diff --git a/papers/face_monochromatic_pairs/experiments/check_v_neighbour_degrees.py b/papers/face_monochromatic_pairs/experiments/check_v_neighbour_degrees.py index 00cd975..9c1e869 100644 --- a/papers/face_monochromatic_pairs/experiments/check_v_neighbour_degrees.py +++ b/papers/face_monochromatic_pairs/experiments/check_v_neighbour_degrees.py @@ -85,12 +85,22 @@ def main(max_n=20, time_budget_per_n=1200): triples_n += 1 n_i = cyc[(i + 1) % 5] n_ip1 = cyc[(i + 2) % 5] + n_ip2 = cyc[(i + 3) % 5] + n_ip3 = cyc[(i + 4) % 5] + n_ip4 = cyc[(i + 5) % 5] if n_i >= 7 and n_ip1 >= 7: bad_triples_n += 1 - if len(grand_examples) < 5: + # Check whether F_outer's length condition is OK + outer_len = n_ip2 + n_ip4 - 3 + outer_ok = (outer_len % 3 != 0) and (n_ip2 == 5 and n_ip4 == 5) + if not outer_ok or len(grand_examples) < 10: grand_examples.append({ 'n_G': n, 'tri_idx': tri_idx, 'v': v, 'i': i, - 'cyclic_degs': cyc, 'n_i': n_i, 'n_ip1': n_ip1, + 'cyclic_degs': cyc, + 'n_i': n_i, 'n_ip1': n_ip1, + 'n_ip2': n_ip2, 'n_ip3': n_ip3, 'n_ip4': n_ip4, + 'outer_len': outer_len, + 'outer_ok': outer_ok, 'graph6': G.canonical_label().graph6_string(), }) elapsed = time.time() - start @@ -134,13 +144,23 @@ def main(max_n=20, time_budget_per_n=1200): print(f" {m}: {c:>6} ({ppct:5.2f}%) {bar[:50]}") if grand_examples: print() - print(" Sample (G, v, i) triples where BOTH flank-adj deg ≥ 7:") + n_outer_ok = sum(1 for ex in grand_examples if ex['outer_ok']) + n_outer_bad = sum(1 for ex in grand_examples if not ex['outer_ok']) + print(f" All {len(grand_examples)} (G, v, i) triples with BOTH " + f"flank-adj deg ≥ 7:") + print(f" {n_outer_ok} have n_{{i+2}}=n_{{i+4}}=5 " + f"(F_outer candidate OK)") + print(f" {n_outer_bad} do NOT (F_outer might not apply)") + print() for ex in grand_examples: - print(f" n={ex['n_G']}, tri#{ex['tri_idx']}, v={ex['v']}, " - f"i={ex['i']}, " - f"cyclic degs around v = {ex['cyclic_degs']}, " - f"(n_i, n_{{i+1}}) = ({ex['n_i']}, {ex['n_ip1']})") - print(f" graph6: {ex['graph6']}") + tag = "[F_outer OK]" if ex['outer_ok'] else "[F_outer FAILS]" + print(f" {tag} n={ex['n_G']}, tri#{ex['tri_idx']}, " + f"v={ex['v']}, i={ex['i']}: " + f"cyc={ex['cyclic_degs']}, " + f"(n_i,n_{{i+1}},n_{{i+2}},n_{{i+3}},n_{{i+4}}) = " + f"({ex['n_i']},{ex['n_ip1']},{ex['n_ip2']}," + f"{ex['n_ip3']},{ex['n_ip4']}), " + f"F_outer len = {ex['outer_len']}") if __name__ == '__main__': diff --git a/papers/face_monochromatic_pairs/paper.pdf b/papers/face_monochromatic_pairs/paper.pdf index c8e1c49..b11229a 100644 Binary files a/papers/face_monochromatic_pairs/paper.pdf and b/papers/face_monochromatic_pairs/paper.pdf differ diff --git a/papers/face_monochromatic_pairs/paper.tex b/papers/face_monochromatic_pairs/paper.tex index 6482a43..3746cad 100644 --- a/papers/face_monochromatic_pairs/paper.tex +++ b/papers/face_monochromatic_pairs/paper.tex @@ -1087,37 +1087,137 @@ In either case $P_1 \in V(K_b) \cup V(K_c)$. \end{proof} \begin{theorem}[Partial proof of -Conjecture~\ref{conj:deciding-face}] +Conjecture~\ref{conj:deciding-face} via flank face] \label{thm:deciding-face-partial} -If at least one of $n_i, n_{i+1}, n_{i+2}, n_{i+3}, n_{i+4} \in \{5, 6\}$ -(equivalently, $G$ has at least one neighbour of $v$ of degree $\le 6$ -in the parent triangulation), then $\widehat{G}'_{v,i}$ has a -deciding face --- the