From b5a9030b987cc4ba239546c55c3c69fb7020b905 Mon Sep 17 00:00:00 2001 From: didericis Date: Sat, 9 May 2026 13:04:46 -0400 Subject: [PATCH] Define plane diamond coloring directly via parity-separation Replaces the scaffold-mediated definition with the equivalent direct condition (two color classes contained in opposite-parity BFS layers from some root) and removes the scaffold definition, 2-colorability theorem, connectedness lemma, and equivalence proposition that existed solely to translate between the two formulations. Updates the refutation proof to invoke the new definition directly. Co-Authored-By: Claude Opus 4.7 --- papers/plane_diamond_coloring/paper.aux | 10 +-- .../plane_diamond_coloring/paper.fdb_latexmk | 8 +-- papers/plane_diamond_coloring/paper.fls | 7 +- papers/plane_diamond_coloring/paper.log | 61 +++++++++--------- papers/plane_diamond_coloring/paper.pdf | Bin 233959 -> 215556 bytes papers/plane_diamond_coloring/paper.tex | 50 ++------------ 6 files changed, 46 insertions(+), 90 deletions(-) diff --git a/papers/plane_diamond_coloring/paper.aux b/papers/plane_diamond_coloring/paper.aux index c89260c..4e82dd7 100644 --- a/papers/plane_diamond_coloring/paper.aux +++ b/papers/plane_diamond_coloring/paper.aux @@ -1,11 +1,13 @@ \relax -\@writefile{toc}{\contentsline {section}{\tocsection {}{}{Notation}}{1}{}\protected@file@percent } -\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Definitions}}{1}{}\protected@file@percent } -\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{Results}}{1}{}\protected@file@percent } \citation{appel1977every} \citation{robertson1997four} \citation{mckaygraph6} +\@writefile{toc}{\contentsline {section}{\tocsection {}{}{Notation}}{1}{}\protected@file@percent } +\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Definitions}}{1}{}\protected@file@percent } +\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{Results}}{1}{}\protected@file@percent } \bibcite{appel1977every}{1} +\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces The unique smallest maximal planar graph with no plane diamond coloring; it has $13$ vertices and degree sequence $(6,6,6,6,6,6,6,5,5,4,4,3,3)$.}}{2}{}\protected@file@percent } +\newlabel{fig:counterexample}{{1}{2}} \bibcite{robertson1997four}{2} \bibcite{mckaygraph6}{3} \newlabel{tocindent-1}{0pt} @@ -13,7 +15,5 @@ \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} -\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces The unique smallest maximal planar graph with no plane diamond coloring; it has $13$ vertices and degree sequence $(6,6,6,6,6,6,6,5,5,4,4,3,3)$.}}{3}{}\protected@file@percent } -\newlabel{fig:counterexample}{{1}{3}} \@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{3}{}\protected@file@percent } \gdef \@abspage@last{3} diff --git a/papers/plane_diamond_coloring/paper.fdb_latexmk b/papers/plane_diamond_coloring/paper.fdb_latexmk index 98dc15e..aea99d3 100644 --- a/papers/plane_diamond_coloring/paper.fdb_latexmk +++ b/papers/plane_diamond_coloring/paper.fdb_latexmk @@ -1,5 +1,5 @@ # Fdb version 3 -["pdflatex"] 1778345508 "paper.tex" "paper.pdf" "paper" 1778345509 +["pdflatex"] 1778346213 "paper.tex" "paper.pdf" "paper" 1778346214 "/usr/local/texlive/2022/texmf-dist/fonts/map/fontname/texfonts.map" 1577235249 3524 cb3e574dea2d1052e39280babc910dc8 "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm" 1246382020 1004 54797486969f23fa377b128694d548df "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex8.tfm" 1246382020 988 bdf658c3bfc2d96d3c8b02cfc1c94c20 "" @@ -18,7 +18,6 @@ "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmsy6.tfm" 1136768653 1116 933a60c408fc0a863a92debe84b2d294 "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmsy8.tfm" 1136768653 1120 8b7d695260f3cff42e636090a8002094 "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmti10.tfm" 1136768653 1480 aa8e34af0eb6a2941b776984cf1dfdc4 "" - 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wNkQg^qqTTnl);F&83FDY^M{9pAD}814A-gak^p=-7MAY^4MRpIDlZ1}KY*|G8UO$Q diff --git a/papers/plane_diamond_coloring/paper.tex b/papers/plane_diamond_coloring/paper.tex