= 4: K_4 itself. Coloring (1,2)=red, (3,4)=red, (1,3)=blue,
(2,4)=blue, (1,4)=green, (2,3)=green; sage's CW embedding
gives h_φ ≡ -1 on all 4 vertices. K_{red,blue} = 4-cycle
1-2-4-3 and K_{red,green} = 4-cycle 1-2-3-4 share both red
edges; both constant.
n= 6: no counterexample (only the triangular prism).
n= 8: a 12-edge cubic planar graph (graph6 G}GOW[) on 8 vertices.
Both Kempe cycles are 8-cycles visiting every vertex.
n=10: 8 cubic planar graphs checked, no counterexample.
So K_4 is the smallest counterexample to Conjecture 5.5 as stated,
but both K_4 and the n=8 example are structurally trivial: K_0 and
K_1 jointly cover V(H). The user's 40-vertex counterexample (paper
Figure) is the smallest non-trivial example found so far, with 24
vertices outside V(K_0) ∪ V(K_1).
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