diff --git a/papers/even_level_graph_generators/paper.aux b/papers/even_level_graph_generators/paper.aux index b0d584b..7392478 100644 --- a/papers/even_level_graph_generators/paper.aux +++ b/papers/even_level_graph_generators/paper.aux @@ -18,46 +18,46 @@ \providecommand\HyField@AuxAddToCoFields[2]{} \@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{section.1}\protected@file@percent } \@writefile{toc}{\contentsline {section}{\tocsection {}{2}{Definitions}}{2}{section.2}\protected@file@percent } -\newlabel{def:edge-switch}{{2.4}{2}{Edge switch}{theorem.2.4}{}} \@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces BFS levels from the degree-$3$ vertex source $S = \{4\}$. The source is level $0$, its three neighbours are level $1$, and the remaining vertices are level $2$. Colour encodes the level.}}{3}{figure.1}\protected@file@percent } \newlabel{fig:levels}{{1}{3}{BFS levels from the degree-$3$ vertex source $S = \{4\}$. The source is level $0$, its three neighbours are level $1$, and the remaining vertices are level $2$. Colour encodes the level}{figure.1}{}} \@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces A level cycle in the triangulation of Figure\nonbreakingspace \ref {fig:levels}. The triangle $1\!-\!2\!-\!3$ is a simple cycle whose three vertices all lie at level $1$, so it is a level cycle at level $1$.}}{3}{figure.2}\protected@file@percent } \newlabel{fig:level-cycle}{{2}{3}{A level cycle in the triangulation of Figure~\ref {fig:levels}. The triangle $1\!-\!2\!-\!3$ is a simple cycle whose three vertices all lie at level $1$, so it is a level cycle at level $1$}{figure.2}{}} -\@writefile{toc}{\contentsline {section}{\tocsection {}{3}{Outerplanarity of level components}}{3}{section.3}\protected@file@percent } -\newlabel{sec:outerplanar-components}{{3}{3}{Outerplanarity of level components}{section.3}{}} -\newlabel{thm:outerplanar-component}{{3.1}{3}{}{theorem.3.1}{}} +\newlabel{def:edge-switch}{{2.4}{4}{Edge switch}{theorem.2.4}{}} \@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces An edge switch on the level cycle of Figure\nonbreakingspace \ref {fig:level-cycle}. The chosen cycle edge $1\!-\!2$ is shared by the triangular faces $(0,1,2)$ and $(1,2,4)$; the switch deletes $1\!-\!2$ (red, left) and inserts $0\!-\!4$ (green, right). Vertex colours indicate the original levels in $G$.}}{4}{figure.3}\protected@file@percent } \newlabel{fig:edge-switch}{{3}{4}{An edge switch on the level cycle of Figure~\ref {fig:level-cycle}. The chosen cycle edge $1\!-\!2$ is shared by the triangular faces $(0,1,2)$ and $(1,2,4)$; the switch deletes $1\!-\!2$ (red, left) and inserts $0\!-\!4$ (green, right). Vertex colours indicate the original levels in $G$}{figure.3}{}} \@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Parity subgraphs of $G' = T$ with respect to the level structure of Figure\nonbreakingspace \ref {fig:levels} (here we take $G = G' = T$). Left: $T$ with vertices coloured by $\ell _G \nonscript \mskip -\medmuskip \mkern 5mu\mathbin {\mathgroup \symoperators mod}\penalty 900 \mkern 5mu\nonscript \mskip -\medmuskip 2$ (blue $=$ even, orange $=$ odd). Middle: the even parity subgraph $E_{G,S}(G')$, induced on $\{0, 4, 5, 6\}$; only edges with both endpoints even appear. Right: the odd parity subgraph $O_{G,S}(G')$, induced on $\{1, 2, 3\}$; the highlighted triangle shows that $O_{G,S}(G')$ is not bipartite for this choice of $G'$.}}{4}{figure.4}\protected@file@percent } \newlabel{fig:parity-subgraph}{{4}{4}{Parity subgraphs of $G' = T$ with respect to the level structure of Figure~\ref {fig:levels} (here we take $G = G' = T$). Left: $T$ with vertices coloured by $\ell _G \bmod 2$ (blue $=$ even, orange $=$ odd). Middle: the even parity subgraph $E_{G,S}(G')$, induced on $\{0, 4, 5, 6\}$; only edges with both endpoints even appear. Right: the odd parity subgraph $O_{G,S}(G')$, induced on $\{1, 2, 3\}$; the highlighted triangle shows that $O_{G,S}(G')$ is not bipartite for this choice of $G'$}{figure.4}{}} +\@writefile{toc}{\contentsline {section}{\tocsection {}{3}{Outerplanarity of level components}}{5}{section.3}\protected@file@percent } +\newlabel{sec:outerplanar-components}{{3}{5}{Outerplanarity of level components}{section.3}{}} +\newlabel{thm:outerplanar-component}{{3.1}{5}{}{theorem.3.1}{}} \@writefile{toc}{\contentsline {section}{\tocsection {}{4}{Even Level Graphs}}{5}{section.4}\protected@file@percent } \newlabel{sec:even-level-graphs}{{4}{5}{Even Level Graphs}{section.4}{}} \newlabel{def:even-level-graph}{{4.1}{5}{Even Level Graph}{theorem.4.1}{}} \newlabel{thm:even-level-4colorable}{{4.2}{5}{}{theorem.4.2}{}} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{Enumeration for small $n$}}{5}{section*.1}\protected@file@percent } -\newlabel{def:derived-level-graph}{{4.3}{5}{Derived level graph}{theorem.4.3}{}} -\newlabel{def:bridge-switch}{{4.4}{5}{Bridge switch}{theorem.4.4}{}} \@writefile{lot}{\contentsline {table}{\numberline {1}{\ignorespaces Even Level Graph counts for $4 \leq n \leq 11$. The \emph {triangulations} column is the number of plane-triangulation iso classes (OEIS A000109). \emph {ELG iso classes} counts pairs $(G, S)$ up to isomorphism; \emph {flag-rooted ELGs} is the automorphism-free count $\DOTSB \sum@ \slimits@ _G \genfrac {}{}{}1{4E}{|\mathrm {Aut}(G)|}\,s(G)$.}}{6}{table.1}\protected@file@percent } \newlabel{tab:elg-counts}{{1}{6}{Even Level Graph counts for $4 \leq n \leq 11$. The \emph {triangulations} column is the number of plane-triangulation iso classes (OEIS A000109). \emph {ELG iso classes} counts pairs $(G, S)$ up to isomorphism; \emph {flag-rooted ELGs} is the automorphism-free count $\sum _G \tfrac {4E}{|\mathrm {Aut}(G)|}\,s(G)$}{table.1}{}} +\newlabel{def:derived-level-graph}{{4.3}{6}{Derived level graph}{theorem.4.3}{}} +\newlabel{def:bridge-switch}{{4.4}{6}{Bridge switch}{theorem.4.4}{}} \newlabel{def:bridge-derived-level-graph}{{4.5}{6}{Bridge-derived level graph}{theorem.4.5}{}} -\newlabel{def:intertwining-tree}{{4.6}{6}{Intertwining tree}{theorem.4.6}{}} -\newlabel{thm:intertwining-iff-hamiltonian-dual}{{4.7}{6}{}{theorem.4.7}{}} -\citation{holton-mckay} \citation{holton-mckay} +\newlabel{def:intertwining-tree}{{4.6}{7}{Intertwining tree}{theorem.4.6}{}} +\newlabel{thm:intertwining-iff-hamiltonian-dual}{{4.7}{7}{}{theorem.4.7}{}} \newlabel{conj:every-triangulation-derived}{{4.8}{7}{}{theorem.4.8}{}} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{The boundary case $n = 21$}}{7}{section*.2}\protected@file@percent } -\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{The cyclically-$5$-connected case: $n = 24$}}{7}{section*.3}\protected@file@percent } +\citation{holton-mckay} \@writefile{lot}{\contentsline {table}{\numberline {2}{\ignorespaces The six Holton--McKay duals at $n = 21$, the first triangulations that are not intertwining trees. Each is a bridge-derived level graph: duals $1$ and $2$ are Even Level Graphs outright (zero switches), and the remaining four reach an Even Level Graph in $1$--$4$ bridge switches. All witnesses are step-verified.}}{8}{table.2}\protected@file@percent } \newlabel{tab:n21}{{2}{8}{The six Holton--McKay duals at $n = 21$, the first triangulations that are not intertwining trees. Each is a bridge-derived level graph: duals $1$ and $2$ are Even Level Graphs outright (zero switches), and the remaining four reach an Even Level Graph in $1$--$4$ bridge switches. All witnesses are step-verified}{table.2}{}} -\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces The six Holton--McKay duals, drawn as crossing-free planar graphs and coloured by parity (blue even, orange odd, with respect to the fixed level-parity labelling). The solid green edges are the bridge edges introduced by the bridge switches from each dual's witness Even Level Graph. Each green edge is a bridge of its parity subgraph, so no new cycle -- and in particular no odd cycle -- is created; duals $1$ and $2$ coincide with their Even Level Graphs and have no added edge.