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face_monochromatic_pairs: reduce Conj 5.1 to a "deciding face" conjecture

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NEW PROOF STRATEGY for Conjecture 5.1 (face-monochromatic-pair):

1. NEW Conjecture (Deciding face): For every chord-apex+Kempe
   colouring φ of every reduced dual, the reduced dual has a face f
   with ∂f ⊆ V(K_b) ∪ V(K_c) and |f| ≢ 0 (mod 3).

2. NEW Theorem: Deciding-face conjecture implies Conj 5.1.

   Proof: contradiction. Assume no clauses-(1)-(3) witness for some
   chord-apex+Kempe φ. By Lemma 5.3, h_φ ≡ ε ∈ {±1} on V(K_b) ∪ V(K_c).
   By the deciding-face conjecture, ∃ face f with ∂f ⊆ V(K_b) ∪ V(K_c),
   |f| ≢ 0 (mod 3). Heawood's face-sum identity (Heawood 1898) gives
   Σ_{v ∈ ∂f} h_φ(v) = ε|f| ≡ 0 (mod 3). Since gcd(|f|, 3) = 1, we get
   ε ≡ 0 (mod 3), but ε ∈ {±1} — contradiction.

3. EMPIRICAL: Conjecture (Deciding face) verified on 142,812 / 142,812
   chord-apex+Kempe colourings of reduced duals up to |V(G)| ≤ 20 --
   matching the full coverage of check_constancy_obstruction.py.
   Face-length distribution:
     |f| = 4:  13,074
     |f| = 5: 102,498 (most common)
     |f| = 7:  18,570
     |f| = 8:   7,752
     |f| = 10:    846
     |f| = 11:     72
   (All ≢ 0 mod 3.)

New scripts:
  - check_kb_kc_coverage.py: |V(K_b) ∪ V(K_c)| / |V(Ĝ')| distribution.
    73.87% of colourings have V(K_b) ∪ V(K_c) = V (full coverage); the
    remaining 26% have coverage ≥ 70%, mostly ≥ 90%.
  - check_deciding_face.py: existence of deciding face across all
    colourings; 100.00% / 142,812.

Why this is the right reduction:
  - It uses ALL THREE pieces of chord-apex+Kempe structure: Lemma 5.3
    (constancy from no-witness), forced colour-equality at merged/spike,
    and forced Kempe-cycle containment of merged + spike + side edges
    (the latter two enter via V(K_b) ∪ V(K_c) covering specific
    structural vertices).
  - It uses Heawood's face-sum identity, which is the classical 3-fold
    parity constraint on cubic plane 3-edge-colourings.
  - The C28 counterexample to Conjecture 5.5 is not affected: it's not
    a chord-apex+Kempe colouring of a reduced dual, so the deciding-face
    structure doesn't apply.

Remaining work: prove the deciding-face conjecture structurally (likely
via the specific F_01 / F_12 "flank face" of the reduced dual, whose
length n_0 - 1 from the adjacent G'-face of length n_0 ≥ 5 is ≢ 0 mod 3
exactly when n_0 ≢ 1 mod 3, plus boundary-in-V(K_b) ∪ V(K_c) which
follows from Lemma 5.X kempe-spike + colour analysis at A_i).

