From d7e9b6af2f620434b1610e81016a02e39dd461b7 Mon Sep 17 00:00:00 2001 From: didericis Date: Mon, 25 May 2026 04:12:09 -0400 Subject: [PATCH] face_monochromatic_pairs: reduce Conj 5.1 to a "deciding face" conjecture MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit 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 --- .../experiments/check_deciding_face.py | 161 ++++++++++++++++++ .../experiments/check_kb_kc_coverage.py | 132 ++++++++++++++ papers/face_monochromatic_pairs/paper.pdf | Bin 1253770 -> 1260470 bytes papers/face_monochromatic_pairs/paper.tex | 102 +++++++++++ 4 files changed, 395 insertions(+) create mode 100644 papers/face_monochromatic_pairs/experiments/check_deciding_face.py create mode 100644 papers/face_monochromatic_pairs/experiments/check_kb_kc_coverage.py diff --git a/papers/face_monochromatic_pairs/experiments/check_deciding_face.py b/papers/face_monochromatic_pairs/experiments/check_deciding_face.py new file mode 100644 index 0000000..47c78c0 --- /dev/null +++ b/papers/face_monochromatic_pairs/experiments/check_deciding_face.py @@ -0,0 +1,161 @@ +"""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() diff --git a/papers/face_monochromatic_pairs/experiments/check_kb_kc_coverage.py b/papers/face_monochromatic_pairs/experiments/check_kb_kc_coverage.py new file mode 100644 index 0000000..bfa2ec0 --- /dev/null +++ b/papers/face_monochromatic_pairs/experiments/check_kb_kc_coverage.py @@ -0,0 +1,132 @@ +"""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}: 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b/papers/face_monochromatic_pairs/paper.tex index 619ccfe..a4035c0 100644 --- a/papers/face_monochromatic_pairs/paper.tex +++ b/papers/face_monochromatic_pairs/paper.tex @@ -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