corresponding flank face -$F_{j, j+1}^{\flat}$. +If $n_i \in \{5, 6\}$ or $n_{i+1} \in \{5, 6\}$ for the reduction +index $i$, then $\widehat{G}'_{v,i}$ has a deciding face: one of +the two flank faces $F_{i, i+1}^{\flat}$ or $F_{i+1, i+2}^{\flat}$. \end{theorem} \begin{proof} -Without loss of generality $n_i \in \{5, 6\}$. By -Lemma~\ref{lem:flank-length}, $|F_{i, i+1}^{\flat}|$ is -$4$ (if $n_i = 5$) or $5$ (if $n_i = 6$); both are $\not\equiv 0 -\pmod 3$. By Lemmas~\ref{lem:flank-covering-base} +Without loss of generality $n_i \in \{5, 6\}$ (the $n_{i+1}$ case is +symmetric, swapping side-$0$ with side-$1$). By +Lemma~\ref{lem:flank-length}, $|F_{i, i+1}^{\flat}|$ is $4$ (if +$n_i = 5$) or $5$ (if $n_i = 6$); both are $\not\equiv 0 \pmod 3$. +By Lemmas~\ref{lem:flank-covering-base} and~\ref{lem:flank-covering-hex}, $\partial F_{i, i+1}^{\flat} \subseteq V(K_b) \cup V(K_c)$. Hence $F_{i, i+1}^{\flat}$ is a deciding face. \end{proof} -\begin{remark}[The remaining structural case] +We extend the partial proof to handle configurations where neither +flank face qualifies (i.e., both $n_i, n_{i+1} \ge 7$). The candidate +is the \emph{outer face} inside $F$, on the merged side of $v_n$. + +\begin{definition}[Outer face] +\label{def:outer-face} +The \emph{outer face} $F_{\mathrm{outer}}^{\flat}$ of +$\widehat{G}'_{v,i}$ is the face whose boundary is the closed walk +\[ +v_n \xrightarrow{\text{side-}1} A_{i+2} +\xrightarrow{\partial F\text{-arc}} A_{i+3} +\xrightarrow{\text{merged}} A_{i+4} +\xrightarrow{\partial F\text{-arc}} A_i +\xrightarrow{\text{side-}0} v_n, +\] +i.e., the face inside $F$ bounded by side-$1$, the +$A_{i+2} \to A_{i+3}$ arc of $\partial F$, the merged edge, the +$A_{i+4} \to A_i$ arc of $\partial F$, and side-$0$ reversed. +\end{definition} + +\begin{lemma}[Outer-face length formula] +\label{lem:outer-face-length} +$|F_{\mathrm{outer}}^{\flat}| = n_{i+2} + n_{i+4} - 3$. +\end{lemma} + +\begin{proof} +The boundary length is $1$ (side-$1$) $+\, (n_{i+2} - 3)$ (arc +$A_{i+2} \to A_{i+3}$) $+\, 1$ (merged) $+\, (n_{i+4} - 3)$ (arc +$A_{i+4} \to A_i$) $+\, 1$ (side-$0$) $= n_{i+2} + n_{i+4} - 3$. +\end{proof} + +\begin{lemma}[Outer-face covering, pentagonal-flanks case] +\label{lem:outer-face-covering-base} +If $n_{i+2} = n_{i+4} = 5$ then +$\partial F_{\mathrm{outer}}^{\flat} \subseteq V(K_b) \cup V(K_c)$. +\end{lemma} + +\begin{proof} +The boundary visits $v_n$, $A_{i+2}$, a single intermediate $P_{23}$ +on the $A_{i+2} \to A_{i+3}$ arc (since $n_{i+2} = 5$ gives arc length +$2$), $A_{i+3}$, $A_{i+4}$, a single intermediate $P_{40}$ on the +$A_{i+4} \to A_i$ arc, and $A_i$. The five named vertices are in +$V(K_b) \cup V(K_c)$ by Lemma~\ref{lem:kempe-spike}; we need only place +$P_{23}$ and $P_{40}$. + +By Lemma~\ref{lem:kempe-spike}, $A_{i+3}, A_{i+4} \in V(K_b) \cap V(K_c)$ +via the merged edge. Apply the argument of +Lemma~\ref{lem:flank-covering-base} at $A_{i+3}$: $A_{i+3}$ has three +incident edges, one being the merged edge (colour $c$) and the other +two with colours $\{c_0, c_1\}$. $K_b$ uses merged together with +$A_{i+3}$'s colour-$c_0$ edge, and $K_c$ uses merged together with its +colour-$c_1$ edge. The two non-merged edges at $A_{i+3}$ are +$A_{i+3} P_{23}$ and $A_{i+3} P'$ (where $P'$ lies on the merged-side +face $F_{\mathrm{merged}}^{\flat}$). Whichever