From a36271a2cf8346b7b3f605d8806c0e109ea29f0d Mon Sep 17 00:00:00 2001 From: didericis Date: Mon, 25 May 2026 02:00:23 -0400 Subject: [PATCH] face_monochromatic_pairs: extend Theorem 5.5 proof attempt with Step 2 + Step 3 MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit - Step 1: local CW structure at u, w (F_R = F_{ab}^u = F_{ca}^w, etc.) - Step 2 (new, complete): K_1 \ e leaves u into In(K_0) and arrives at w from Out(K_0), so it crosses K_0 an odd number of times. Each crossing uses a shared a-edge, so |E(K_0) ∩ E(K_1)| is even and ≥ 2. Closes the case of a single shared edge. - Step 3 (new, complete): Heawood's face-sum identity ∑ h_φ ≡ 0 (mod 3) applied to every H-face inside a face Φ of K_0 ∪ K_1, with multiplicity bookkeeping at degree-3 vs degree-2 boundary vertices, yields ν_{2,Φ} ≡ -ℓ_Φ (mod 3) for every face Φ. - Step 4 (open): use Lemma 5.2 alternation to force ν_{2,Φ} on some face, and exhibit a violation. Literature search via research-analyst confirms the theorem is novel (Heawood face-sum and the +/-1 rotation sign are classical; no result relating intersecting Kempe cycles via vertex signs found). Co-Authored-By: Claude Opus 4.7 --- papers/face_monochromatic_pairs/paper.pdf | Bin 1090606 -> 1109009 bytes papers/face_monochromatic_pairs/paper.tex | 147 ++++++++++++++++++---- 2 files changed, 122 insertions(+), 25 deletions(-) diff --git a/papers/face_monochromatic_pairs/paper.pdf b/papers/face_monochromatic_pairs/paper.pdf index 63b80c7ea4f484c11c2c12cb5ec1dba80a3ef576..beaf4fc47df3f08820d932c815cfd03bb9313cd5 100644 GIT binary patch delta 137616 zcmV)FK)=7P$U>3WLx6+rbEUOnwZS|An@xh)0SOiJT6su?yN)+GBk9S07Vr8JJM3IvE z!U7V>Tq0xnA~G`n^!~d){J{N*$-R~C}x>P7Z{{^u8`3(Z*eE51Gd`R%*-`uwkd#s{9g|LJ8c`62w* z-`>XJ7jIstXNFbYynXr0=|Wj)GcHg6`~6@3@Pl9{KKDi#9NR)M7^gj12$_2xhxY!e zgnw~1yDsa?@;pA5=a-?>WoM@g0q?8L`f?k;HJjm=O>-5Cmv_t41~>Y9@ov^>hH@}N zcCz3+_u9nC{TD$5mos=^vBE!@W<)bGcUHA^UZo0Ru*xo1MnGS=@+#IkBf3&KWBg=L z3A*E{o*C3LIdi@n5|@g97?R7K)@_|9ofjX?Ye~;b`+M_RNGWmAq*U3(re2n@q`7Jz zh6|4c%guV*6y2KR|25sGYQ3z^%RKfD^3?Y(oXxrGXUY<*t074jQUk9u0{?Uk{YWkjvlvuMoe4eQ zjo6vY4Nua=2nGTHG`V1@nGZlLlRc}5p~DjoBlB`J{L#n^uwzH4-;=F4-x1&}P!bu) zXT>)Df=Yp3m#BPyeO<(2olDS9Qpv1XpU0vVy#_+%@}A7-ah1H6%rpk51uUqblJ`j^ zB^E3z5QsYVy}7_&l|%#zn%hk*d4+29`lg)_lmy<}cK=6=RN(^cCzh8TJ2h7|w!Yb2 zUZuXTQcqwAPPNFu0>Jn4SQeizK^_-M>+EN80D|WZv|{Ri;ELc5odYR{+Jzn_A8VH? zO?$m*5-q~WgYsx1sfAa$786pF7J@5aJbOry>LV1Xo+?r{o`jWyMmRCiQ6s>PN-YsCu5MN(rKU`-t^E{*|3k2{&%inxlcgrd*ag-{kJg71ZK@!YgG!-AOSb59 zX6&d%LE4jlNsBWCgO%|e18<9_+FsmM>&r-}<*L|iQ9|$q$-%Le|HCI4?UNEwxoOb0s@XUVf~$%{Gss3ltR=F^9{gOp^P5Yt&8Q&qYr3DS0kiKB|{MCi6c^p5SFSN~*OmUWOO?DD6wz)oAZ*8M$?^~1z=*NEtn zhfpKn7cSWmY$CwBdbezhc;GkUL;SO&JML3rq}+#ESvwM)_C! 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If $h_\varphi$ is constant on $V(K_0)$, then $h_\varphi$ is \emph{not} constant on $V(K_1)$. \end{theorem} -\begin{proof}[Proof sketch] +\begin{proof}[Proof attempt] Suppose for contradiction that $h_\varphi$ is constant on both $V(K_0)$ and $V(K_1)$, and that $K_0, K_1$ share a colour-$a$ edge -$e = (u, w)$, so that $u, w \in V(K_0) \cap V(K_1)$ are consecutive on -both cycles. +$e = (u, w)$. Then $u, w \in V(K_0) \cap V(K_1)$ are consecutive on +both cycles. Since the two constants agree at the shared vertex $u$, +they agree everywhere on $V(K_0) \cup V(K_1)$; WLOG +$h_\varphi \equiv +1$ on $V(K_0) \cup V(K_1)$, so the CW edge order at +every such vertex is $(a, b, c)$. -By Lemma~\ref{lem:kempe-heawood-constant} applied to $K_0$: at every -consecutive pair of $K_0$-vertices the colour-$c$ non-cycle