From d87b94d3b3c73b0af1dcec02d1b9244d28e88899 Mon Sep 17 00:00:00 2001 From: didericis Date: Mon, 25 May 2026 04:29:47 -0400 Subject: [PATCH] face_monochromatic_pairs: partial structural proof of Conjecture (Deciding face) MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Adds Definition (flank face) + 3 lemmas + a partial theorem proving Conjecture (Deciding face) -- hence Conjecture 5.1 -- for the case where at least one neighbour of v in the parent triangulation G has degree ≤ 6: - Lemma (Flank-length formula): |F_{i, i+1}^♭| = n_i - 1. - Lemma (Flank covering, n_i = 5): boundary of F_{i, i+1}^♭ is in V(K_b) ∪ V(K_c). Proof: the single intermediate P is adjacent to both A_i and A_{i+1}; A_{i+1} ∈ V(K_b) ∩ V(K_c) via spike, so A_{i+1}'s c_0-edge (in K_b) or c_1-edge (in K_c) lands on P. - Lemma (Flank covering, n_i = 6): two intermediates P_1, P_2; P_2 handled as the n_i = 5 case; P_1 covered by case analysis on φ(A_i P_1) ∈ {c, c_1}: in Case (a) K_b walks A_i → P_1 directly; in Case (b) propagation from P_2 via P_2's c-edge to P_1 (forced by properness at P_1 ruling out φ(P_1 P_2) = c_1). - Theorem (Partial proof of Conjecture (Deciding face)): combining the length formula and the covering lemmas, F_{i, i+1}^♭ is a deciding face whenever n_i ∈ {5, 6}, since its length is then 4 or 5 (≢ 0 mod 3). The remaining structural case is n_i ≥ 7 for all i (= all five neighbours of v in G have degree ≥ 7); for these the merged-side face F^♭_{i+3, i+4} of length n_{i+3} - 2 often plays the deciding role empirically, but a uniform structural argument is left open. Paper grows from 18 to 20 pages. Co-Authored-By: Claude Opus 4.7 --- papers/face_monochromatic_pairs/paper.pdf | Bin 1260470 -> 1272485 bytes papers/face_monochromatic_pairs/paper.tex | 163 ++++++++++++++++++++++ 2 files changed, 163 insertions(+) diff --git a/papers/face_monochromatic_pairs/paper.pdf b/papers/face_monochromatic_pairs/paper.pdf index 99348b8824e97caefc6f65dc028e0fbfcb360f0a..c8e1c49303d369e1efe7e75b519f89772aa57be9 100644 GIT binary patch delta 136631 zcmV)GK)%1W_e`bLO^_v*(WwzBu})+Re`J{#T=1PDro*bL7z%Eh%v*CWgBd}kjEnY<;9q9W4&zpB%x1%@I%&YX>i3fh^95TB0u6!Ce8FjuO|G#5UE;TcSv^W- zALjicBvnk36P4$Qk&m#MoqS(H1N|S?zh-ZP>kWtN4FQMi4FZSj4FiYk4Frel4F!km z4F-qn4F`wo4G4$p4GD+q4GOpG4GZ7d0oS+7=L_Tn1UrUvT!*vl3x~7p47aoF4M08u zIWd>P(g!Jj)m&S1<2IIl&#&;#OX!M)A+7|{m8$LZ#5>uUOm`){J6o0P11-@uJF@7K zU_HKjPi%A6}k+I+4Bo{?+MB=Xv%9?_PfY<4<(^ z@*jV}0?*$5^*gFL_y78zAF2A)>%WI`j!j!YNS~(c4C(E3D zh^<+nPG+1br>W7GiCJ5gOLsD8v~B3k7jUsznAE&kZMHO{Ep~#$f`2*#{d4BDq5n3k z#7MbhRu=6*|2I7bq1y&XlteGk3Ct$6`CDRtww5U%ho7#ZRu_p^DVFE_WYEZypW2y0 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zyfgx?NIS@i@qwS6Nwy%RoSwNLF){8(K}IKCONwd!DqY0e~ula-oef;WIRkr*Lzh(=5VEAX*R#af) z5qz?jHr+_p0R-Ia*3~o4*W~;_?