From 88031c9d730167c3d54679b552487396e05e03dc Mon Sep 17 00:00:00 2001 From: didericis Date: Sat, 9 May 2026 12:52:30 -0400 Subject: [PATCH] Replace counterexample with minimal n=13 graph found by exhaustive search Adds search_counterexample_comprehensive iterating Sage's planar_graphs generator across all maximal planar graphs of bounded order. Exhaustive enumeration through order 13 (9150+49566 triangulations) yields exactly one graph with no plane diamond coloring, at order 13. Updates Theorem 2.6 to assert minimality and uniqueness, and replaces the figure and edge list with the smaller counterexample. Co-Authored-By: Claude Opus 4.7 --- .../plane_diamond_coloring/counterexample.png | Bin 38090 -> 31680 bytes papers/plane_diamond_coloring/paper.aux | 2 +- .../plane_diamond_coloring/paper.fdb_latexmk | 8 +-- papers/plane_diamond_coloring/paper.log | 60 +++++++----------- papers/plane_diamond_coloring/paper.pdf | Bin 236597 -> 233959 bytes papers/plane_diamond_coloring/paper.tex | 21 +++--- plane_diamond_coloring.py | 41 ++++++++++-- 7 files changed, 74 insertions(+), 58 deletions(-) diff --git a/papers/plane_diamond_coloring/counterexample.png b/papers/plane_diamond_coloring/counterexample.png index ec373784455dcc770d5014e0b270e88ccb7ca3ca..d57c692dea9d19fec9bc6f0c92c1f2092428153b 100755 GIT binary patch literal 31680 zcmaI8cU)6T*ES4DGl0}k1SulYl_H=BqBIqdnuJgU0@6a4Dn(G~MMR`Z5fFqVNbgb% zMWjeG6lnsAG(nW2QhaB}bMEJUzVH3Lf1KkvviIznHEY(o*0ru7#!&w%6YMyQf`WqS z`ZY~s3JOXR_-B-!7W|EnG`ku2L)k|Q?PKEM;Ny?EZBKCnvYn=XI@x!c8j>ayT9(XAOxDkCZw^{G4^R+=l%#q)HLR}d_kEJhsLN2o;v 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@@ \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} -\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces A maximal planar graph on $16$ vertices with no plane diamond coloring.}}{3}{}\protected@file@percent } +\@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 35d370d..98dc15e 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"] 1778344761 "paper.tex" "paper.pdf" "paper" 1778344761 +["pdflatex"] 1778345508 "paper.tex" "paper.pdf" "paper" 1778345509 "/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 "" @@ -62,9 +62,9 @@ "/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map" 1647878959 4410336 7d30a02e9fa9a16d7d1f8d037ba69641 "" "/usr/local/texlive/2022/texmf-var/web2c/pdftex/pdflatex.fmt" 1665017617 2826443 7e98410c533054b636c6470db83a27bc "" "/usr/local/texlive/2022/texmf.cnf" 1647878952 577 209b46be99c9075fd74d4c0369380e8c "" - "counterexample.png" 1778344642 38090 f4e4209c691fb5c14386207f1bb29f68 "" - "paper.aux" 1778344761 957 a638007d0f0ce9d55a02d7823fdbddc2 "pdflatex" - "paper.tex" 1778344755 9964 4b14437133e6c4b02a0c45d6a8579fa8 "" + "counterexample.png" 1778345471 31680 49acd944c061e194d20aac1d5cd8ef86 "" + "paper.aux" 1778345509 1030 d76673a7c450eca67268187399b24cac "pdflatex" + "paper.tex" 1778345503 10589 51592aa33a281bbb7d4895e76f9e26b3 "" (generated) "paper.aux" "paper.log" diff --git a/papers/plane_diamond_coloring/paper.log b/papers/plane_diamond_coloring/paper.log index 0b57add..5912c64 100644 --- a/papers/plane_diamond_coloring/paper.log +++ b/papers/plane_diamond_coloring/paper.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 9 MAY 2026 12:39 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 9 MAY 2026 12:51 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -193,51 +193,39 @@ File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Liv e )) 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z6(J%++tP&pfKq=Epe9hw^ZTqv9kcGuZW;a!x{6@$euD<0t5#=tKM}pkpzSzYoiH1* z%D=6%VHs#^u}mH0=GQC@G~=^vN?@x}15R-356vA%;DmIbYXgA;Bc_JEUzTW5fS!`o4+Ga@=MHrp8>Vs;>TS FTmYQ)?VJDr diff --git a/papers/plane_diamond_coloring/paper.tex b/papers/plane_diamond_coloring/paper.tex index d2876b3..37d0c2b 100644 --- a/papers/plane_diamond_coloring/paper.tex +++ b/papers/plane_diamond_coloring/paper.tex @@ -151,35 +151,36 @@ Every maximal planar graph $G$ has a plane diamond coloring. \end{conjecture} \begin{theorem} -Conjecture 2.5 is false. +Conjecture 2.5 is false. Moreover, the smallest counterexample has order $13$, and is unique up to isomorphism among triangulations of order at most $13$. \end{theorem} \begin{proof} -Let $G$ be the maximal planar graph on $16$ vertices with graph6 string\footnote{We use the standard graph6 encoding of McKay; see \cite{mckaygraph6}.} +Let $G$ be the maximal planar graph on $13$ vertices with graph6 string\footnote{We use the standard graph6 encoding of McKay; see \cite{mckaygraph6}.