didericis/math-research
1
0
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity

face_monochromatic_pairs: stronger structural regularity at |S|=8

Browse Source 
experiments/check_S8_hit8_pG.py finds that all 30 |S|=8 bad
colourings with hit = 8 have p_G = 11 EXACTLY. Not p_G ∈ {9, 10, 11}
as I'd expected, but always 11.

This means: when |S| = 8 and 8 G'-pentagons are hit, the parent
triangulation v has NO degree-5 neighbours (= all 5 neighbours have
degree ≥ 6), and hence the reduced dual has 12 - 1 - 0 = 11
G'-pentagons. Three G'-pentagons are uncovered, not merely one.

Updated Remark (gprime-pigeonhole-stop) in paper to reflect this
stronger regularity: the size of S = V \ (V(K_b) ∪ V(K_c)) is
structurally tied to the count of pentagonal F_k adjacent to F_v in
chord-apex+Kempe colourings. A non-empirical proof of this is open.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
didericis
2026-05-25 07:37:12 -04:00
parent b2aa14cefa
commit e8d3d9e5d0
3 changed files with 165 additions and 8 deletions
Download Patch File Download Diff File Expand all files Collapse all files
papers/face_monochromatic_pairs/paper.tex
+14 -8
View File
@@ -1347,14 +1347,20 @@ $8$ & $252$ & $8$ & $\boldsymbol{9}$\,$^\dag$ & $1$ \\
$10$ & $36$ & $7$ & $8$ & $1$ \\
\end{tabular}
\end{center}
$^\dag$ Empirically, the combination $|S| = 8$ with \# pent. hit $= 8$
\emph{and} $p_G = 8$ never occurs --- so although both $\text{hit} = 8$
and $p_G = 8$ are individually possible at $|S| = 8$, they are never
simultaneous. (\texttt{experiments/check\_30\_residual\_v2.py}.)
This is a structural fact about chord-apex+Kempe colourings that we
do not have a non-empirical proof of, but its empirical confirmation
shows the G'-pentagon fallback closes the $|S| = 8$ case along with
the other $|S|$ values.
$^\dag$ Empirically, every chord-apex+Kempe colouring with $|S| = 8$
and \# pent. hit $= 8$ has $p_G = 11$ exactly --- not merely $p_G \ge 9$.
That is, every one of the $30$ cases that achieves hit $= 8$ at $|S| = 8$
has all $5$ of $v$'s neighbours in the parent triangulation of
degree $\ge 6$, so no $F_k$ adjacent to $F_v$ is pentagonal, leaving
$p_G = 12 - 1 - 0 = 11$ G'-pentagons in the reduced dual. So when
hit $= 8$, $3$ G'-pentagons are uncovered ---
\emph{not merely~$1$}. (\texttt{experiments/check\_S8\_hit8\_pG.py}.)
This is a much stronger structural regularity than we expected,
suggesting that in chord-apex+Kempe colourings, the size of
$S = V \setminus (V(K_b) \cup V(K_c))$ is structurally tied to
the count of pentagonal $F_k$ adjacent to $F_v$ in a way that always
leaves G'-pentagons uncovered. A structural (non-empirical) proof of
this regularity is open.
Combining Theorem~\ref{thm:deciding-face-partial-extended} with the
empirical structural facts above, the $G'$-pentagon fallback