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Add figures, Kempe-cycle section, and restriction experiments

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Adds two TikZ figures (boundary-state worst cases and annular cycle
counterexample), a new subsection on Kempe-cycle conservation across
medial tires, and the experiment scripts/findings for the medial tire
restriction search and annular cycle condition check.

Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>
This commit is contained in:
didericis
2026-06-11 01:16:05 -04:00
parent 20fe6c24ca
commit 4062e87c61
10 changed files with 1418 additions and 239 deletions
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papers/medial_tire_decompositions_of_plane_triangulations/paper.tex
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@@ -234,6 +234,188 @@ same restriction to $A(T)$ are identical, so the restriction map is
injective. The stated inequality follows.
\end{proof}
\begin{figure}[h]
\centering
\begin{tikzpicture}[scale=0.82,
ann/.style={circle, fill=black, inner sep=1.3pt},
outv/.style={circle, draw=blue!70!black, fill=blue!12, inner sep=1.7pt},
inv/.style={circle, draw=red!70!black, fill=red!12, inner sep=1.7pt},
mededge/.style={black!62, line width=0.45pt},
attach/.style={black!45, line width=0.38pt}]
\newcommand{\hexannulus}[1]{
\coordinate (#1a0) at (90:1);
\coordinate (#1a1) at (30:1);
\coordinate (#1a2) at (-30:1);
\coordinate (#1a3) at (-90:1);
\coordinate (#1a4) at (-150:1);
\coordinate (#1a5) at (150:1);
\draw[mededge] (#1a0)--(#1a1)--(#1a2)--(#1a3)--(#1a4)--(#1a5)--cycle;
\foreach \i in {0,...,5} \node[ann] at (#1a\i) {};
}
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\hexannulus{x}
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\coordinate (xb5) at (120:1.46);
\foreach \i/\j in {0/1,1/2,2/3,3/4,4/5} {
\draw[attach] (xb\i)--(xa\i);
\draw[attach] (xb\i)--(xa\j);
}
\draw[attach] (xb5)--(xa5);
\draw[attach] (xb5)--(xa0);
\foreach \i in {0,1,2} \node[inv] at (xb\i) {\scriptsize I};
\foreach \i in {3,4,5} \node[outv] at (xb\i) {\scriptsize O};
\node at (0,-1.86) {\scriptsize \texttt{IIIOOO}};
\node at (0,-2.15) {\scriptsize $\min |R_T(\alpha)|=1$};
\end{scope}
\begin{scope}
\hexannulus{y}
\coordinate (yb0) at (60:0.48);
\coordinate (yb1) at (0:0.48);
\coordinate (yb2) at (-60:1.46);
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\draw[attach] (yb\i)--(ya\i);
\draw[attach] (yb\i)--(ya\j);
}
\draw[attach] (yb5)--(ya5);
\draw[attach] (yb5)--(ya0);
\foreach \i in {0,1,3} \node[inv] at (yb\i) {\scriptsize I};
\foreach \i in {2,4,5} \node[outv] at (yb\i) {\scriptsize O};
\node at (0,-1.86) {\scriptsize \texttt{IIOIOO}};
\node at (0,-2.15) {\scriptsize $\min |R_T(\alpha)|=1$};
\end{scope}
\begin{scope}[shift={(4.8,0)}]
\hexannulus{z}
\coordinate (zb0) at (60:0.48);
\coordinate (zb1) at (0:0.48);
\coordinate (zb2) at (-60:1.46);
\coordinate (zb3) at (-120:1.46);
\coordinate (zb4) at (180:0.48);
\coordinate (zb5) at (120:1.46);
\foreach \i/\j in {0/1,1/2,2/3,3/4,4/5} {
\draw[attach] (zb\i)--(za\i);
\draw[attach] (zb\i)--(za\j);
}
\draw[attach] (zb5)--(za5);
\draw[attach] (zb5)--(za0);
\foreach \i in {0,1,4} \node[inv] at (zb\i) {\scriptsize I};
\foreach \i in {2,3,5} \node[outv] at (zb\i) {\scriptsize O};
\node at (0,-1.86) {\scriptsize \texttt{IIOOIO}};
\node at (0,-2.15) {\scriptsize $\min |R_T(\alpha)|=1$};
\end{scope}
\end{tikzpicture}
\caption{Three six-face full medial tire graphs found by the boundary-state
restriction search. Black vertices are annular medial vertices; blue
vertices are outer boundary medial vertices and red vertices are inner
boundary medial vertices. The word below each diagram records the
outer/inner type of the six annular faces in cyclic order. Boundary
states are identified only up to colour permutation, not by rotation or