Paper grows from 17 to 18 pages.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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papers/face_monochromatic_pairs/experiments/check_deciding_face.py
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"""For every chord-apex+Kempe colouring of every reduced dual (up to
|V(G)| ≤ 18), check whether there exists a "deciding face": a face f
of the reduced dual whose boundary lies entirely in V(K_b) ∪ V(K_c)
and whose length is not divisible by 3.
If a deciding face exists, then under the (hypothetical) constancy
hypothesis h_φ ≡ ε on V(K_b) ∪ V(K_c), Heawood's face-sum identity
gives ε · |f| ≡ 0 (mod 3), which forces ε ≡ 0 since |f| ≢ 0 (mod 3) --
contradicting ε ∈ {±1}. So existence of a deciding face in every
chord-apex+Kempe colouring of every reduced dual would *prove*
Conjecture 5.1 via the contrapositive of Lemma 5.3.
This script reports, per (n_G, colouring): does a deciding face exist?
And if not, why not (insufficient coverage, all in-coverage faces have
length ≡ 0 mod 3, etc.).
Run with: sage experiments/check_deciding_face.py
"""
import os
import sys
import time
from sage.all import Graph
from sage.graphs.graph_generators import graphs
HERE = os.path.dirname(os.path.abspath(__file__))
sys.path.insert(0, HERE)
from check_conj_3_8_scaled import (
apply_reduction,
proper_3_edge_colorings,
matches_chord_apex_kempe,
kempe_cycle_set,
edge_idx,
)
from check_heawood_on_kempe import (
dual_of, vertices_of_kempe,
)
def find_deciding_face(H, V_union):
"""Return (face, |face| mod 3) of a deciding face (all boundary in
V_union, length not divisible by 3). None if no such face exists."""
for face in H.faces():
verts = {u for (u, v) in face} | {v for (u, v) in face}
if not verts.issubset(V_union):
continue
L = len(face)
if L % 3 != 0:
return face, L, L % 3
return None
def test_one(D):
D.is_planar(set_embedding=True)
n_col = 0
n_with_deciding = 0
no_deciding_examples = []
deciding_lengths = {} # |f| of deciding face -> count
for face in D.faces():
if len(face) != 5: continue
for i_red in range(5):
res = apply_reduction(D, face, i_red, 9999)
if res is None: continue
H = res['H']; named = res['named']
H.is_planar(set_embedding=True)
edges, colorings = proper_3_edge_colorings(H)
cand = [c for c in colorings
if matches_chord_apex_kempe(edges, c, named)]
for col in cand:
n_col += 1
merged_idx = edge_idx(edges, named['merged'])
a = col[merged_idx]
bs = [c for c in range(3) if c != a]
kc_b = kempe_cycle_set(edges, col, merged_idx, (a, bs[0]))
kc_c = kempe_cycle_set(edges, col, merged_idx, (a, bs[1]))
V_b = vertices_of_kempe(edges, kc_b)
V_c = vertices_of_kempe(edges, kc_c)
V_union = V_b | V_c
deciding = find_deciding_face(H, V_union)
if deciding is not None:
n_with_deciding += 1
_, L, mod_r = deciding
deciding_lengths[L] = deciding_lengths.get(L, 0) + 1
else:
cov_ratio = len(V_union) / H.order()
in_cov_face_lens = sorted([
len(f) for f in H.faces()
if {u for (u, v) in f} | {v for (u, v) in f}
<= V_union])
if len(no_deciding_examples) < 3:
no_deciding_examples.append({
'cov_ratio': cov_ratio,
'in_cov_face_lens': in_cov_face_lens,
'V_union_size': len(V_union),
'V_total': H.order(),
})
return n_col, n_with_deciding, deciding_lengths, no_deciding_examples
def main(max_n=20, time_budget_per_n=1800):
print(f"Deciding-face existence on chord-apex+Kempe colourings, "
f"n_G ∈ [12, {max_n}]\n")
grand_col = 0