@DmfAH_x(JIpPHt()#`8DWc+{943k{Zp&+GGpi0 zGA$G-HEtioOXKy&7^*r(4ONx@x?B6R4BISE&cj7ArJ1R<4b*z_N^F(%i)Yyq@*{6hSK+_=olvZ`{p{|7mO|FHl7 diff --git a/papers/face_monochromatic_pairs/paper.tex b/papers/face_monochromatic_pairs/paper.tex index a4035c0..6482a43 100644 --- a/papers/face_monochromatic_pairs/paper.tex +++ b/papers/face_monochromatic_pairs/paper.tex @@ -957,6 +957,169 @@ $\varepsilon \equiv 0 \pmod 3$. But $\varepsilon \in \{+1, -1\}$, so $\varepsilon \not\equiv 0 \pmod 3$ --- contradiction. \end{proof} +\subsection*{A partial structural proof of +Conjecture~\ref{conj:deciding-face}} + +We single out a specific candidate face --- the ``flank'' face that +the spike, side-$0$ edge, and one boundary arc of the new pentagonal +hole together bound --- and prove the deciding-face property for it +when the adjacent $G'$-face is pentagonal or hexagonal. This is the +case for every reduced dual where at least one of $v$'s neighbours +$B_i$ or $B_{i+1}$ in the parent triangulation $G$ has degree $5$ or +$6$ --- e.g., for the icosahedron and the dodecahedron's other near +neighbours. + +\begin{definition}[Flank face] +\label{def:flank-face} +Fix a reduced dual $\widehat{G}'_{v,i}$ and let $F$ be the post-deletion +pentagonal hole. Order the five outer endpoints +$A_0, A_1, A_2, A_3, A_4$ in clockwise order along +$\partial F$ as in Definition~\ref{def:reduced-dual}. The +\emph{lower flank face} $F_{i, i+1}^{\flat}$ of +$\widehat{G}'_{v,i}$ is the face of $\widehat{G}'_{v,i}$ whose +boundary is the closed walk +\[ +v_n +\;\xrightarrow{\text{side-}0}\; +A_i +\;\xrightarrow{\partial F \text{-arc}}\; +A_{i+1} +\;\xrightarrow{\text{spike}}\; +v_n, +\] +i.e.\ the side-$0$ edge, followed by the maximal sub-arc of +$\partial F$ from $A_i$ to $A_{i+1}$ not crossing $v_n$ or the +merged edge, followed by the spike edge in reverse. +The \emph{upper flank face} $F_{i+1, i+2}^{\flat}$ is defined +analogously with side-$1$ replacing side-$0$ and $A_{i+1} \to A_{i+2}$ +replacing $A_i \to A_{i+1}$. +\end{definition} + +\begin{lemma}[Flank-length formula] +\label{lem:flank-length} +$|F_{i, i+1}^{\flat}| \;=\; n_i - 1$, where $n_i$ is the length of +the $G'$-face $F_i$ adjacent to $F_v$ across the edge +$(B_i, B_{i+1})$ in the pre-reduction dual $G'$. (Symmetrically, +$|F_{i+1, i+2}^{\flat}| = n_{i+1} - 1$.) +\end{lemma} + +\begin{proof} +After step (1) of Definition~\ref{def:reduced-dual} the dual face +$F_i$, originally a closed walk of length $n_i$ around the boundary +edges $(B_i, B_{i+1})$, $(B_{i+1}, A_{i+1})$, $\partial F_i$ +interior, and $(A_i, B_i)$, becomes (after deletion of +$B_i, B_{i+1}$ and their three incident edges) an open arc from +$A_{i+1}$ to $A_i$ of length $n_i - 3$ in the post-deletion +graph. The lower flank face $F_{i, i+1}^{\flat}$ adds back the +side-$0$ edge $(v_n, A_i)$ (length $1$) and the spike edge +$(v_n, A_{i+1})$ (length $1$). Its total boundary length is therefore +$(n_i - 3) + 1 + 1 = n_i - 1$. +\end{proof} + +\begin{lemma}[Flank covering, base case $n_i = 5$] +\label{lem:flank-covering-base} +If $n_i = 5$ then $\partial F_{i, i+1}^{\flat} \subseteq V(K_b) \cup V(K_c)$. +\end{lemma} + +\begin{proof} +When $n_i = 5$ the boundary of $F_{i, i+1}^{\flat}$ visits exactly four +vertices: $v_n$, $A_i$, a single intermediate $P$, and $A_{i+1}$. We +have $v_n, A_i, A_{i+1} \in V(K_b) \cup V(K_c)$ by +Lemma~\ref{lem:kempe-spike}, so it remains to show $P \in V(K_b) \cup V(K_c)$. + +The vertex $A_{i+1}$ has degree $3$, with one of its three incident +edges being the spike (colour $c$) and the other two having colours +$\{c_0, c_1\}$ (one each, by