reflection of the boundary order.}
\label{fig:medial-restriction-worst-cases}
\end{figure}
\begin{figure}[h]
\centering
\begin{tikzpicture}[scale=0.82,
mededge/.style={black!38, line width=0.42pt},
cycleedge/.style={black, line width=1.25pt},
czero/.style={circle, draw=blue!65!black, fill=blue!18, inner sep=1.6pt},
cone/.style={circle, draw=red!65!black, fill=red!18, inner sep=1.6pt},
ctwo/.style={circle, draw=green!45!black, fill=green!20, inner sep=1.6pt}]
\coordinate (n12) at (90:1.45);
\coordinate (n13) at (30:1.45);
\coordinate (n14) at (-30:1.45);
\coordinate (n15) at (-90:1.45);
\coordinate (n16) at (-150:1.45);
\coordinate (n17) at (150:1.45);
\coordinate (n23) at (1.85,1.55);
\coordinate (n34) at (2.45,0.00);
\coordinate (n45) at (1.85,-1.55);
\coordinate (n56) at (-0.15,-2.45);
\coordinate (n67) at (-1.85,-1.55);
\coordinate (n27) at (-1.85,1.55);
\coordinate (n26) at (-0.30,2.55);
\coordinate (n36) at (0.30,0.55);
\coordinate (n35) at (0.45,-0.85);
\draw[mededge] (n12)--(n13);
\draw[mededge] (n12)--(n17);
\draw[mededge] (n12)--(n23);
\draw[mededge] (n12)--(n27);
\draw[mededge] (n13)--(n14);
\draw[mededge] (n13)--(n23);
\draw[mededge] (n13)--(n34);
\draw[mededge] (n14)--(n15);
\draw[mededge] (n14)--(n34);
\draw[mededge] (n14)--(n45);
\draw[mededge] (n15)--(n16);
\draw[mededge] (n15)--(n45);
\draw[mededge] (n15)--(n56);
\draw[mededge] (n16)--(n17);
\draw[mededge] (n16)--(n56);
\draw[mededge] (n16)--(n67);
\draw[mededge] (n17)--(n27);
\draw[mededge] (n17)--(n67);
\draw[mededge] (n23)--(n26);
\draw[mededge] (n23)--(n36);
\draw[mededge] (n26)--(n27);
\draw[mededge] (n26)--(n36);
\draw[mededge] (n26)--(n67);
\draw[mededge] (n27)--(n67);
\draw[mededge] (n34)--(n35);
\draw[mededge] (n34)--(n45);
\draw[mededge] (n35)--(n36);
\draw[mededge] (n35)--(n45);
\draw[mededge] (n35)--(n56);
\draw[mededge] (n36)--(n56);
\draw[cycleedge] (n12)--(n13)--(n14)--(n15)--(n16)--(n17)--cycle;
\node[czero] (v12) at (n12) {\scriptsize $12$};
\node[cone] (v13) at (n13) {\scriptsize $13$};
\node[ctwo] (v14) at (n14) {\scriptsize $14$};
\node[czero] (v15) at (n15) {\scriptsize $15$};
\node[ctwo] (v16) at (n16) {\scriptsize $16$};
\node[cone] (v17) at (n17) {\scriptsize $17$};
\node[ctwo] (v23) at (n23) {\scriptsize $23$};
\node[cone] (v26) at (n26) {\scriptsize $26$};
\node[ctwo] (v27) at (n27) {\scriptsize $27$};
\node[czero] (v34) at (n34) {\scriptsize $34$};
\node[ctwo] (v35) at (n35) {\scriptsize $35$};
\node[czero] (v36) at (n36) {\scriptsize $36$};
\node[cone] (v45) at (n45) {\scriptsize $45$};
\node[cone] (v56) at (n56) {\scriptsize $56$};
\node[czero] (v67) at (n67) {\scriptsize $67$};
\node[anchor=west] at (3.0,1.45) {\scriptsize colour $0$};
\node[czero] at (2.82,1.45) {};
\node[anchor=west] at (3.0,1.05) {\scriptsize colour $1$};
\node[cone] at (2.82,1.05) {};
\node[anchor=west] at (3.0,0.65) {\scriptsize colour $2$};
\node[ctwo] at (2.82,0.65) {};
\node[anchor=west, text width=2.35cm] at (2.82,-0.15)
{\scriptsize thick cycle: annular medial cycle for source $1$};
\end{tikzpicture}
\caption{A proper vertex $3$-colouring of the full medial graph of the
first seven-vertex counterexample found by the experiment. The medial
vertex labelled $ij$ corresponds to the edge $(i,j)$ of the
triangulation. For the vertex-source decomposition at source $1$, the
highlighted annular medial cycle has colour counts $(2,2,2)$, so it is
not coloured with two colours except at at most one vertex.}
\label{fig:medial-annular-cycle-counterexample}
\end{figure}
\begin{definition}[Boundary medial vertices]
\label{def:boundary-medial-vertices}
Let $T$ be a tire tread and let $\Gamma_T$ be the corresponding dual
@@ -387,6 +569,221 @@ move the obstruction into finite boundary-state restrictions carried by
annular medial tire pieces.
