didericis
d121d2d3b6
Adds Proposition 1.13 (Edge-vertex coloring bijection for D(T)): for
a tire graph T satisfying the spoke-only hypothesis of Prop 1.8 (so
D(T) ~= C_{n+m} ∘ K_1), the number of proper 3-edge-colorings of D(T)
equals the number of proper 3-vertex-colorings of its interior dual
subgraph Γ ~= C_{n+m}, and both equal 2^{n+m} + 2 · (-1)^{n+m}.
Proof: Two bijection steps.
Step 1: Restriction is a bijection between proper 3-edge-colorings
of D(T) and proper 3-edge-colorings of the cycle C_L (where
L = n+m), because at each d_f the leaf's color is forced to be
the unique third color absent from the two cycle edges, and
leaves impose no further constraint.
Step 2: Proper 3-edge-colorings of C_L = proper 3-vertex-colorings
of L(C_L) = proper 3-vertex-colorings of C_L (since L(C_L) ~= C_L).
Step 3: Chromatic polynomial of C_L at k=3 is 2^L + 2 · (-1)^L.
Adds Remark 1.14 noting the closed form depends only on n+m, not
on the specific spoke-only annular triangulation or chord structure
of O.
Empirically verified for L in [3, 10] via Sage's chromatic
polynomials: edge-3-colorings of D(T) = vertex-3-colorings of C_L
= formula in every case.
Paper grows from 7 to 8 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