level_resolutions: add n=7 missing-isomorphism figures, rebuild PDF

Add the figures for the n=7, idx=2 missing-isomorphism case
(missing_iso_n7_idx2.png is included in paper.tex), plus its
4-coloring and level-decomposition companions and the G-for-T
preimage graph. Rebuild paper.pdf and its LaTeX aux/log/out.

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>

papers/level_resolutions_of_maximal_planar_graphs/paper.aux

@@ -29,7 +29,7 @@
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 \newlabel{obs:preimage}{{5.1}{3}{}{theorem.5.1}{}}
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@@ -37,7 +37,7 @@
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 \newlabel{obs:icosa}{{5.2}{4}{}{theorem.5.2}{}}
 \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{5.3}{Restatement of the resolution-preimage conjecture}}{4}{subsection.5.3}\protected@file@percent }
-\newlabel{conj:md4}{{5.3}{4}{$\mathrm {md}_4$ surjectivity}{theorem.5.3}{}}
+\newlabel{conj:md5}{{5.3}{4}{$\mathrm {md}_5$ surjectivity}{theorem.5.3}{}}
 \@writefile{toc}{\contentsline {section}{\tocsection {}{6}{An edge-flip resolution algorithm}}{5}{section.6}\protected@file@percent }
 \newlabel{sec:flip-algorithm}{{6}{5}{An edge-flip resolution algorithm}{section.6}{}}
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@@ -51,16 +51,16 @@
 \newlabel{def:simple-level-resolution}{{6.4}{6}{Simple level resolution}{theorem.6.4}{}}
 \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{6.6}{Empirical status}}{6}{subsection.6.6}\protected@file@percent }
 \newlabel{obs:empirical-lk-bipartite}{{6.5}{6}{}{theorem.6.5}{}}
-\@writefile{toc}{\contentsline {paragraph}{\tocparagraph {}{}{Coverage test for Conjecture\nonbreakingspace \ref  {conj:simple-md4}.}}{6}{section*.2}\protected@file@percent }
+\@writefile{toc}{\contentsline {paragraph}{\tocparagraph {}{}{Coverage test.}}{6}{section*.2}\protected@file@percent }
-\newlabel{obs:md4-simple-resolution}{{6.6}{7}{}{theorem.6.6}{}}
+\@writefile{lot}{\contentsline {table}{\numberline {2}{\ignorespaces Simple-resolution coverage under the algorithm of Section\nonbreakingspace \ref  {sec:flip-algorithm}. ``Algorithm output (any)'' counts iso-classes that appear as the iso-class of some labelled triangulation output of the algorithm; ``Simple level resolutions'' counts iso-classes that additionally satisfy the bipartite-parity condition of Definition\nonbreakingspace \ref  {def:simple-level-resolution}.}}{7}{table.2}\protected@file@percent }
-\newlabel{conj:simple-md4}{{6.7}{7}{Simple-resolution $\mathrm {md}_4$ surjectivity}{theorem.6.7}{}}
+\newlabel{tab:simple-coverage}{{2}{7}{Simple-resolution coverage under the algorithm of Section~\ref {sec:flip-algorithm}. ``Algorithm output (any)'' counts iso-classes that appear as the iso-class of some labelled triangulation output of the algorithm; ``Simple level resolutions'' counts iso-classes that additionally satisfy the bipartite-parity condition of Definition~\ref {def:simple-level-resolution}}{table.2}{}}
-\newlabel{sec:contraction-lift}{{6.7}{7}{Towards a proof: a contraction--lift strategy}{subsection.6.7}{}}
+\newlabel{obs:md5-simple-resolution}{{6.6}{7}{}{theorem.6.6}{}}
-\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{6.7}{Towards a proof: a contraction--lift strategy}}{7}{subsection.6.7}\protected@file@percent }
+\newlabel{q:terminate-all-n}{{6.7}{7}{}{theorem.6.7}{}}
-\newlabel{lem:good-contraction}{{6.8}{7}{Good contraction}{theorem.6.8}{}}
+\@writefile{toc}{\contentsline {section}{\tocsection {}{7}{Discussion and open questions}}{7}{section.7}\protected@file@percent }
-\newlabel{lem:lift}{{6.9}{7}{Lift}{theorem.6.9}{}}
+\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces At $n = 7$, iso-class $T$ with degree sequence $(6, 5, 5, 5, 3, 3, 3)$ is not a simple level resolution. Left: $T$ drawn with face source $S = \{0, 1, 5\}$ outlined; vertices are labelled by their BFS level. Middle: the even-parity subgraph $E_{T,S}(T)$, induced on $\{0, 1, 5\}$, contains the source-face triangle (highlighted) — an odd cycle. Right: the odd-parity subgraph $O_{T,S}(T)$, induced on $\{2, 3, 4, 6\}$, contains a triangle on $\{2, 3, 6\}$ (highlighted). The obstruction is identical for every face source of $T$; vertex sources at degree-$3$ vertices produce $L_1$ triangles in the odd-parity subgraph.}}{8}{figure.1}\protected@file@percent }
-\@writefile{toc}{\contentsline {paragraph}{\tocparagraph {}{}{Inductive scheme.}}{8}{section*.3}\protected@file@percent }
+\newlabel{fig:missing-iso}{{1}{8}{At $n = 7$, iso-class $T$ with degree sequence $(6, 5, 5, 5, 3, 3, 3)$ is not a simple level resolution. Left: $T$ drawn with face source $S = \{0, 1, 5\}$ outlined; vertices are labelled by their BFS level. Middle: the even-parity subgraph $E_{T,S}(T)$, induced on $\{0, 1, 5\}$, contains the source-face triangle (highlighted) — an odd cycle. Right: the odd-parity subgraph $O_{T,S}(T)$, induced on $\{2, 3, 4, 6\}$, contains a triangle on $\{2, 3, 6\}$ (highlighted). The obstruction is identical for every face source of $T$; vertex sources at degree-$3$ vertices produce $L_1$ triangles in the odd-parity subgraph}{figure.1}{}}
-\newlabel{q:terminate-all-n}{{6.10}{8}{}{theorem.6.10}{}}
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-\@writefile{toc}{\contentsline {section}{\tocsection {}{7}{Discussion and open questions}}{8}{section.7}\protected@file@percent }
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 \bibcite{appelhaken}{1}
 \bibcite{rsst}{2}
 \bibcite{tutte}{3}
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papers/level_resolutions_of_maximal_planar_graphs/paper.log

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 \OT1/cmr/m/n/10 Equiv-a-lently, ev-ery max-i-mal pla-nar graph (tri-an-gu-la-ti
 on) is 4-colorable. The Appel--
  []
 
 
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 \OT1/cmr/m/n/10 Haken proof [[]] and sub-se-quent Robertson--Sanders--Seymour--
 Thomas re-fine-ment [[]]
  []
 
 
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 \OT1/cmr/m/n/10 rely on dis-charg-ing ar-gu-ments and computer-verified re-duci
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papers/level_resolutions_of_maximal_planar_graphs/paper.out

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