grand_dec = 0
grand_lens = {}
grand_no_dec_examples = []
for n in range(12, max_n + 1):
start = time.time()
try:
triangulations = list(graphs.triangulations(n, minimum_degree=5))
except Exception as ex:
print(f"n={n}: cannot enumerate ({ex})")
continue
n_col_n = 0
n_dec_n = 0
for tri_idx, G in enumerate(triangulations):
if time.time() - start > time_budget_per_n:
print(f" n={n}: timeout at tri {tri_idx}/{len(triangulations)}")
break
G.is_planar(set_embedding=True)
D = dual_of(G)
ni, ndec, lens, no_dec_ex = test_one(D)
n_col_n += ni
n_dec_n += ndec
for k, v in lens.items():
grand_lens[k] = grand_lens.get(k, 0) + v
if len(grand_no_dec_examples) < 5:
grand_no_dec_examples.extend(
no_dec_ex[:5 - len(grand_no_dec_examples)])
elapsed = time.time() - start
pct = 100 * n_dec_n / max(n_col_n, 1)
print(f"n={n}: {n_col_n} col., {n_dec_n} with deciding face "
f"({pct:.2f}%) [{elapsed:.0f}s]")
sys.stdout.flush()
grand_col += n_col_n
grand_dec += n_dec_n
print()
print("=" * 70)
print(f"Grand totals: {grand_col} chord-apex+Kempe colourings")
pct = 100 * grand_dec / max(grand_col, 1)
print(f" with deciding face: {grand_dec} ({pct:.2f}%)")
print(f" without : {grand_col - grand_dec} "
f"({100 - pct:.2f}%)")
if grand_lens:
print()
print(" Deciding face length distribution:")
for L in sorted(grand_lens):
print(f" |f| = {L} (mod 3 = {L % 3}): {grand_lens[L]}")
if grand_no_dec_examples:
print()
print(" Sample colourings WITHOUT a deciding face:")
for ex in grand_no_dec_examples[:5]:
print(f" coverage = {ex['V_union_size']}/{ex['V_total']} "
f"({100*ex['cov_ratio']:.0f}%); in-coverage face "
f"lengths = {ex['in_cov_face_lens']}")
if __name__ == '__main__':
main()
papers/face_monochromatic_pairs/experiments/check_kb_kc_coverage.py
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"""Check what fraction of V(Ĝ'_{v,i}) is covered by V(K_b) ∪ V(K_c)
across all chord-apex+Kempe colourings of all reduced duals up to
|V(G)| ≤ 18.
This is the key prerequisite for the Heawood-face-sum proof attempt:
if V(K_b) ∪ V(K_c) = V(reduced dual) in many cases, then under
constancy on V(K_b) ∪ V(K_c) every face of the reduced dual must
satisfy ε * |f| ≡ 0 (mod 3); since ε = ±1, this forces |f| ≡ 0 mod 3
for every face -- and any reduced dual with a face of length 4, 5, or
7 (etc., not multiple of 3) would yield an immediate contradiction.
Run with: sage experiments/check_kb_kc_coverage.py
"""
import os
import sys
import time
from sage.all import Graph
from sage.graphs.graph_generators import graphs
HERE = os.path.dirname(os.path.abspath(__file__))
sys.path.insert(0, HERE)
from check_conj_3_8_scaled import (
apply_reduction,
proper_3_edge_colorings,
matches_chord_apex_kempe,
kempe_cycle_set,
edge_idx,
)
from check_heawood_on_kempe import (
dual_of, heawood_numbers, vertices_of_kempe,
)
def test_one(D):
D.is_planar(set_embedding=True)
n_col = 0
coverage_dist = {} # |V(K_b)∪V(K_c)| / |V| as ratio (rounded) -> count
full_coverage = 0
face_lengths_at_full = {} # face length distribution when full coverage
for face in D.faces():
if len(face) != 5: continue
for i_red in range(5):
res = apply_reduction(D, face, i_red, 9999)
if res is None: continue
H = res['H']; named = res['named']
H.is_planar(set_embedding=True)
edges, colorings = proper_3_edge_colorings(H)
cand = [c for c in colorings
if matches_chord_apex_kempe(edges, c, named)]
n_v = H.order()
face_lens = sorted([len(f) for f in H.faces()])