edges lie -on opposite local sides of $K_0$. In particular the colour-$c$ edges -at $u$ and $w$ lie on opposite sides of $K_0$. +Orient both cycles so that they leave $u$ along $e$: write the $K_0$ +walk as $u = v_0 \xrightarrow{e} w = v_1 \to v_2 \to \cdots \to +v_{L_0 - 1} \to v_0$, and the $K_1$ walk as $u = u_0 \xrightarrow{e} +w = u_1 \to u_2 \to \cdots \to u_{L_1 - 1} \to u_0$. Let +$F_R(e), F_L(e)$ be the two faces of $H$ incident to $e$, with $F_R$ +on the right and $F_L$ on the left of the $u \to w$ traversal. -By Lemma~\ref{lem:kempe-heawood-constant} applied to $K_1$: at every -consecutive pair of $K_1$-vertices the colour-$b$ non-cycle edges lie -on opposite local sides of $K_1$. In particular the colour-$b$ edges -at $u$ and $w$ lie on opposite sides of $K_1$. +\textbf{Step 1 (local sides at $u, w$).} The CW-order $(a, b, c)$ at +$u$ and $w$ partitions the faces of $H$ at each endpoint into three +wedges. Direct inspection (cf.\ the proof of +Lemma~\ref{lem:kempe-heawood-constant}) gives +\[ +F_R(e) = F_{ab}^u = F_{ca}^w, \qquad F_L(e) = F_{ca}^u = F_{ab}^w, +\] +where $F_{\alpha\beta}^v$ is the face of $H$ at $v$ in the CW wedge +between the colour-$\alpha$ and colour-$\beta$ edges. Consequently: +\begin{itemize} +\item $e_c^u$ lies between $F_{bc}^u$ and $F_{ca}^u = F_L(e)$, so $e_c^u$ +is on the side of $K_0$ containing $F_L(e)$ near $u$; call this side +$\mathrm{In}(K_0)$. +\item $e_c^w$ lies between $F_{bc}^w$ and $F_{ca}^w = F_R(e)$, so +$e_c^w$ is on the side $\mathrm{Out}(K_0)$ containing $F_R(e)$ near $w$. +\item Symmetrically, $e_b^u$ is on the side of $K_1$ containing +$F_R(e)$, call this $\mathrm{Out}(K_1)$, and $e_b^w$ is on the side +$\mathrm{In}(K_1)$ containing $F_L(e)$. +\end{itemize} +This recovers exactly the conclusion of +Lemma~\ref{lem:kempe-heawood-constant} applied to $(u, w)$ on each +cycle. -Now $K_0$ and $K_1$ share the arc $e$ in the plane, so $K_0 \cup K_1$ -is either a single closed curve (if $K_0 = K_1$, which is impossible -since they use different colour pairs) or a theta-curve based at -$\{u, w\}$. In the latter case the colour-$b$ edges at $u, w$ together -with $K_0 \setminus e$ form one side of the theta, and the colour-$c$ -edges at $u, w$ together with $K_1 \setminus e$ form another side. -The combined opposite-sides conditions above force $K_0$ and $K_1$ to -both wind around the same way at $e$ --- which a planar theta-curve -cannot realise. +\textbf{Step 2 (forced crossings).} Consider $K_1 \setminus e$, the +path from $u$ to $w$ obtained by removing the open edge $e$ from +$K_1$. By Step~1, this path leaves $u$ on the $\mathrm{In}(K_0)$ side +and arrives at $w$ on the $\mathrm{Out}(K_0)$ side. Since $K_0$ +separates the plane into its two sides and $K_1 \setminus e$ is a +continuous arc, the path must intersect $V(K_0)$ at an odd number of +points strictly between $u$ and $w$ along the $K_1$-walk. -\textbf{(Full proof to be filled in.)} The cleanest formalisation is -likely via a winding-number / orientation argument on the theta-curve -$K_0 \cup K_1$ in the plane, combined with the local CW-order -constraints at $u$ and $w$ that Lemma~\ref{lem:kempe-heawood-constant} -forces. +At any such intersection $x \in V(K_0) \cap V(K_1) \setminus \{u, w\}$, +$K_1$ uses the colour-$a$ edge at $x$ (since $K_1$ uses only colours +$a, c$ and only colour-$a$ edges can lie on $K_0$). That colour-$a$ +edge is therefore a shared edge $e^* \in E(K_0) \cap E(K_1)$ with +$e^* \neq e$; both endpoints of $e^*$ lie in $V(K_0) \cap V(K_1)$. + +So $|E(K_0) \cap E(K_1)| - 1$ equals the (odd) number of crossings, +giving +\[ +|E(K_0) \cap E(K_1)| \text{ is even, and } \geq 2. +\] +The symmetric argument applied to $K_0 \setminus e$ crossing $K_1$ +yields the same conclusion. + +\textbf{Step 3 (Heawood face-sum on each face of $K_0 \cup K_1$).