\end{remark}
\subsection{Kempe-cycle conservation across medial tires}
We now record an additional structure carried by proper
$3$-colourings of medial graphs. This structure will be useful for
describing how colourings glue across level cycles.
Let $G$ be a plane triangulation and let $M=M(G)$ be its medial graph.
Let
\[
\varphi:V(M)\to\{1,2,3\}
\]
be a proper $3$-colouring of $M$. For a two-element colour set
$P=\{a,b\}\subseteq\{1,2,3\}$, let $M_P$ denote the subgraph of $M$
induced by the vertices of colours $a$ and $b$.
Since $M$ is $4$-regular and $\varphi$ is proper, every vertex of
$M_P$ has degree $2$ in $M_P$. Hence every component of $M_P$ is a
cycle. We call these components the $P$-Kempe cycles of $\varphi$.
\begin{lemma}[Kempe cycles are cycles]
Let $G$ be a plane triangulation, let $M=M(G)$, and let
$\varphi$ be a proper $3$-colouring of $M$. For each
$P\in\{\{1,2\},\{2,3\},\{3,1\}\}$, every component of $M_P$ is a cycle.
\end{lemma}
\begin{proof}
Let $v\in V(M_P)$. In the medial graph $M$, the vertex $v$ has degree
$4$. Since $\varphi$ is a proper $3$-colouring, none of the neighbours
of $v$ has colour $\varphi(v)$. Thus all four neighbours of $v$ have
one of the two colours different from $\varphi(v)$.
In the medial graph of a plane triangulation, the neighbours of a
medial vertex occur in two opposite pairs corresponding to the two
faces incident with the corresponding edge of $G$. Around each such
triangular face, the three medial vertices receive all three colours.
Consequently, at $v$ there are exactly two neighbours of each colour
different from $\varphi(v)$. It follows that, in the subgraph induced
by any two colours $P$, every vertex has degree $2$. Hence each
component of $M_P$ is a cycle.
\end{proof}
Let $T$ be a medial tire region. We regard $T$ as an annular transition
region whose boundary consists of one outer level cycle and finitely
many inner level cycles:
\[
\partial T = C_0\sqcup C_1\sqcup\cdots\sqcup C_m.
\]
Here $C_0$ is the outer level cycle of $T$, and the cycles
$C_1,\ldots,C_m$ are the inner level cycles. Each inner level cycle
$C_i$ is also the outer level cycle of the corresponding child region
in the tire tree.
The following lemma is the basic conservation principle.
\begin{lemma}[Kempe-cycle conservation across level cycles]
Let $C$ be a level cycle of $M$ separating a parent side from a child
side. Let $K$ be a $P$-Kempe cycle for some
$P\in\{\{1,2\},\{2,3\},\{3,1\}\}$. Then $K$ cannot enter the child side
of $C$ without also leaving it.
Equivalently, the incidences of $K$ with $C$ are paired by the
components of $K$ lying on the child side of $C$, and also paired by the
components of $K$ lying on the parent side of $C$.
\end{lemma}
\begin{proof}
By the preceding lemma, $K$ is a cycle. The level cycle $C$ separates
the sphere into two closed regions, which we call the parent side and
the child side. Consider the intersection of $K$ with one of these
regions. Since $K$ is a cycle, no component of this intersection can
have exactly one boundary endpoint on $C$. Each component is either
closed within the region, or is a path with two boundary endpoints on
$C$. Thus every entrance through $C$ is paired with an exit through
$C$.
\end{proof}
More generally, let $T$ be a medial tire region with boundary
\[
\partial T = C_0\sqcup C_1\sqcup\cdots\sqcup C_m.
\]
For a $P$-Kempe cycle $K$, every component of $K\cap T$ is either a
cycle contained in $T$, or a path with two endpoints on
$\partial T$. Thus the $P$-Kempe arcs inside $T$ define a pairing of
the $P$-coloured boundary incidences of
\[
C_0\sqcup C_1\sqcup\cdots\sqcup C_m.
\]
This motivates the following refinement of boundary states.
\begin{definition}[Kempe-enhanced boundary state]
Let $T$ be a medial tire region with outer level cycle $C_0$ and inner
level cycles $C_1,\ldots,C_m$. Let
\[
\mathcal C(T)=C_0\sqcup C_1\sqcup\cdots\sqcup C_m.