}}{8}{figure.5}\protected@file@percent } -\newlabel{fig:n21-duals}{{5}{8}{The six Holton--McKay duals, drawn as crossing-free planar graphs and coloured by parity (blue even, orange odd, with respect to the fixed level-parity labelling). The solid green edges are the bridge edges introduced by the bridge switches from each dual's witness Even Level Graph. Each green edge is a bridge of its parity subgraph, so no new cycle -- and in particular no odd cycle -- is created; duals $1$ and $2$ coincide with their Even Level Graphs and have no added edge}{figure.5}{}} +\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{The cyclically-$5$-connected case: $n = 24$}}{8}{section*.3}\protected@file@percent } \bibcite{holton-mckay}{1} +\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces The six Holton--McKay duals, drawn as crossing-free planar graphs and coloured by parity (blue even, orange odd, with respect to the fixed level-parity labelling). The solid green edges are the bridge edges introduced by the bridge switches from each dual's witness Even Level Graph. Each green edge is a bridge of its parity subgraph, so no new cycle -- and in particular no odd cycle -- is created; duals $1$ and $2$ coincide with their Even Level Graphs and have no added edge.}}{9}{figure.5}\protected@file@percent } +\newlabel{fig:n21-duals}{{5}{9}{The six Holton--McKay duals, drawn as crossing-free planar graphs and coloured by parity (blue even, orange odd, with respect to the fixed level-parity labelling). The solid green edges are the bridge edges introduced by the bridge switches from each dual's witness Even Level Graph. Each green edge is a bridge of its parity subgraph, so no new cycle -- and in particular no odd cycle -- is created; duals $1$ and $2$ coincide with their Even Level Graphs and have no added edge}{figure.5}{}} \newlabel{tocindent-1}{0pt} \newlabel{tocindent0}{14.69437pt} \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} -\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{9}{section*.4}\protected@file@percent } \@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces The $24$-vertex dual $T$ of the unique $44$-vertex non-Hamiltonian cyclically $5$-connected cubic planar graph (Holton--McKay Fig.\nonbreakingspace 2.10), drawn crossing-free and coloured by the fixed parity labelling (blue even, orange odd). $T$ is $5$-connected and not an intertwining tree, yet is a bridge-derived level graph: the two solid green edges $\{6,19\}$ and $\{20,22\}$ are the bridge edges introduced by the two bridge switches carrying its witness Even Level Graph (source $19$) to $T$. Each green edge is a bridge of its parity subgraph -- $\{6, 19\}$ in the even subgraph, $\{20,22\}$ in the odd -- so no new cycle, and in particular no odd cycle, is created.}}{10}{figure.6}\protected@file@percent } \newlabel{fig:n24-dual}{{6}{10}{The $24$-vertex dual $T$ of the unique $44$-vertex non-Hamiltonian cyclically $5$-connected cubic planar graph (Holton--McKay Fig.~2.10), drawn crossing-free and coloured by the fixed parity labelling (blue even, orange odd). $T$ is $5$-connected and not an intertwining tree, yet is a bridge-derived level graph: the two solid green edges $\{6,19\}$ and $\{20,22\}$ are the bridge edges introduced by the two bridge switches carrying its witness Even Level Graph (source $19$) to $T$. Each green edge is a bridge of its parity subgraph -- $\{6, 19\}$ in the even subgraph, $\{20,22\}$ in the odd -- so no new cycle, and in particular no odd cycle, is created}{figure.6}{}} +\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{10}{section*.4}\protected@file@percent } \gdef \@abspage@last{10} diff --git a/papers/even_level_graph_generators/paper.log b/papers/even_level_graph_generators/paper.log index 5d934f5..17d73f1 100644 --- a/papers/even_level_graph_generators/paper.log +++ b/papers/even_level_graph_generators/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) 22 MAY 2026 13:04 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 22 MAY 2026 16:17 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -354,113 +354,100 @@ File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Liv e )) [1{/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map}] - +[2] + File: fig_levels.png Graphic file (type png) -Package pdftex.def Info: fig_levels.png used on input line 189. +Package pdftex.def Info: fig_levels.png used on input line 213. 