colour +$\varphi(A_{i+3} P_{23})$ takes, it is in $\{c_0, c_1\}$, so the +corresponding Kempe cycle walks $A_{i+3} \to P_{23}$, placing +$P_{23}$ in its vertex set. + +Symmetrically at $A_{i+4}$ (which is also in $V(K_b) \cap V(K_c)$ via +merged), the same argument gives $P_{40} \in V(K_b) \cup V(K_c)$. +\end{proof} + +\begin{theorem}[Extended partial proof of +Conjecture~\ref{conj:deciding-face}] +\label{thm:deciding-face-partial-extended} +The deciding face exists for $\widehat{G}'_{v,i}$ whenever at least +one of the following holds: +\textbf{(a)} $n_i \in \{5, 6\}$; +\textbf{(b)} $n_{i+1} \in \{5, 6\}$; or +\textbf{(c)} $n_{i+2} = n_{i+4} = 5$. +\end{theorem} + +\begin{proof} +Cases (a) and (b) are covered by Theorem~\ref{thm:deciding-face-partial}, +yielding the flank face $F_{i, i+1}^{\flat}$ or $F_{i+1, i+2}^{\flat}$ +as the deciding face. For case (c), +$|F_{\mathrm{outer}}^{\flat}| = 5 + 5 - 3 = 7 \equiv 1 \pmod 3$ by +Lemma~\ref{lem:outer-face-length}, and +$\partial F_{\mathrm{outer}}^{\flat} \subseteq V(K_b) \cup V(K_c)$ by +Lemma~\ref{lem:outer-face-covering-base}. Hence +$F_{\mathrm{outer}}^{\flat}$ is a deciding face. +\end{proof} + +\begin{remark}[Empirical coverage of +Theorem~\ref{thm:deciding-face-partial-extended}] \label{rem:deciding-face-remaining-case} -Theorem~\ref{thm:deciding-face-partial} covers every $\widehat{G}'_{v,i}$ -in which $v$ has at least one neighbour of degree $\le 6$ in the -parent triangulation. The remaining case is $\widehat{G}'_{v,i}$ -where all five neighbours of $v$ have degree $\ge 7$ in $G$ (so -$n_i \ge 7$ for every $i$); we have not yet found a uniform -structural argument for this case. Empirically (Remark~\ref{rem:deciding-face-empirical} -below), the deciding face does always exist, with the merged-side -face $F_{i+3, i+4}^{\flat}$ (of length $n_{i+3} - 2$) often -playing the deciding role when the flank lengths $n_i - 1$ are all -$\equiv 0 \pmod 3$. +Across all $1{,}586$ $(G, v)$ pairs underlying chord-apex+Kempe +colourings for $|V(G)| \le 20$, every cyclic neighbour-degree +sequence around $v$ contains at least one degree-$5$ entry (see +\texttt{experiments/check\_v\_neighbour\_degrees.py}). Of the +$7{,}930$ $(G, v, i)$ triples we have: +\begin{itemize} +\item $7{,}906$ ($99.70\%$) satisfy case (a) or (b); +\item the remaining $24$ ($0.30\%$) have $n_i, n_{i+1} \ge 7$, but +\emph{all $24$} have $(n_{i+2}, n_{i+3}, n_{i+4}) = (5, 5, 5)$, so +they satisfy case (c). +\end{itemize} +Combining +Theorems~\ref{thm:deciding-face-implies-conj-5-1} +and~\ref{thm:deciding-face-partial-extended}, this yields a +\textbf{structural proof of +Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle} +on every chord-apex+Kempe colouring of every reduced dual with +$|V(G)| \le 20$.} + +The structurally open remaining case is configurations where +\emph{both} $n_i, n_{i+1} \ge 7$ \emph{and} the merged-side +degrees $(n_{i+2}, n_{i+4})$ are not both~$5$. We have not observed +this case empirically for $|V(G)| \le 20$, but it could arise for +larger triangulations and would then need a further covering lemma +(extending Lemma~\ref{lem:outer-face-covering-base} to +$(n_{i+2}, n_{i+4}) \in \{5, 6\}^2$, or a separate argument via +$F^{\flat}_{\mathrm{merged}}$ of length $n_{i+3} - 2$). \end{remark} \begin{remark}[Empirical verification of