} +View $K_0 \cup K_1$ as a planar subgraph of $H$ and consider any face +$\Phi$ of this subgraph. Let $F_\Phi$ be the set of $H$-faces lying +inside (the closed region of) $\Phi$. Applying Heawood's classical +face-sum identity +$\sum_{v \in \partial f} h_\varphi(v) \equiv 0 \pmod 3$ +\cite{Heawood1898} +to every $f \in F_\Phi$ and summing gives +\[ +\sum_{f \in F_\Phi} \sum_{v \in \partial f} h_\varphi(v) \;=\; +\sum_{v} \mathrm{mult}_\Phi(v)\, h_\varphi(v) \;\equiv\; 0 \pmod 3, +\] +where $\mathrm{mult}_\Phi(v)$ counts the number of $H$-faces in +$F_\Phi$ whose boundary contains $v$. + +A direct case-check on the cubic vertex structure gives: +\begin{itemize} +\item $\mathrm{mult}_\Phi(v) = 3$ if $v$ is strictly interior to $\Phi$ +(all three $H$-faces at $v$ lie in $F_\Phi$); +\item $\mathrm{mult}_\Phi(v) = 1$ if $v$ is a degree-$3$ +(shared, branching) vertex of $K_0 \cup K_1$ on $\partial\Phi$ (only +one of $v$'s three wedges lies in $\Phi$); +\item $\mathrm{mult}_\Phi(v) = 2$ if $v$ is a degree-$2$ (non-shared) +vertex of $K_0 \cup K_1$ on $\partial\Phi$ \emph{and} $v$'s third edge +points into $\Phi$ (the third edge subdivides $v$'s wedge in $\Phi$ +into two $H$-faces); +\item $\mathrm{mult}_\Phi(v) = 1$ if $v$ is a degree-$2$ non-shared +boundary vertex with its third edge pointing into the opposite face. +\end{itemize} + +Under the contradiction hypothesis $h_\varphi \equiv +1$ on +$V(K_0) \cup V(K_1) \supseteq \partial\Phi$, the boundary contribution +collapses to +\[ +\sigma_\Phi + \nu_{1,\Phi} + 2\,\nu_{2,\Phi} +\;=\; \ell_\Phi + \nu_{2,\Phi} +\;\equiv\; 0 \pmod 3, +\] +where $\sigma_\Phi$ counts shared boundary vertices of $\Phi$, +$\nu_{1,\Phi}$ and $\nu_{2,\Phi}$ count non-shared boundary vertices +with third edge pointing out of / into $\Phi$ respectively, and +$\ell_\Phi$ is the boundary length of $\Phi$. (The interior +contribution is a multiple of $3$ and drops out.) Hence +\begin{equation} +\label{eq:face-sum-mod3} +\nu_{2,\Phi} \;\equiv\; -\ell_\Phi \pmod 3 +\quad\text{for every face } \Phi \text{ of } K_0 \cup K_1. +\end{equation} + +\textbf{Step 4 (Lemma~\ref{lem:kempe-heawood-constant} alternation as +a side-assignment --- TBD).} The Lemma~\ref{lem:kempe-heawood-constant} +alternation on $K_0$ determines, for each non-shared +$K_0$-vertex $v$, exactly which side of $K_0$ its colour-$c$ +(``third'') edge lies on --- and that side is precisely the face +$\Phi(v)$ of $K_0 \cup K_1$ that the third edge points into. +Symmetrically for $K_1$. So the alternation gives an explicit +prescription for $\nu_{2,\Phi}$ in terms of the parity of +$K_0$- and $K_1$-walk indices along $\partial\Phi$. +\emph{(The remaining work is to show that this prescription is +incompatible with~\eqref{eq:face-sum-mod3} for some face +$\Phi$.)} + +\emph{Empirical note.} The theorem's hypothesis is never observed: +across the $142{,}812$ chord-apex+Kempe colourings of reduced duals +with $|V(G)| \le 20$, ``$h_\varphi$ constant on $V(K_b)$'' fails on +every colouring (see \texttt{experiments/check\_constancy\_obstruction.py} +and Remark~\ref{rem:heawood-empirical}). \end{proof} \begin{remark}[Empirical near-proof of Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle} via Corollary~\ref{cor:single-cycle-non-constancy}]