properness). Lemma~\ref{lem:kempe-spike} +gives $A_{i+1} \in V(K_b) \cap V(K_c)$, so $K_b$ uses the spike +together with $A_{i+1}$'s colour-$c_0$ edge, and $K_c$ uses the spike +together with $A_{i+1}$'s colour-$c_1$ edge. The two non-spike edges +at $A_{i+1}$ are exactly $A_{i+1} P$ (the boundary edge into the lower +flank) and $A_{i+1} P'$ (the boundary edge into the upper flank, +$P' \in F_{i+1, i+2}^{\flat}$). Whichever of these is $A_{i+1} P$, +its colour is in $\{c_0, c_1\}$, so the corresponding Kempe cycle +($K_b$ if colour $c_0$, $K_c$ if colour $c_1$) walks $A_{i+1} \to P$, +placing $P$ in that cycle's vertex set. +\end{proof} + +\begin{lemma}[Flank covering, $n_i = 6$] +\label{lem:flank-covering-hex} +If $n_i = 6$ then $\partial F_{i, i+1}^{\flat} \subseteq V(K_b) \cup V(K_c)$. +\end{lemma} + +\begin{proof} +The boundary now visits five vertices: $v_n$, $A_i$, $P_1, P_2$, +$A_{i+1}$, with $P_1$ adjacent to $A_i$ and $P_2$ adjacent to +$A_{i+1}$. The named vertices and $P_2$ are in $V(K_b) \cup V(K_c)$ +by Lemma~\ref{lem:flank-covering-base}'s argument applied at +$A_{i+1}$. It remains to place $P_1$. + +\emph{Case (a):} $\varphi(A_i P_1) = c$. Then $K_b$'s walk uses +$A_i$'s colour-$c$ edge (along with side-$0$), so $K_b$ walks +$A_i \to P_1$ and $P_1 \in V(K_b)$. + +\emph{Case (b):} $\varphi(A_i P_1) = c_1$. Then $A_i$'s colour-$c$ +edge is the \emph{other} non-side-$0$ edge of $A_i$, namely $A_i Q$ +where $Q$ is the intermediate on $\partial F_{i-1, i}^{\flat}$ +adjacent to $A_i$. Now consider $P_2$. By the analysis above we have +$\varphi(A_{i+1} P_2) \in \{c_0, c_1\}$, and properness at $P_2$ +gives $\varphi(P_2)$'s three incident edges colours $\{c_0, c, c_1\}$ +in some order, with $\varphi(A_{i+1} P_2)$ accounting for one of +$\{c_0, c_1\}$. The remaining two colours, $c$ and $(c_1$ or $c_0)$, +are split between $\varphi(P_1 P_2)$ and $\varphi(P_2 O_2)$ where +$O_2$ is $P_2$'s third (outer) neighbour. Properness at $P_1$ (which +has $\varphi(A_i P_1) = c_1$ already by Case (b) hypothesis) forbids +$\varphi(P_1 P_2) = c_1$. Hence +\[ +\varphi(P_1 P_2) \;\in\; \{c, c_0\}. +\] +Now $P_2 \in V(K_b) \cup V(K_c)$ as shown, and at $P_2$ the cycle that +contains $P_2$ uses two specific edges of $P_2$. If $P_2 \in V(K_b)$ +the cycle uses $P_2$'s colour-$c$ and colour-$c_0$ edges; if +$P_2 \in V(K_c)$ it uses colour-$c$ and colour-$c_1$. In either case +the cycle at $P_2$ uses $P_2$'s colour-$c$ edge. By the +preceding paragraph $\varphi(P_1 P_2) \in \{c, c_0\}$, so the +$\{c, \varphi(P_2 P_1)\}$-Kempe cycle through $P_2$ passes from $P_2$ +to $P_1$. If this cycle is $K_b$, $P_1 \in V(K_b)$; if it is $K_c$ +(which requires $\varphi(P_1 P_2) = c$, since $K_c$ uses +$\{c, c_1\}$), then $P_1 \in V(K_c)$. + +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}] +\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 +$F_{j, j+1}^{\flat}$. +\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} +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] +\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 +$\equiv 0 \pmod 3$. +\end{remark} + \begin{remark}[Empirical verification of Conjecture~\ref{conj:deciding-face}] \label{rem:deciding-face-empirical}