} \[ - \texttt{O???IAGKEBEQQYHdplW\{n} + \texttt{LQhO?SFIwnTYTi} \] shown in Figure~\ref{fig:counterexample}. Equivalently, $G$ has edge set \begin{align*} -&\{0,8\}, \{0,10\}, \{0,14\}, \{1,7\}, \{1,10\}, \{1,12\}, \{2,9\}, \{2,11\}, \{2,13\}, \\ -&\{3,9\}, \{3,11\}, \{3,15\}, \{4,8\}, \{4,12\}, \{4,14\}, \{4,15\}, \{5,6\}, \{5,13\}, \\ -&\{5,14\}, \{5,15\}, \{6,11\}, \{6,13\}, \{6,15\}, \{7,10\}, \{7,12\}, \{7,14\}, \{8,10\}, \\ -&\{8,12\}, \{8,14\}, \{9,11\}, \{9,13\}, \{9,15\}, \{10,12\}, \{10,14\}, \{11,13\}, \\ -&\{11,15\}, \{12,13\}, \{12,14\}, \{12,15\}, \{13,14\}, \{13,15\}, \{14,15\}. +&\{0,2\}, \{0,4\}, \{0,11\}, \{1,3\}, \{1,5\}, \{1,12\}, \{2,4\}, \{2,9\}, \{2,11\}, \\ +&\{3,5\}, \{3,10\}, \{3,12\}, \{4,7\}, \{4,9\}, \{4,11\}, \{5,8\}, \{5,10\}, \{5,12\}, \\ +&\{6,7\}, \{6,8\}, \{6,9\}, \{6,10\}, \{6,11\}, \{6,12\}, \{7,8\}, \{7,9\}, \{7,10\}, \\ +&\{7,11\}, \{8,9\}, \{8,10\}, \{8,12\}, \{9,11\}, \{10,12\}. \end{align*} -We have $|V(G)| = 16$ and $|E(G)| = 42 = 3 \cdot 16 - 6$, so $G$ is a triangulation. +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 $u$. 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 $H_u$. 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. + +For minimality and uniqueness, we exhaustively enumerated every maximal planar graph of order at most $13$ using \texttt{Sage}'s \texttt{graphs.planar\_graphs} generator (with \texttt{minimum\_connectivity=3} and \texttt{maximum\_face\_size=3}). The numbers of triangulations $1, 1, 2, 5, 14, 50, 233, 1249, 7595, 49566$ at orders $4, 5, \dots, 13$ respectively (matching OEIS A000109) were each tested for the existence of a plane diamond coloring, and exactly one --- the graph $G$ above, occurring at order $13$ --- was found to lack one. \end{proof} \begin{figure}[htbp] \centering \includegraphics[width=0.45\textwidth]{counterexample.png} - \caption{A maximal planar graph on $16$ vertices with no plane diamond coloring.} + \caption{The unique smallest maximal planar graph with no plane diamond coloring; it has $13$ vertices and degree sequence $(6,6,6,6,6,6,6,5,5,4,4,3,3)$.} \label{fig:counterexample} \end{figure} diff --git a/plane_diamond_coloring.py b/plane_diamond_coloring.py index 988c8b2..b64d803 100644 --- a/plane_diamond_coloring.py +++ b/plane_diamond_coloring.py @@ -62,13 +62,40 @@ def search_counterexample(n: int, num_trials: int) -> Graph | None: return None +def search_counterexample_comprehensive(max_order: int, min_order: int = 4) -> list[Graph]: + """ + Iterate through every maximal planar graph of order in [min_order, max_order] + and return all those without a plane diamond coloring. + """ + counterexamples: list[Graph] = [] + for n in range(min_order, max_order + 1): + checked = 0 + for g in graphs.planar_graphs(n, minimum_connectivity=3, maximum_face_size=3): + checked += 1 + if not has_plane_diamond_coloring(g): + print(f"Counterexample at order {n} (graph #{checked}): {g.graph6_string()}") + counterexamples.append(g) + if checked % 100 == 0: + print(f" order {n}: checked {checked} graphs, {len(counterexamples)} counterexamples so far") + print(f"order {n} done: {checked} triangulations checked") + return counterexamples + + if __name__ == "__main__": import sys - n = int(sys.argv[1]) if len(sys.argv) > 1 else 12 - num_trials = int(sys.argv[2]) if len(sys.argv) > 2 else 100 - counterexample = search_counterexample(n, num_trials) - if counterexample is None: - print(f"No counterexample found in {num_trials} random triangulations of order {n}") + if len(sys.argv) > 1 and sys.argv[1] == "comprehensive": + max_order = int(sys.argv[2]) if len(sys.argv) > 2 else 10 + min_order = int(sys.argv[3]) if len(sys.argv) > 3 else 4 + counterexamples = search_counterexample_comprehensive(max_order, min_order) + print(f"Found {len(counterexamples)} counterexamples in orders {min_order}..{max_order}") + for g in counterexamples: + canonize_and_save_graph(g) else: - canonical, graph_dir = canonize_and_save_graph(counterexample) - print(f"Counterexample saved to {graph_dir}") + n = int(sys.argv[1]) if len(sys.argv) > 1 else 12 + num_trials = int(sys.argv[2]) if len(sys.argv) > 2 else 100 + counterexample = search_counterexample(n, num_trials) + if counterexample is None: + print(f"No counterexample found in {num_trials} random triangulations of order {n}") + else: + canonical, graph_dir = canonize_and_save_graph(counterexample) + print(f"Counterexample saved to {graph_dir}")