didericis
e47f89918a
User correctly pointed out:
(1) The Figure 4 partial-tire-dual interior structure is not a "theta
graph" in the K_{2,3} sense (which requires all three paths of
length ≥ 2). It is θ(1, 6, 6): a 12-cycle with one chord.
(2) θ(1, p, q) IS outerplanar (just a polygon with one chord), so it
belongs IN the menagerie, not outside it.
Revisions:
- Section 6 ("2-connected outerplanar with Δ ≤ 3"): previously claimed
the class is just cycles; corrected to "cycle, possibly with a
matching of chords." Added explicit description of θ(1, p, q) and
a closed-form for its proper 3-edge-coloring count:
P_e(θ(1,p,q), 3) = (2^{p+q} - 2^p (-1)^q - 2^q (-1)^p + 10 (-1)^{p+q}) / 3.
Verified against Sage's chromatic polynomial for all p, q ∈ {2..6}.
- "Outside the menagerie" section: previously said "theta graphs (all
flavours) are not outerplanar." Corrected to clarify that only
θ(p, q, r) with all three paths of length ≥ 2 (= K_{2,3} subdivisions)
is not outerplanar. Explicitly noted that the bridge-case partial
tire dual gives θ(1, p, q) which IS in the menagerie, with edge-3-
coloring count given by the closed form.
The Figure 4 partial-tire-dual (m=4 outer cycle + barbell O with
bridge) has θ(1, 6, 6) as its interior dual subgraph and so admits
exactly 1326 proper 3-edge-colorings on the interior cycle-with-
chord; leaves contribute their forced colors as in the spoke-only
case.
Paper unchanged. This is a correction within the notes/ subdir only.
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