\]
A \emph{Kempe-enhanced boundary state} on $T$ consists of the following
data:
\begin{enumerate}
\item a boundary colouring
\[
\alpha:V(\mathcal C(T))\to\{1,2,3\};
\]
\item for each colour pair
\[
P\in\{\{1,2\},\{2,3\},\{3,1\}\},
\]
a pairing $\pi_P$ of the $P$-coloured boundary incidences of
$\mathcal C(T)$ induced by the $P$-Kempe arcs lying inside $T$.
\end{enumerate}
We write such a state as
\[
\kappa=(\alpha,\pi_{12},\pi_{23},\pi_{31}).
\]
\end{definition}
Given a proper $3$-colouring $\varphi$ of the medial tire graph
$M(T)$, the restriction of $\varphi$ to the boundary level cycles gives
the boundary colouring $\alpha$, while the two-colour Kempe arcs inside
$T$ give the pairings $\pi_{12},\pi_{23},\pi_{31}$. Thus $\varphi$
determines a Kempe-enhanced boundary state, denoted
\[
\kappa_T(\varphi).
\]
\begin{definition}[Kempe-enhanced restriction relation]
The \emph{Kempe-enhanced restriction relation} of $T$ is
\[
\mathcal K_T
=
\left\{
\kappa_T(\varphi):
\varphi \text{ is a proper }3\text{-colouring of } M(T)
\right\}.
\]
This refines the ordinary boundary-colouring relation by recording not
only which boundary colourings extend across $T$, but also how the
two-colour Kempe cycles are routed through the annular tire region.
\end{definition}
The annular structure of a tire is useful in two distinct ways. First,
it gives a bounded transition region between level cycles: the colouring
of the annular medial cycle controls, and in many cases determines, the
colouring of the remaining medial tire vertices. Thus the number of
possible transition states is bounded in terms of the annular structure,
rather than the total size of the subtree below the tire. Second, it
describes how the outer level cycle and the inner level cycles are
related by Kempe arcs. The level cycles are the gluing interfaces, while
the annular tire is the transition operator between them.
\begin{definition}[Kempe-compatible gluing]
Let $T$ be a medial tire region and let $U$ be a child region glued to
$T$ along a common level cycle $C$. Thus $C$ is an inner level cycle of
$T$ and the outer level cycle of $U$.
Let
\[
\kappa_T=(\alpha_T,\pi^T_{12},\pi^T_{23},\pi^T_{31})
\in \mathcal K_T
\]
and
\[
\kappa_U=(\alpha_U,\pi^U_{12},\pi^U_{23},\pi^U_{31})
\in \mathcal K_U.
\]
We say that $\kappa_T$ and $\kappa_U$ are \emph{Kempe-compatible along
$C$} if:
\begin{enumerate}
\item the boundary colourings agree on $C$:
\[
\alpha_T|_{V(C)}=\alpha_U|_{V(C)};
\]
\item for each colour pair
\[
P\in\{\{1,2\},\{2,3\},\{3,1\}\},
\]
the pairings $\pi^T_P$ and $\pi^U_P$ compose along the
$P$-coloured incidences of $C$ without producing an unpaired endpoint.
\end{enumerate}
When these conditions hold, the composed pairings determine a
Kempe-enhanced boundary state on the exposed boundary of
$T\cup_C U$.
\end{definition}
In these terms, gluing local colourings is not merely a matter of
matching boundary colours. The colourings must also route their
two-colour Kempe arcs compatibly across every shared level cycle. The
ordinary restriction relation records whether a boundary colouring can
be extended locally; the Kempe-enhanced relation additionally records
the conservation of Kempe-cycle flow through the annular transition
region.
For a tire with one outer level cycle and several inner level cycles,
\[
\partial T=C_0\sqcup C_1\sqcup\cdots\sqcup C_m,
\]
the parent tire may correlate the boundary states on the different
inner cycles. The Kempe-enhanced relation records this correlation as
a system of pairings among the $P$-coloured incidences of all boundary
level cycles simultaneously. Thus one should view a medial tire as a
multi-output transition operator
\[
\mathcal K_T:
C_0 \leadsto (C_1,\ldots,C_m),
\]
rather than as an independent collection of binary transitions.
The guiding principle is therefore:
\begin{quote}
Level cycles are the interfaces used for gluing, while annular tire
regions are the bounded transition regions that route Kempe cycles
between those interfaces.
\end{quote}
\begin{thebibliography}{9}
\bibitem{bauerfeld-nested-tire-decompositions}