%% filename: amsart-template.tex
%% version: 1.1
%% date: 2014/07/24
%%
%% American Mathematical Society
%% Technical Support
%% Publications Technical Group
%% 201 Charles Street
%% Providence, RI 02904
%% USA
%% tel: (401) 455-4080
%%      (800) 321-4267 (USA and Canada only)
%% fax: (401) 331-3842
%% email: tech-support@ams.org
%%
%% Copyright 2008-2010, 2014 American Mathematical Society.
%%
%% This work may be distributed and/or modified under the
%% conditions of the LaTeX Project Public License, either version 1.3c
%% of this license or (at your option) any later version.
%% The latest version of this license is in
%%   http://www.latex-project.org/lppl.txt
%% and version 1.3c or later is part of all distributions of LaTeX
%% version 2005/12/01 or later.
%%
%% This work has the LPPL maintenance status `maintained'.
%%
%% The Current Maintainer of this work is the American Mathematical
%% Society.
%%
%% ====================================================================

%     AMS-LaTeX v.2 template for use with amsart
%
%     Remove any commented or uncommented macros you do not use.

\documentclass{amsart}

\usepackage{hyperref}
\usepackage{enumitem}
\usepackage{graphicx}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{xca}[theorem]{Exercise}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{question}[theorem]{Question}
\newtheorem{observation}[theorem]{Observation}

\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}

\numberwithin{equation}{section}

\begin{document}

\title{Even Level Graph Generators}

%    Remove any unused author tags.

%    author one information
\author{Eric Bauerfeld}
\address{}
\curraddr{}
\email{}
\thanks{}


\subjclass[2010]{Primary }

\keywords{}

\date{}

\dedicatory{}

\begin{abstract}
\end{abstract}

\maketitle

\section{Introduction}

\section{Definitions}

Throughout, $G = (V, E)$ is a plane maximal planar graph (a triangulation)
with a fixed planar embedding $\Pi_G$. We write $|V| = n$, so $|E| = 3n - 6$
and $G$ has $2n - 4$ triangular faces.

\begin{definition}[Level source]
A \emph{level source} of $G$ is any vertex $v \in V$; we write
$S = \{v\}$ for the level-0 source.
\end{definition}

\begin{definition}[Levels]
Given a level source $S \subseteq V$, the \emph{level} of $v \in V$ is
$\ell_G(v) = \mathrm{dist}_G(v, S)$, the graph distance from $v$ to the nearest
source vertex.
\end{definition}

\begin{figure}[h]
\centering
\includegraphics[width=0.55\textwidth]{fig_levels.png}
\caption{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.}
\label{fig:levels}
\end{figure}

\begin{definition}[Level cycle]
A \emph{level cycle} of $G$ (with respect to a level source $S$) is a
simple cycle in $G$ all of whose vertices have the same level.
\end{definition}

\begin{figure}[h]
\centering
\includegraphics[width=0.55\textwidth]{fig_level_cycle.png}
\caption{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 $1$.}
\label{fig:level-cycle}
\end{figure}

\begin{definition}[Edge switch]
\label{def:edge-switch}
Let $G$ be a triangulation with level source $S$, and let $e = uv$ be an
edge of a level cycle of $G$. The \emph{edge switch} at $e$ is the edge
flip on $e$: writing $uvw$ and $uvx$ for the two triangular faces of $G$
containing $e$, the edge $uv$ is removed and the edge $wx$ is added. As
with any edge flip, the result is a triangulation on the same vertex set
provided $w$ and $x$ are non-adjacent in $G$.
\end{definition}

\begin{figure}[h]
\centering
\includegraphics[width=0.95\textwidth]{fig_edge_switch.png}
\caption{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 $G$.}
\label{fig:edge-switch}
\end{figure}

\begin{definition}[Parity subgraph]
Let $G$ be a triangulation with level source $S$, and let $G'$ be a triangulation
on the same vertex set as $G$. The \emph{even parity subgraph} $E_{G,S}(G')$ is
the subgraph of $G'$ induced by $\{v \in V : \ell_G(v) \equiv 0 \pmod 2\}$. The
\emph{odd parity subgraph} is defined analogously for odd $\ell_G$.
\end{definition}

\begin{figure}[h]
\centering
\includegraphics[width=\textwidth]{fig_parity_subgraph.png}
\caption{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 $G'$.}
\label{fig:parity-subgraph}
\end{figure}

\section{Outerplanarity of level components}
\label{sec:outerplanar-components}

For each integer $k \geq 0$ and each $(G, S)$, write $L_k$ for the
subgraph of $G$ induced by the level-$k$ vertices. A \emph{level
component} of $G$ (with respect to $S$) is a connected component of
some $L_k$.

\begin{theorem}
\label{thm:outerplanar-component}
For every plane triangulation $G$ and every level source $S$ of $G$,
every level component of $G$ is outerplanar.
\end{theorem}

\begin{proof}
Since every subgraph of an outerplanar graph is outerplanar, it suffices
to show that each level subgraph $L_k$ is outerplanar. For $k = 0$,
$L_0 = S$ is a single vertex and is trivially outerplanar.