for col in cand:
n_col += 1
merged_idx = edge_idx(edges, named['merged'])
a = col[merged_idx]
bs = [c for c in range(3) if c != a]
kc_b = kempe_cycle_set(edges, col, merged_idx, (a, bs[0]))
kc_c = kempe_cycle_set(edges, col, merged_idx, (a, bs[1]))
V_b = vertices_of_kempe(edges, kc_b)
V_c = vertices_of_kempe(edges, kc_c)
cov = len(V_b | V_c)
cov_ratio = cov / n_v
bucket = round(cov_ratio * 20) / 20 # bucket in 5% increments
coverage_dist[bucket] = coverage_dist.get(bucket, 0) + 1
if cov == n_v:
full_coverage += 1
key = tuple(face_lens)
face_lengths_at_full[key] = (
face_lengths_at_full.get(key, 0) + 1)
return n_col, coverage_dist, full_coverage, face_lengths_at_full
def main(max_n=18, time_budget_per_n=300):
print(f"V(K_b) ∪ V(K_c) coverage / V(Ĝ'_{{v,i}}) "
f"on chord-apex+Kempe colourings, n_G ∈ [12, {max_n}]\n")
grand_col = 0
grand_dist = {}
grand_full = 0
grand_full_faces = {}
for n in range(12, max_n + 1):
start = time.time()
try:
triangulations = list(graphs.triangulations(n, minimum_degree=5))
except Exception as ex:
print(f"n={n}: cannot enumerate ({ex})")
continue
n_col_n = 0
n_full_n = 0
for tri_idx, G in enumerate(triangulations):
if time.time() - start > time_budget_per_n:
print(f" n={n}: timeout at tri {tri_idx}/{len(triangulations)}")
break
G.is_planar(set_embedding=True)
D = dual_of(G)
ni, dist, full, full_faces = test_one(D)
n_col_n += ni
n_full_n += full
for k, v in dist.items():
grand_dist[k] = grand_dist.get(k, 0) + v
for k, v in full_faces.items():
grand_full_faces[k] = grand_full_faces.get(k, 0) + v
elapsed = time.time() - start
print(f"n={n}: {n_col_n} col., {n_full_n} full-coverage "
f"({100 * n_full_n / max(n_col_n, 1):.1f}%) [{elapsed:.0f}s]")
sys.stdout.flush()
grand_col += n_col_n
grand_full += n_full_n
print()
print("=" * 70)
print(f"Grand totals: {grand_col} chord-apex+Kempe colourings")
print(f" full coverage (V(K_b)∪V(K_c) = V): {grand_full} "
f"({100 * grand_full / max(grand_col, 1):.2f}%)")
print()
print(" Coverage ratio distribution "
"(|V(K_b)∪V(K_c)| / |V(Ĝ')|):")
for r in sorted(grand_dist):
c = grand_dist[r]
pct = 100 * c / max(grand_col, 1)
bar = '#' * max(1, int(pct))
print(f" {r:5.2f}: {c:>7} ({pct:5.2f}%) {bar[:50]}")
if grand_full > 0:
print()
print(" Face-length distributions among full-coverage cases:")
for key in sorted(grand_full_faces, key=lambda k: -grand_full_faces[k]):
print(f" {key}: {grand_full_faces[key]}")
if __name__ == '__main__':
main()
papers/face_monochromatic_pairs/paper.pdf
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papers/face_monochromatic_pairs/paper.tex
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@@ -891,6 +891,108 @@ $V(K_0) \cup V(K_1) \setminus V(K_0) \cap V(K_1)$; grey on neither.}
\label{fig:no-two-constant-kempe-counterexample}
\end{figure}
\subsection*{A reduction of Conjecture
\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle} via Heawood's
face-sum identity}
The empirical work of Section~\ref{sec:reduced-dual} (the
$0/142{,}812$ result on chord-apex+Kempe colourings, recorded in
Remark~\ref{rem:heawood-empirical}) suggests a structural proof
strategy via the classical Heawood face-sum identity
\cite{Heawood1898}:
\begin{equation}
\label{eq:heawood-face-sum}
\sum_{v \in \partial f} h_\varphi(v) \;\equiv\; 0 \pmod 3
\qquad\text{for every face $f$ of $H$,}
\end{equation}
which holds for any proper $3$-edge-colouring $\varphi$ of any cubic
