From 1305006b50ac41a47757ee8f3a942695a486f685 Mon Sep 17 00:00:00 2001 From: didericis Date: Mon, 25 May 2026 03:44:41 -0400 Subject: [PATCH] =?UTF-8?q?face=5Fmonochromatic=5Fpairs:=2028-vertex=20C28?= =?UTF-8?q?=20fullerene=20is=20the=20smallest=20counterexample=20to=20Conj?= =?UTF-8?q?=205.5=20(face-length=20=E2=89=A5=205=20form)?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit - Conjecture 5.5 marked FALSE; disproof remark rewritten to use the 28-vertex C28 fullerene as the primary counterexample. - Main figure swapped from the ad-hoc 40-vertex graph to the 28-vertex fullerene (figures/min-face-5-counterexample.png). The 40-vertex graph and K_4 are now mentioned only as smaller counterexamples outside the face-length-≥-5 class. - Added the structural fact that face-length ≥ 5 is already the strongest face-length restriction admitting any cubic plane graphs on the sphere; face-length ≥ 6 would force 0 ≥ 12 via Euler. So no further face-length strengthening can save the conjecture. Co-Authored-By: Claude Opus 4.7 --- papers/face_monochromatic_pairs/paper.pdf | Bin 1192277 -> 1253770 bytes papers/face_monochromatic_pairs/paper.tex | 102 +++++++++++++--------- 2 files changed, 59 insertions(+), 43 deletions(-) diff --git a/papers/face_monochromatic_pairs/paper.pdf b/papers/face_monochromatic_pairs/paper.pdf index 3948784831c969c768555a929f41258c80ccf6e7..781053b8e215168630ce21cf6cd600fc7dc3ba3e 100644 GIT binary patch delta 249517 zcmZ5nbySth*9HR=5mZtnq`Ny0sKgE;C~>F(~5?naPKDQS@I?)+Xk-23x> z{^4RR_B*p@X3w*qXYXNTUc|RW+|>i#H_CP9o^5Exv4o0@H z->)cudR0~1DYj6Tlfr@VC@K9xmTkC&urz7(4@~ddlqYNLi&os$Gl#{`QwhgIS-BxD zc~W!o2013w2d|$${X&r}m3^Kob^7}%EQ{XGMrW``)b{Xt{_CqaoQJG*V(JL;erc$k ze%f?877jG_7ROG<-EobLoQ_ABK>gX$AqQF~aFNs(s2^LrsCa38J~U~2cyPD_(Q+f- z`M{eMi*=oMV=p$MZ{~uc>m@N$=X~A0C$4OWMi+FmcWoy5;gL;vM$B`Bb~ah$Hp3vj zH<>T9QtCfo?)sE=)@0N9J&f>c%_y|)Jlhj*+8MF2R356;<8~@FiIn4QOd#mDE8$v( z0&bq?s@RbO(E5xM-60nMIcXaANt6eBtDgBo=kGff?jt5Gz|ex8a$m6nR7X)4d&dS% 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edge-sharing Kempe cycles, large-face cubic plane graphs --- \textbf{FALSE}] \label{conj:no-two-constant-kempe-cycles} Let $H$ be a cubic plane graph in which every face has length at least $5$, with a proper $3$-edge-colouring $\varphi$. Fix a colour @@ -826,52 +826,68 @@ share at least one colour-$a$ edge). If $h_\varphi$ is constant on $V(K_0)$, then $h_\varphi$ is \emph{not} constant on $V(K_1)$. \end{conjecture} -\begin{remark}[On the face-length hypothesis] -\label{rem:no-two-constant-kempe-face-length-hypothesis} -Without the hypothesis that every face has length $\ge 5$, the -conjecture is \emph{false}, with very small counterexamples: -\begin{itemize} -\item \textbf{$K_4$ (the tetrahedron, $n = 4$):} The standard -proper $3$-edge-colouring has $h_\varphi$ constant (all $-1$ or -all $+1$, depending on planar embedding orientation) on every vertex, -and every pair of distinct Kempe cycles shares colour-edges. (All -four faces of $K_4$ are triangles.) -\item \textbf{An $n = 8$ cubic plane graph with girth $3$ -(graph6 \texttt{G\}GOW[}):} Found by the brute-force enumeration in -\texttt{experiments/search\_smaller\_counterexample.py}; both Kempe -cycles are $8$-cycles visiting every vertex, and $h_\varphi$ is -constant on each. -\item \textbf{An ad-hoc $n = 40$ cubic plane graph with girth $3$:} -The graph in \texttt{constant\_heawood\_counterexample.tikz} (face -lengths $\{3{:}2,\,4{:}4,\,5{:}10,\,6{:}1,\,7{:}1,\,8{:}2,\,9{:}1,\, -10{:}1\}$, $|V| = 40$) has an $8$-cycle and a $12$-cycle that share a -colour-red edge and both have $h_\varphi \equiv -1$, with $24$ of the -$40$ vertices ($60\%$) lying outside $V(K_0) \cup V(K_1)$ --- so this -is a structurally non-trivial counterexample, unlike $K_4$ and the -$n = 8$ example where the two Kempe cycles already cover all -vertices. -\end{itemize} -All three counterexamples have at least one face of length -$\le 4$. The face-length-$\ge 5$ hypothesis above is the smallest -restriction that simultaneously kills all currently known -counterexamples; whether it suffices to make the conjecture true is -open. The smallest cubic plane graphs satisfying this hypothesis have -$|V(H)| \ge 20$ (with $|V| = 20$ forced to be the dodecahedron, all -faces pentagonal); brute-force search up to some moderate $|V|$ is in -\texttt{experiments/search\_smaller\_counterexample.py}. +\begin{remark}[Disproof of +Conjecture~\ref{conj:no-two-constant-kempe-cycles}] +\label{rem:no-two-constant-kempe-cycles-counterexample} +Conjecture~\ref{conj:no-two-constant-kempe-cycles} is \emph{false}. +The smallest counterexample is a cubic plane graph $H$ on $28$ +vertices with $12$ pentagonal and $4$ hexagonal faces (a $C_{28}$ +fullerene). It is the planar dual of the third element (in +\texttt{Sage}'s order) of +\texttt{graphs.triangulations(16, minimum\_degree=5)}, with canonical +\texttt{graph6} string +\begin{center} +\small\ttfamily +[kG[A?\_A?\_?\_?K?D?@\_CO?o?@\_??A??@C??O??AG?C????`???a???W???A\_???F. +\end{center} +A proper $3$-edge-colouring of $H$ (colours red/blue/green, +see Figure~\ref{fig:no-two-constant-kempe-counterexample}) makes both +\begin{align*} +K_{\mathrm{red},\mathrm{blue}} + &= \text{a $12$-cycle on $H$,} \\ +K_{\mathrm{red},\mathrm{green}} + &= \text{a different $12$-cycle on $H$,} +\end{align*} +sharing the colour-red edge $(0, 1)$ and satisfying +$h_\varphi \equiv -1$ on the vertex set of each. Globally +$h_\varphi$ takes value $+1$ on $4$ vertices and $-1$ on $24$; the +four $+1$-vertices and a further four lie outside +$V(K_0) \cup V(K_1)$, which has size $20$. +The construction, verification, and rendering are in +\texttt{experiments/verify\_28\_vertex\_counterexample.py}, and the +exhaustive search that found it is in +\texttt{experiments/search\_min\_face5\_counterexample.py}. + +\smallskip +The face-length-$\ge 5$ hypothesis is in fact the strongest +face-length hypothesis admitting any cubic plane graphs at all on the +sphere: by Euler's formula, if every face had length $\ge 6$ then +the sum of face lengths would satisfy $3|V(H)| \ge 6(|V(H)|/2 + 2)$, +i.e.\ $0 \ge 12$, contradiction. So strengthening the conjecture by +raising the minimum face length further is impossible. Without the +face-length hypothesis there are far smaller counterexamples, +including the tetrahedron $K_4$ at $|V| = 4$ (every Kempe cycle is a +$4$-cycle visiting every vertex, with $h_\varphi$ constant on all of +them by vertex-transitivity) and an $8$-vertex example (graph6 +\texttt{G\}GOW[}) found by +\texttt{experiments/search\_smaller\_counterexample.py}; both have +girth $3$. An ad-hoc $40$-vertex counterexample with the same +``two intersecting Kempe cycles $\equiv -1$, large region outside'' +flavour is in +\texttt{constant\_heawood\_counterexample.tikz}. \end{remark} \begin{figure}[h] \centering -\includegraphics[width=0.7\textwidth]{figures/no-two-constant-kempe-counterexample.png} -\caption{The $n = 40$ counterexample to the unrestricted version of -Conjecture~\ref{conj:no-two-constant-kempe-cycles} (cf.\ -Remark~\ref{rem:no-two-constant-kempe-face-length-hypothesis}). Both -the outer red/blue $8$-cycle and the red/green $12$-cycle (outer -frame plus the upper-left ladder side) have $h_\varphi \equiv -1$ and -share the colour-red edge $(0, 7)$. The face lengths of this drawing -are $\{3, 4, 5, 6, 7, 8, 9, 10\}$ with two triangles, so it does not -satisfy the face-length-$\ge 5$ hypothesis above.} +\includegraphics[width=0.7\textwidth]{figures/min-face-5-counterexample.png} +\caption{Smallest counterexample to +Conjecture~\ref{conj:no-two-constant-kempe-cycles}: a $C_{28}$ +fullerene-style cubic plane graph (12 pentagons + 4 hexagons) with a +proper $3$-edge-colouring on which $h_\varphi$ is simultaneously +constant ($\equiv -1$) on the red/blue $12$-cycle and the red/green +$12$-cycle, which share the colour-red edge $(0, 1)$. Light-shaded +nodes are on $V(K_0) \cap V(K_1)$; medium-shaded on +$V(K_0) \cup V(K_1) \setminus V(K_0) \cap V(K_1)$; grey on neither.} \label{fig:no-two-constant-kempe-counterexample} \end{figure}