diff --git a/papers/face_monochromatic_pairs/experiments/counterexample_conj_5_5.py b/papers/face_monochromatic_pairs/experiments/counterexample_conj_5_5.py index 9bfaaf6..960d760 100644 --- a/papers/face_monochromatic_pairs/experiments/counterexample_conj_5_5.py +++ b/papers/face_monochromatic_pairs/experiments/counterexample_conj_5_5.py @@ -314,6 +314,9 @@ def main(): G = build_graph(EDGES) col = edge_colour_map(EDGES) print(f" |V(H)| = {G.order()}, |E(H)| = {G.size()}") + print(f" canonical graph6 = {G.canonical_label().graph6_string()}") + print(f" girth = {G.girth()}, |Aut| = {G.automorphism_group().order()}, " + f"hamiltonian = {G.is_hamiltonian()}") degs = sorted(set(G.degree())) print(f" degree set = {degs}") diff --git a/papers/face_monochromatic_pairs/paper.pdf b/papers/face_monochromatic_pairs/paper.pdf index 7069c7c..24fde31 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 b6f36b2..8b302de 100644 --- a/papers/face_monochromatic_pairs/paper.tex +++ b/papers/face_monochromatic_pairs/paper.tex @@ -844,20 +844,23 @@ $h_\varphi \equiv -1$ on the vertex set of each. Globally $h_\varphi$ takes value $+1$ on $16$ vertices and $-1$ on $24$ vertices, with all of the $+1$-vertices concentrated in the inner ``tilted ladder'' region, so that both Kempe cycles miss them entirely. -The construction and verification (cubic, planar, proper -$3$-edge-colouring, Kempe-cycle tracing, Heawood-number computation -via the CW rotation at each vertex) are in -\texttt{experiments/counterexample\_conj\_5\_5.py}, with the source -drawing in \texttt{constant\_heawood\_counterexample.tikz}. -The partial proof attempt below establishes -$|E(K_0) \cap E(K_1)| \geq 2$ (Step~2) and closes the sub-case where -two shared a-edges are consecutive on \emph{both} cycles (``Case~A'' -/ Step~4). The general claim fails because in the counterexample no -pair of shared a-edges is consecutive on both cycles --- the -$K_1$-arc between two $K_0$-consecutive shared a-edges itself passes -through other shared a-edges, so the lune-face assumption of Step~4 -does not hold. +\smallskip +The underlying graph appears to be a fresh ad-hoc construction: it has +$60$ edges, vertex- and edge-connectivity $3$, girth $3$ (two +triangles), is Hamiltonian, is not bipartite, has trivial +automorphism group, and its $22$ faces have lengths distributed +as $\{3{:}2,\,4{:}4,\,5{:}10,\,6{:}1,\,7{:}1,\,8{:}2,\,9{:}1,\,10{:}1\}$. +Its canonical \texttt{graph6} string (via +\texttt{G.canonical\_label().graph6\_string()}) is +\begin{center} +\small\ttfamily +ggE\_?C?O\_B?B?@C?P???B??\_?@???\_??Sa???W????@a??\_B????G\_???@?????O??@\_???@?\_???K?????\_????K?????o???B?????AG?????B?????G?????AG?????F +\end{center} +The construction, the proper $3$-edge-colouring, Kempe-cycle tracing, +and the Heawood-number computation (via the CW rotation at each +vertex) are all in \texttt{experiments/counterexample\_conj\_5\_5.py}; +the source drawing is in \texttt{constant\_heawood\_counterexample.tikz}. \end{remark} \begin{figure}[h] @@ -873,185 +876,6 @@ $(0, 7)$.} \label{fig:no-two-constant-kempe-counterexample} \end{figure} -\begin{proof}[Partial proof attempt (now superseded by -Remark~\ref{rem:no-two-constant-kempe-cycles-counterexample})] -\textbf{Note.} -Conjecture~\ref{conj:no-two-constant-kempe-cycles} is false (see the -counterexample in -Remark~\ref{rem:no-two-constant-kempe-cycles-counterexample} / -Figure~\ref{fig:no-two-constant-kempe-counterexample}). The argument -below is preserved as partial progress: Steps~1--2 are unconditional -and Step~4 closes the sub-case where some pair of shared a-edges is -consecutive on both $K_0$ and $K_1$. - -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 -$e = (u, w)$. 