index 37d0c2b..0963d9c 100644 --- a/papers/plane_diamond_coloring/paper.tex +++ b/papers/plane_diamond_coloring/paper.tex @@ -96,54 +96,16 @@ Equivalently, $L_i = \{v \in V(G) : d(v, u) = i\}$, where $d(v, u)$ denotes the \end{definition} \begin{definition} -Let $G$ be a maximal planar graph with a plane embedding, and let $\{L_0, L_1, L_2, \dots\}$ be the distance partition of $G$ from some $u \in V(G)$. The \emph{diamond scaffold} of $G$ relative to $u$ is the spanning subgraph $G^\diamond \subseteq G$ obtained by removing every edge $\{x, y\} \in E(G)$ such that $x, y \in L_i$ for some $i$. -\end{definition} - -\begin{definition} -Let $G$ be a maximal planar graph. A \emph{plane diamond coloring} of $G$ is a proper $4$-coloring $C$ of $G$ such that there exist two colors $c_a, c_b$ and a diamond scaffold $G^\diamond$ of $G$ with a proper $2$-coloring $C^\diamond : V(G) \to \{c_a, c_b\}$ satisfying +Let $G$ be a maximal planar graph. A \emph{plane diamond coloring} of $G$ is a proper $4$-coloring $C$ of $G$ for which there exist a vertex $u \in V(G)$ and two distinct colors $c_a, c_b$ such that, with respect to the distance partition $\{L_0, L_1, L_2, \dots\}$ of $G$ from $u$, \[ - C^\diamond(v) = c_a \quad \text{for every } v \in C^{-1}(c_a), -\] -\[ - C^\diamond(v) = c_b \quad \text{for every } v \in C^{-1}(c_b). + C^{-1}(c_a) \subseteq \bigcup_{i \text{ even}} L_i \qquad \text{and} \qquad C^{-1}(c_b) \subseteq \bigcup_{i \text{ odd}} L_i. \] \end{definition} \section{Results} -\begin{theorem} -The diamond scaffold of any maximal planar graph $G$ is $2$-colorable. -\end{theorem} - -\begin{proof} -Let $\{L_0, L_1, L_2, \dots\}$ be the distance partition of $G$ from the chosen vertex $u$, and let $G^\diamond$ be the resulting diamond scaffold. We show $G^\diamond$ is bipartite by exhibiting a proper $2$-coloring. - -For any edge $\{x, y\} \in E(G)$, the depths of $x$ and $y$ differ by at most $1$: if $x \in L_i$, then prepending the edge $\{y, x\}$ to a shortest path from $x$ to $u$ gives a walk of length $i + 1$ from $y$ to $u$, so $y \in L_j$ for some $j \leq i + 1$, and symmetrically $i \leq j + 1$. Hence $|i - j| \leq 1$. - -By construction, $G^\diamond$ contains no edge with both endpoints in the same level $L_i$. Combined with the bound above, every edge of $G^\diamond$ joins some $L_i$ to $L_{i+1}$. Color each vertex $v \in L_i$ by the parity of $i$. Every edge of $G^\diamond$ connects vertices of opposite parity, so this is a proper $2$-coloring. -\end{proof} - -\begin{lemma} -The diamond scaffold $G^\diamond$ of a maximal planar graph $G$ relative to $u$ is connected. -\end{lemma} - -\begin{proof} -Let $\{L_0, L_1, L_2, \dots\}$ be the distance partition of $G$ from $u$. We show by induction on $i$ that every vertex of $L_i$ is connected to $u$ in $G^\diamond$. The base case $i = 0$ is immediate, since $L_0 = \{u\}$. For $i \geq 1$, let $v \in L_i$. By definition of $L_i$, there is a shortest path from $v$ to $u$ of length $i$ in $G$, whose penultimate vertex $w$ lies in $L_{i-1}$. The edge $\{v, w\}$ joins $L_i$ to $L_{i-1}$, hence is not a level edge, hence belongs to $G^\diamond$. By the inductive hypothesis $w$ is connected to $u$ in $G^\diamond$, so $v$ is as well. -\end{proof} - -\begin{proposition} -A maximal planar graph $G$ has a plane diamond coloring if and only if there exist a proper $4$-coloring $C$ of $G$, a vertex $u \in V(G)$, and two distinct colors $c_a, c_b$ such that, with respect to the distance partition $\{L_0, L_1, L_2, \dots\}$ of $G$ from $u$, -\[ - C^{-1}(c_a) \subseteq \bigcup_{i \text{ even}} L_i \qquad \text{and} \qquad C^{-1}(c_b) \subseteq \bigcup_{i \text{ odd}} L_i. -\] -\end{proposition} - -\begin{proof} -Since $G^\diamond$ is connected and bipartite (Theorem 2.1 and Lemma 2.2), its proper $2$-coloring is unique up to swapping the two colors, and is given by the parity of level. Hence a proper $2$-coloring $C^\diamond : V(G) \to \{c_a, c_b\}$ of $G^\diamond$ exists with $C^\diamond(v) = c_a$ on the even-parity layers and $C^\diamond(v) = c_b$ on the odd-parity