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PDF statistics: 226 PDF objects out of 1000 (max. 8388607) 170 compressed objects within 2 object streams diff --git a/papers/even_level_graph_generators/paper.pdf b/papers/even_level_graph_generators/paper.pdf index 566baaa..71825b9 100644 Binary files a/papers/even_level_graph_generators/paper.pdf and b/papers/even_level_graph_generators/paper.pdf differ diff --git a/papers/even_level_graph_generators/paper.tex b/papers/even_level_graph_generators/paper.tex index d973c44..d527bfa 100644 --- a/papers/even_level_graph_generators/paper.tex +++ b/papers/even_level_graph_generators/paper.tex @@ -59,7 +59,8 @@ \begin{document} -\title{Even Level Graph Generators} +\title[Even Level Graph Generators]{Even Level Graph Generators:\\ +a constructive conjecture stronger than the Four Color Theorem} % Remove any unused author tags. @@ -93,9 +94,14 @@ $2$-coloring of an Even Level Graph. The second family is the sets each inducing a tree, which are $4$-colorable by coloring the two trees from disjoint pairs of colors. We conjecture that every maximal planar graph is a bridge-derived level graph, an intertwining tree, or -both; since both families are $4$-colorable by construction, the -conjecture would give a constructive proof of the four color theorem for -triangulations, and hence for all planar graphs. We show that a +both. Since both families are $4$-colorable by construction, the +conjecture implies the four color theorem for triangulations, and hence +for all planar graphs; in fact it is \emph{strictly stronger}, demanding +not merely that a $4$-coloring exist but that every triangulation be +assembled by one of these two explicit constructions. A proof would +therefore be a new, constructive proof of the four color theorem -- and +correspondingly the conjecture is at least as hard, and very likely +harder, than that theorem. We show that a triangulation is an intertwining tree exactly when its dual is Hamiltonian, so every triangulation on at most $20$ vertices is an intertwining tree and the first possible failures occur at $n = 21$, at @@ -153,7 +159,25 @@ Our central question is whether these two families exhaust all triangulations (Conjecture~\ref{conj:every-triangulation-derived}). As both families consist of $4$-colorable graphs, an affirmative answer would constitute a -constructive proof of the four color theorem for triangulations. +constructive proof of the four color theorem for triangulations, and +hence for all planar graphs. + +We emphasize that the conjecture is a \emph{stronger} statement than the +four color theorem, not an equivalent reformulation of it. A proper +$4$-coloring with its colors grouped as $\{1,2\}\mid\{3,4\}$ is exactly a +partition of the vertices into two parts each inducing a bipartite +subgraph, so the four color theorem is precisely the assertion that every +triangulation admits such a partition. The conjecture asserts strictly +more: that the partition can be realized \emph{constructively} -- as the +level parity of an Even Level Graph reached by bridge switches, or as a +split into two induced \emph{trees}. The four color theorem alone supplies +neither construction; bridge-derivability in particular is a reachability +condition well beyond the bare existence of a $4$-coloring, so the +conjecture implies the four color theorem but is not implied by it. +A proof would accordingly be a new, constructive proof of the four color +theorem, and the conjecture is at least as hard to settle -- and, absent +any structural characterization of the bridge-derived family, very likely +harder. We connect the two constructions through duality: a triangulation is an intertwining tree if and only if its dual is Hamiltonian