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 $(a, b, c)$. - -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. - -\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. - -\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. - -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 (lune-face contradiction, ``Case A'' sub-case).} -Suppose there exist two shared a-edges $e_1 = (p, p')$ and -$e_2 = (q, q')$ which are consecutive on \emph{both} the $K_0$-walk -\emph{and} the $K_1$-walk (so the $K_0$-arc from $p'$ to $q$ and the -$K_1$-arc from $p'$ to $q$ both have all-non-shared interior). In -particular, when $|E(K_0) \cap E(K_1)| = 2$ this is automatic, but in -general it is an extra hypothesis. Let $A_1$ be the $K_0$-arc from -$p'$ to $q$ of length $m$. The arc begins with the colour-$b$ edge at -$p'$ and ends with the colour-$b$ edge at $q$, so $m$ is odd. Two -cases for the cyclic order in which $K_1$ visits $\{p, p', q, q'\}$: - -\begin{description} -\item[Case A.] $K_1$ visits them in the same cyclic order -$p, p', q, q'$ as $K_0$. Then $K_1$ has a $K_1$-arc $B_1$ from $p'$ -to $q$ (between $e_1$ and $e_2$ on the $K_1$-walk), of odd length -$n$, whose intermediate vertices are all non-shared. -\item[Case B.] $K_1$ visits them in the opposite order -$p, p', q', q$. Then $K_1$'s arcs between $e_1, e_2$ go from $p'$ to -$q'$ and from $q$ to $p$; they do not share endpoints with $A_1$. -\end{description} - -\emph{Case A is impossible.} The arcs $A_1$ and $B_1$ share both -endpoints $p', q$ and meet only at those endpoints, so they bound a -``lune'' face $\Phi^*$ of $K_0 \cup K_1$ whose boundary has exactly -two corners (both of wedge type $(b, c)$) and length $m + n$. Now -\begin{itemize} -\item Every intermediate vertex of $B_1$ is a non-shared $K_1$-vertex, -hence not on $K_0$, hence lies in one of the two open regions of -$\mathbb{R}^2 \setminus K_0$. Since $B_1$ is a connected path joining -$p', q \in V(K_0)$ and never visits $V(K_0)$ in its interior, -$B_1 \setminus \{p', q\}$ lies entirely on \emph{one} side of $K_0$. -In particular the colour-$c$ edges at $p'$ and at $q$ (the first -and last edges of $B_1$) point into the \emph{same} side of $K_0$. -\item By Lemma~\ref{lem:kempe-heawood-constant} applied along the -$K_0$-arc from $p'$ to $q$, consecutive colour-$c$ non-cycle edges -alternate sides. After $m$ steps with $m$ odd, the colour-$c$ edges -at $p'$ and at $q$ lie on \emph{opposite} sides of $K_0$. -\end{itemize} -These two conclusions are incompatible, contradicting the assumption -that both $K_0$ and $K_1$ have constant $h_\varphi$ in Case A. - -\textbf{Step 5 (Case B --- TBD).} In Case B, the four faces of -$K_0 \cup K_1$ are each three-corner ``triangles'' bounded by one -$K_0$-arc, one $K_1$-arc, and one shared a-edge (one corner each of -types $(a,b)$, $(b,c)$, $(c,a)$). For such a face the -Step~4 lune argument does not apply: $A_1$ and the corresponding -$K_1$-arc $B_1$ no longer share both endpoints, and both -Lemma~\ref{lem:kempe-heawood-constant} alternations and the mod-$3$ -constraint~\eqref{eq:face-sum-mod3} can be checked to be -consistent on the parity counts. The contradiction in this case is -\emph{open}. - -\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}] \label{rem:heawood-empirical} \sloppy