didericis
7801ce959e
NEW Theorem 1.15 (Tait correspondence for tires):
#{4-colorings of T} / |S_4| = #{3-edge-colorings of Γ} / |S_3|
That is, the number of 4-vertex-colorings of the tire T up to
color permutation equals the number of 3-edge-colorings of the
inner dual Γ up to color permutation.
Proof: standard Tait. Encode 4 colors as Z_2 × Z_2; define
χ*(e*) = c(u) + c(v) for each interior annular edge. The
triangulation constraint guarantees χ* is a proper 3-edge-coloring
of Γ; the lift c → χ* is 4-to-1 (global Z_2 × Z_2 translation).
Quotienting by |S_4| = 24 and |S_3| = 6 gives the stated equality.
NEW Theorem 1.16 (count formula):
(i) For spoke-only tires (Γ ≅ C_n):
#{proper 3-edge-colorings of Γ} = 2^n + 2(-1)^n.
(ii) For single-chord tires (Γ ≅ Θ(1, b, c), b + c = n):
#{proper 3-edge-colorings of Γ} = 6(α_b α_c + β_b β_c),
where α_L = (2^{L-1} + 2(-1)^{L-1})/3,
β_L = (2^{L-1} - (-1)^{L-1})/3.
Verification: Θ(1, 2, 2) = K_4 \ e gives 6.
Proofs:
(i) Standard chromatic polynomial of cycle at k = 3.
(ii) Transfer matrix on the two non-chord paths with chord
color fixed and endpoint configurations enumerated.
Remark 1.17: For more chords, the count depends on the chord
arrangement, not just (n, k). Two outerplanar graphs with the
same vertex and chord counts can have different 3-edge-coloring
counts. But linear-time computation via tree decomposition
(treewidth ≤ 2 for outerplanar) is always available.
Added Tait's 1880 paper as bibitem.
Page count: 11 → 12. Theorem 1.18 (tree structure) renumbered from
1.15 to 1.18 to make room.
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