Fix $k \geq 1$ and let $D_k$ be the drawing of $L_k$ inherited from
$\Pi_G$. Let $F^\ast$ be the face of $D_k$ containing the source.
Suppose for contradiction that some $u \in L_k$ does not lie on
$\partial F^\ast$, so $u$ lies on the boundary of some other face of
$D_k$. Take any path $P$ in $G$ from $v_0 \in S$ to $u$. As a curve in
$\Pi_G$, $P$ starts in $F^\ast$ and ends at a point off $\partial
F^\ast$, so it must transition from $F^\ast$ to a different face of
$D_k$; in a planar embedding this can happen only at a vertex of
$D_k$, that is, at a level-$k$ vertex $w$ on $P$. Either $w \neq u$
(so $P$ has length $\geq \mathrm{dist}_G(S, w) + 1 \geq k + 1$), or
$w = u$ (contradicting $u \notin \partial F^\ast$). Since every
$S$-to-$u$ path has length $\geq k + 1$, $\mathrm{dist}_G(S, u) \geq
k + 1$, contradicting $u \in L_k$.
\end{proof}

\section{Even Level Graphs}
\label{sec:even-level-graphs}

\begin{definition}[Even Level Graph]
\label{def:even-level-graph}
A plane triangulation $G$ with level source $S$ is an \emph{Even Level
Graph} if every level cycle of $G$ has even length.
\end{definition}

\begin{theorem}
\label{thm:even-level-4colorable}
Every Even Level Graph is $4$-colorable.
\end{theorem}

\begin{proof}
Since adjacent vertices in $G$ have levels differing by at most $1$,
any edge between two same-parity endpoints in fact connects two
vertices at the same level. Hence
\[
  E_{G,S}(G) \;=\; \bigsqcup_{i \geq 0} L_{2i},
  \qquad
  O_{G,S}(G) \;=\; \bigsqcup_{i \geq 0} L_{2i+1},
\]
and each $L_k$ is bipartite because its cycles are level cycles of
$G$, which have even length by hypothesis. Choose a $2$-coloring of
$E_{G,S}(G)$ in $\{\text{red}, \text{blue}\}$ and a $2$-coloring of
$O_{G,S}(G)$ in $\{\text{yellow}, \text{green}\}$. Same-parity edges
of $G$ are properly colored by the respective bipartition;
opposite-parity edges connect $\{\text{red}, \text{blue}\}$ to
$\{\text{yellow}, \text{green}\}$. The combined assignment is a
proper $4$-coloring of $G$.
\end{proof}

\begin{definition}[Derived level graph]
\label{def:derived-level-graph}
Let $G$ be an Even Level Graph with level source $S$, and let $E$ and
$O$ denote the edge sets of the even and odd parity subgraphs
$E_{G,S}(G)$ and $O_{G,S}(G)$. A \emph{derived level graph} of $G$ is
a triangulation $G'$ on the same vertex set as $G$ obtained by a
sequence of edge switches (Definition~\ref{def:edge-switch}), each
acting on an edge of $E$ or of $O$. We do not update $E$ or $O$ to
reflect the level structure of intermediate triangulations: throughout
the sequence, an edge is classified as belonging to $E$ (resp.\ $O$) if
and only if both of its endpoints have even (resp.\ odd) level in $G$.

A derived level graph $G'$ is \emph{valid} if both $E_{G,S}(G')$ and
$O_{G,S}(G')$ contain only even cycles.
\end{definition}

\begin{definition}[Bridge switch]
\label{def:bridge-switch}
Let $G'$ be a triangulation reached from an Even Level Graph $G$, with
parity classes inherited from $G$ as in
Definition~\ref{def:derived-level-graph}. An edge switch on an edge
$e \in E \cup O$ of $G'$, replacing $uvw, uvx$ by the edge $wx$, is a
\emph{bridge switch} if either
\begin{itemize}
\item the new edge $wx$ is a cross-parity edge (one endpoint even, the
other odd), so $wx$ enters neither parity subgraph; or
\item $wx$ is a same-parity edge and is a \emph{bridge} in the parity
subgraph it joins -- that is, $w$ and $x$ lie in different connected
components of that parity subgraph, so adding $wx$ creates no new cycle.
\end{itemize}
\end{definition}

\begin{definition}[Bridge-derived level graph]
\label{def:bridge-derived-level-graph}
A \emph{bridge-derived level graph} of an Even Level Graph $G$ is a
triangulation obtained from $G$ by a sequence of bridge switches
(Definition~\ref{def:bridge-switch}).
\end{definition}