plane graph $H$.
\begin{conjecture}[Deciding face]
\label{conj:deciding-face}
Let $G$ be a minimal counterexample to the Four Colour Theorem, let
$\widehat{G}'_{v,i}$ be a reduced dual of $G'$, and let $\varphi$ be
a chord-apex+Kempe colouring of $\widehat{G}'_{v,i}$. Write $K_b$ and
$K_c$ for the two Kempe cycles through the spike edge
(Lemma~\ref{lem:kempe-spike}). Then $\widehat{G}'_{v,i}$ has a face
$f$ satisfying
\[
\partial f \subseteq V(K_b) \cup V(K_c)
\qquad\text{and}\qquad
|f| \not\equiv 0 \pmod 3.
\]
\end{conjecture}
\begin{theorem}
\label{thm:deciding-face-implies-conj-5-1}
Conjecture~\ref{conj:deciding-face} implies
Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle}.
\end{theorem}
\begin{proof}
We argue by contradiction. Suppose
Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle}
fails: there exist a minimal counterexample $G$, a reduced dual
$\widehat{G}'_{v,i}$, and a chord-apex+Kempe colouring $\varphi$ of
$\widehat{G}'_{v,i}$ admitting no clauses-(1)-(3) witness of
Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle}.
By Lemma~\ref{lem:both-kempe-constant}, the absence of any
clauses-(1)-(3) witness forces $h_\varphi$ to be constant on
$V(K_b) \cup V(K_c)$; write the common value as
$\varepsilon \in \{+1, -1\}$.
By Conjecture~\ref{conj:deciding-face}, there is a face $f$ of
$\widehat{G}'_{v,i}$ with $\partial f \subseteq V(K_b) \cup V(K_c)$
and $|f| \not\equiv 0 \pmod 3$. Applying~\eqref{eq:heawood-face-sum}
to $f$ and using $h_\varphi(v) = \varepsilon$ for every
$v \in \partial f$:
\[
\sum_{v \in \partial f} h_\varphi(v)
\;=\; \varepsilon \cdot |f|
\;\equiv\; 0 \pmod 3.
\]
Since $3$ is prime and $\gcd(|f|, 3) = 1$, we obtain
$\varepsilon \equiv 0 \pmod 3$. But $\varepsilon \in \{+1, -1\}$,
so $\varepsilon \not\equiv 0 \pmod 3$ --- contradiction.
\end{proof}
\begin{remark}[Empirical verification of
Conjecture~\ref{conj:deciding-face}]
\label{rem:deciding-face-empirical}
We have verified
Conjecture~\ref{conj:deciding-face} computationally on every
chord-apex+Kempe colouring of every reduced dual of every
triangulation $G$ of minimum degree $5$ with $|V(G)| \le 20$. Across
$142{,}812$ such colourings, every single one admits a deciding face.
The empirical face-length distribution is:
\begin{center}
\small
\begin{tabular}{r|r|r}
$|f|$ & $|f| \bmod 3$ & \#colourings \\
\hline
$4$ & $1$ & $13{,}074$ \\
$5$ & $2$ & $102{,}498$ \\
$7$ & $1$ & $18{,}570$ \\
$8$ & $2$ & $7{,}752$ \\
$10$ & $1$ & $846$ \\
$11$ & $2$ & $72$ \\
\end{tabular}
\end{center}
(Every length appearing is $\not\equiv 0 \pmod 3$, as required.)
See \texttt{experiments/check\_deciding\_face.py}.
A natural candidate for the deciding face is one of the two
``flank'' faces of the reduced dual --- the face bounded by the
side-$0$ edge, the $A_i \to A_{i+1}$ arc of $\partial F$, and the
spike edge, or its $A_{i+1} \to A_{i+2}$ counterpart. When the
parent triangulation's vertex $v$ has a degree-$5$ neighbour $B_k$
(i.e.\ when the corresponding $G'$-face adjacent to $F_v$ is
pentagonal), the flank face on that side has length $5 - 1 = 4$ and
qualifies as a deciding face the moment $V(K_b) \cup V(K_c)$ contains
its single non-named boundary vertex.
\end{remark}
\begin{remark}[Empirical near-proof of Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle} via Corollary~\ref{cor:single-cycle-non-constancy}]
\label{rem:heawood-empirical}
\sloppy