layers (or vice versa). The agreement condition $C(v) = C^\diamond(v)$ on $C^{-1}(c_a) \cup C^{-1}(c_b)$ is then equivalent to the stated containment. -\end{proof} - \begin{remark} -The conjecture below asserts a structural property of $4$-colorings of maximal planar graphs strictly stronger than the conclusion of the Four Color Theorem \cite{appel1977every,robertson1997four}: it requires not merely the existence of a proper $4$-coloring, but the existence of a proper $4$-coloring together with a root $u$ such that two of the four color classes are separated by the parity of the BFS layering from $u$. +Definition 1.2 imposes a structural condition on $4$-colorings of maximal planar graphs strictly stronger than the conclusion of the Four Color Theorem \cite{appel1977every,robertson1997four}: it requires not merely the existence of a proper $4$-coloring, but the existence of a proper $4$-coloring together with a root $u$ such that two of the four color classes are separated by the parity of the BFS layering from $u$. \end{remark} \begin{conjecture} @@ -151,7 +113,7 @@ Every maximal planar graph $G$ has a plane diamond coloring. \end{conjecture} \begin{theorem} -Conjecture 2.5 is false. Moreover, the smallest counterexample has order $13$, and is unique up to isomorphism among triangulations of order at most $13$. +The preceding conjecture is false. Moreover, the smallest counterexample has order $13$, and is unique up to isomorphism among triangulations of order at most $13$. \end{theorem} \begin{proof} @@ -168,9 +130,9 @@ shown in Figure~\ref{fig:counterexample}. Equivalently, $G$ has edge set \end{align*} We have $|V(G)| = 13$ and $|E(G)| = 33 = 3 \cdot 13 - 6$, so $G$ is a triangulation. -By Proposition 2.4, it suffices to verify that for every choice of root $u \in V(G)$ and every pair of distinct colors $c_a, c_b$, no proper $4$-coloring $C$ of $G$ satisfies $C^{-1}(c_a) \subseteq \bigcup_{i \text{ even}} L_i^{(u)}$ and $C^{-1}(c_b) \subseteq \bigcup_{i \text{ odd}} L_i^{(u)}$, where $\{L_i^{(u)}\}$ denotes the distance partition from $u$. +By Definition 1.2, it suffices to show that for every root $u \in V(G)$, no proper $4$-coloring $C$ of $G$ admits two distinct colors $c_a, c_b$ with $C^{-1}(c_a)$ contained in the union of even-indexed levels and $C^{-1}(c_b)$ contained in the union of odd-indexed levels of the distance partition from $u$. -For a fixed root $u$, the existence of such a $4$-coloring $C$ together with colors $c_a, c_b$ is equivalent to $4$-colorability of the auxiliary graph $H_u$ obtained from $G$ by adjoining two new vertices $\alpha, \beta$, joining $\alpha$ to every vertex in $\bigcup_{i \text{ odd}} L_i^{(u)}$, joining $\beta$ to every vertex in $\bigcup_{i \text{ even}} L_i^{(u)}$, and adding the edge $\{\alpha, \beta\}$. Indeed, in any proper $4$-coloring of $H_u$ the colors of $\alpha$ and $\beta$ are distinct and absent from the odd-parity and even-parity layers of $G$ respectively, yielding the required colors $c_a := C(\alpha)$ and $c_b := C(\beta)$. Conversely, given $C, c_a, c_b$ as in the proposition, setting $C(\alpha) := c_a$ and $C(\beta) := c_b$ extends $C$ to a proper $4$-coloring of $H_u$. +For a fixed root $u$, the existence of such a triple $(C, c_a, c_b)$ is equivalent to $4$-colorability of the auxiliary graph $H_u$ obtained from $G$ by adjoining two new vertices $\alpha, \beta$, joining $\alpha$ to every vertex in odd-indexed levels, joining $\beta$ to every vertex in even-indexed levels, and adding the edge $\{\alpha, \beta\}$. Indeed, in any proper $4$-coloring of $H_u$ the colors of $\alpha$ and $\beta$ are distinct and absent from the odd-parity and even-parity layers of $G$ respectively, yielding $c_a := C(\alpha)$ and $c_b := C(\beta)$. Conversely, given a $4$-coloring satisfying the parity-separation condition, setting $C(\alpha) := c_a$ and $C(\beta) := c_b$ extends it to a proper $4$-coloring of $H_u$. A direct computation (using \texttt{Sage}'s \texttt{chromatic\_number}) verifies that $\chi(H_u) > 4$ for every $u \in V(G)$, so $G$ admits no plane diamond coloring.