Because a bridge switch never closes a cycle in a parity subgraph, it
never introduces an odd cycle there. As an Even Level Graph has
bipartite parity subgraphs (every level cycle is even), every
bridge-derived level graph has bipartite parity subgraphs as well, and
so is automatically a valid derived level graph. Equivalently, the
first Betti number of each parity subgraph is non-increasing along any
sequence of bridge switches.

\begin{definition}[Intertwining tree]
\label{def:intertwining-tree}
A maximal planar graph $G$ is an \emph{intertwining tree} if its
vertex set can be partitioned into two sets $A$ and $B$ such that
both induced subgraphs $G[A]$ and $G[B]$ are trees.
\end{definition}

\begin{theorem}
\label{thm:intertwining-iff-hamiltonian-dual}
A maximal planar graph $G$ is an intertwining tree if and only if its
dual $G^\ast$ has a Hamiltonian cycle.
\end{theorem}

\begin{proof}
($\Rightarrow$) Let $V(G) = A \sqcup B$ with $G[A]$ and $G[B]$ trees.
Every triangular face $\{x,y,z\}$ of $G$ meets both $A$ and $B$: if all
three vertices were in $A$ the triangle would be a cycle in the tree
$G[A]$, and likewise for $B$. Draw a closed curve through the faces of
$G$ separating the $A$-vertices from the $B$-vertices within each face.
Since every face is split, the curve visits every face exactly once and
crosses an edge of $G$ precisely when that edge joins $A$ to $B$; it is
therefore a Hamiltonian cycle of $G^\ast$.

($\Leftarrow$) Let $H$ be a Hamiltonian cycle of $G^\ast$. Drawn in the
plane, $H$ is a Jordan curve visiting every face of $G$ once; let $A$
and $B$ be the vertices of $G$ interior and exterior to $H$. The
$2n-4$ edges of $H$ cross exactly the edges of $G$ between $A$ and $B$,
leaving $(3n-6)-(2n-4) = n-2$ edges inside $G[A]$ and $G[B]$ together.
The edges inside $A$ lie in the disk bounded by $H$ and span $A$
without enclosing a face (each face is cut by $H$), so $G[A]$ is a tree;
likewise $G[B]$.
\end{proof}

\begin{conjecture}
\label{conj:every-triangulation-derived}
Every maximal planar graph is a bridge-derived level graph of some Even
Level Graph, an intertwining tree, or both.
\end{conjecture}

Since a bridge-derived level graph is automatically a valid derived level
graph, this is a stronger statement than the corresponding conjecture
phrased with arbitrary $E/O$ switches; it is also the form that the
evidence below actually supports.

By Theorem~\ref{thm:intertwining-iff-hamiltonian-dual}, the
intertwining-tree disjunct fails for $G$ exactly when $G^\ast$ is a
counterexample to Tait's conjecture. The smallest such $G^\ast$ have
$38$ vertices (Holton--McKay~\cite{holton-mckay}, exactly $6$ graphs),
so the smallest triangulations that are not intertwining trees occur at
$n = 21$ and there are exactly $6$ of them. Below $n = 21$ every
maximal planar graph is an intertwining tree, which is why the
disjunction holds trivially in that range.

\subsection*{Empirical status}

For each isomorphism class of maximal planar graphs on $n$ vertices,
we ask whether (i) some isomorphic representative is a bridge-derived
level graph of some Even Level Graph, and/or (ii) it is an intertwining
tree. The conjecture holds for the class iff at least one of (i), (ii)
holds. Below $n = 21$ condition (ii) holds for \emph{every} class, so
the table mainly records how far the bridge-derived disjunct (i) reaches
on its own. We classified bridge-derivability exhaustively for
$n \le 9$, where every backward bridge-orbit can be enumerated in full.

\begin{center}
\begin{tabular}{rcccccc}
$n$ & \# iso & bridge only & inter.\ only & both & missing & status \\\hline
$6$  & $2$    & $0$ & $0$  & $2$    & $0$ & holds \\
$7$  & $5$    & $0$ & $1$  & $4$    & $0$ & holds \\
$8$  & $14$   & $0$ & $2$  & $12$   & $0$ & holds \\
$9$  & $50$   & $0$ & $14$ & $36$   & $0$ & holds \\
\end{tabular}
\end{center}

\noindent
Here ``bridge only'' counts classes that are bridge-derived but not
intertwining trees, ``inter.\ only'' the reverse, and ``both'' the
intersection; ``missing'' counts classes that are neither (a
counterexample). The ``bridge only'' column is $0$ throughout this range
precisely because every class is an intertwining tree for $n \le 20$;
the ``inter.\ only'' counts ($1,2,14$) are the classes that the
bridge-derived disjunct alone does not yet reach, showing that
bridge-derivability is strictly weaker than ``intertwining tree'' here
and that the two disjuncts genuinely complement one another.

\subsection*{The boundary case $n = 21$}

The first triangulations that are \emph{not} intertwining trees are the
six duals of the Holton--McKay graphs, at $n = 21$. For the disjunction
to survive at $n = 21$, each of these six must be a valid derived level
graph. We find:
\begin{itemize}
\item All six duals are confirmed not intertwining trees (exhaustive
check of all $2^{20}-1$ vertex bipartitions), consistent with
Theorem~\ref{thm:intertwining-iff-hamiltonian-dual}.
\item Two of the six are themselves Even Level Graphs (for a suitable
source vertex), hence trivially valid derived level graphs. So the
disjunction holds for them through the derived-level-graph disjunct --
the first instances where that disjunct does work the intertwining-tree
disjunct cannot.
\item The remaining four are not Even Level Graphs for any source, and
their full $E/O$-orbits ($\sim\!10^8$ states per source labelling) are
far too large to exhaust. Restricting to \emph{bridge switches}
(Definition~\ref{def:bridge-switch}) shrinks the relevant orbits by
roughly two orders of magnitude and, crucially, keeps every reachable
triangulation valid. A backward bridge-switch search over the valid
parity partitions found an Even Level Graph witness for each of the
four, so all four are \emph{bridge-derived level graphs}
(Definition~\ref{def:bridge-derived-level-graph}) and hence valid
derived level graphs. The witnessing orbits are small -- between a few
hundred and $\sim\!1.7\times 10^5$ states -- even though other parity
partitions of the same triangulations have orbits exceeding $10^6$;
finding one good partition suffices. Each witness is in fact only a
\emph{handful} of bridge switches from its dual: the explicit Even Level
Graph, parity labelling, and bridge-switch sequence are recorded for all
six -- path lengths $3,1,2,4$ for these four and $0$ for the two that are
Even Level Graphs outright -- and each step has been verified to be a
valid bridge switch.
\end{itemize}
Thus at $n = 21$ the disjunction is confirmed for all six critical iso
classes: two are Even Level Graphs outright, and the other four are
bridge-derived level graphs. The bridge-switch restriction is what made
the search tractable -- it both shrinks the orbit and guarantees
validity, so any Even Level Graph located in a backward orbit is an
immediate witness. Table~\ref{tab:n21} records the outcome for each dual.

\begin{table}[ht]
\centering
\begin{tabular}{cccc}
dual & intertwining tree & Even Level Graph source & bridge switches to ELG \\\hline
$0$ & no & --       & $3$ \\
$1$ & no & $10$     & $0$ \\
$2$ & no & $9$      & $0$ \\
$3$ & no & --       & $1$ \\
$4$ & no & --       & $2$ \\
$5$ & no & --       & $4$ \\
\end{tabular}
\caption{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.}
\label{tab:n21}
\end{table}

\begin{figure}[ht]
\centering
\includegraphics[width=\textwidth]{figures/n21_witnesses.png}
\caption{The witness Even Level Graph for each of the six Holton--McKay
duals, drawn radially by level (source at the centre, level-$k$ vertices on
ring $k$) and coloured by parity (blue even, orange odd). Dashed red edges
are the same-parity edges flipped by the bridge switches; solid green edges
are the bridge edges they introduce; applying the switches turns each Even
Level Graph into the corresponding dual. Duals $1$ and $2$ are Even Level
Graphs outright, so no switch is shown.}
\label{fig:n21-witnesses}
\end{figure}

\begin{thebibliography}{9}

\bibitem{holton-mckay}
D.~A. Holton and B.~D. McKay.
\emph{The smallest non-Hamiltonian 3-connected cubic planar graphs have
38 vertices}.
Journal of Combinatorial Theory, Series B, 45(3):305--319, 1988.

\end{thebibliography}

\end{document}
