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Add level-cycle coloring conjecture

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2026-06-01 01:41:11 -04:00
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papers/coloring_nested_tire_graphs/paper.tex
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@@ -159,13 +159,13 @@ planar region these faces cover.
\begin{definition}[Tire graph] \begin{definition}[Tire graph]
\label{def:tire-graph} \label{def:tire-graph}
A \emph{tire graph} consists of a plane graph $T$ together with an A \emph{tire graph} consists of a plane graph $T$ together with an
\emph{outer boundary} $B_{\mathrm{out}} \subseteq T$ and an \emph{inner \emph{outer boundary} $B_{\mathrm{out}} \subseteq T$ and a
outerplanar graph} $O \subseteq T$ with $V(B_{\mathrm{out}}) \cap V(O) \emph{connected inner outerplanar graph} $O \subseteq T$ with
= \emptyset$, where $V(B_{\mathrm{out}}) \cap V(O) = \emptyset$, where
\begin{itemize} \begin{itemize}
\item $B_{\mathrm{out}}$ is either a simple cycle of length $\geq 3$ \item $B_{\mathrm{out}}$ is either a simple cycle of length $\geq 3$
or a single vertex (a \emph{degenerate outer boundary}); or a single vertex (a \emph{degenerate outer boundary});
\item $O$ is an outerplanar graph; its \emph{inner boundary} \item $O$ is a connected outerplanar graph; its \emph{inner boundary}
$B_{\mathrm{in}}$ is the closed walk in $O$ that traces the $B_{\mathrm{in}}$ is the closed walk in $O$ that traces the
boundary of $O$'s outer face in the inherited embedding, boundary of $O$'s outer face in the inherited embedding,
which is a simple cycle when $O$ is $2$-connected and a which is a simple cycle when $O$ is $2$-connected and a
@@ -194,8 +194,8 @@ planar region that may fail to be a $2$-manifold at cut-vertices of
$O$ (where two ``lobes'' of the depth-$d$ region meet at a single $O$ (where two ``lobes'' of the depth-$d$ region meet at a single
vertex); the inner boundary $B_{\mathrm{in}}$ is then a non-simple vertex); the inner boundary $B_{\mathrm{in}}$ is then a non-simple
closed walk that visits the cut-vertex multiple times. The relaxed closed walk that visits the cut-vertex multiple times. The relaxed
definition accommodates outerplanar inner graphs with bridges, definition accommodates outerplanar inner graphs with bridges or
cut-vertices, or multiple connected components. When either cut-vertices, while $O$ itself remains connected. When either
boundary is degenerate, the tread is a closed disk with that vertex boundary is degenerate, the tread is a closed disk with that vertex
as apex. as apex.
@@ -304,7 +304,7 @@ embedding from $\Pi_G$.
Then $T_{C'}$, with the inherited embedding, is a tire graph in the sense of Definition~\ref{def:tire-graph}. Its outer boundary Then $T_{C'}$, with the inherited embedding, is a tire graph in the sense of Definition~\ref{def:tire-graph}. Its outer boundary
$B_{\mathrm{out}}$ is the side of $R_{C'}$ closer to $S$ in $\Pi_G$, $B_{\mathrm{out}}$ is the side of $R_{C'}$ closer to $S$ in $\Pi_G$,
namely the level-$d$ subgraph $G[V_{C'} \cap L_d]$ (a simple cycle or namely the level-$d$ subgraph $G[V_{C'} \cap L_d]$ (a simple cycle or
single vertex); its inner outerplanar graph is $O = G[V_{C'} \cap single vertex); its connected inner outerplanar graph is $O = G[V_{C'} \cap
L_{d+1}]$, and its inner boundary $B_{\mathrm{in}}$ is the outer-face L_{d+1}]$, and its inner boundary $B_{\mathrm{in}}$ is the outer-face
boundary closed walk of $O$ in the inherited embedding (a simple cycle boundary closed walk of $O$ in the inherited embedding (a simple cycle
when $O$ is $2$-connected, a non-simple closed walk in general). The when $O$ is $2$-connected, a non-simple closed walk in general). The
@@ -386,12 +386,10 @@ inherited embedding; this outer-face boundary is a single closed walk
that traces around $O$ from the outside, traversing any bridge edge that traces around $O$ from the outside, traversing any bridge edge
twice and visiting cut-vertices multiple times. This walk is the twice and visiting cut-vertices multiple times. This walk is the
inner boundary $B_{\mathrm{in}}$. No further restriction on $O$'s inner boundary $B_{\mathrm{in}}$. No further restriction on $O$'s
internal structure is needed: when $R_{C'}$ has more than two internal structure is needed: when the inner side has several lobes
boundary components in the surface-classification sense (i.e.\ meeting through cut-vertices or bridges of $O$, the outer-face
several disjoint simple cycles on $L_{d+1}$), these correspond boundary closed walk of the connected graph $O$ captures them by
precisely to the multiple connected components or bridge crossings of revisiting those vertices or traversing those bridges twice.
$O$, and the outer-face boundary closed walk of $O$ captures them
collectively.
\emph{Tire structure.} The triangular faces of $T_{C'}$ inside the closed \emph{Tire structure.} The triangular faces of $T_{C'}$ inside the closed
boundary region are by construction the depth-$d$ faces in $F_{C'}$, boundary region are by construction the depth-$d$ faces in $F_{C'}$,
@@ -480,13 +478,12 @@ $O = G[V_{C'} \cap L_{d+1}]$, and the inner boundary closed walk
$B_{\mathrm{in}}$ then visits $v$ multiple times --- once for each $B_{\mathrm{in}}$ then visits $v$ multiple times --- once for each
arc. No additional hypothesis is needed. arc. No additional hypothesis is needed.
\emph{Multi-hole topology of $R_{C'}$.} Even when $R_{C'}$ encloses \emph{Non-$2$-connected inner topology.} Even when the inner side of
several disjoint depth-$>d$ sub-regions, the inner outerplanar graph $R_{C'}$ has several lobes joined at cut-vertices or across bridges,
$O$ captures the multi-hole structure as a disconnected or the connected inner outerplanar graph $O$ captures this structure as
non-$2$-connected outerplanar graph (with bridges or multiple a non-$2$-connected outerplanar graph, and its outer-face boundary
components), and its outer-face boundary closed walk serves as closed walk serves as $B_{\mathrm{in}}$ by traversing bridges twice
$B_{\mathrm{in}}$ traversing bridges twice and visiting cut-vertices and visiting cut-vertices multiple times.
multiple times.
In the special case $d = 0$ with single-vertex source $S = \{v_0\}$, In the special case $d = 0$ with single-vertex source $S = \{v_0\}$,
$R_{C'}$ is the star of $v_0$, a topological closed disk with one $R_{C'}$ is the star of $v_0$, a topological closed disk with one
@@ -955,8 +952,7 @@ A parent tire $T_p$ has multiple children precisely when its
inner outerplanar graph $O^{(T_p)}$ has multiple bounded faces with inner outerplanar graph $O^{(T_p)}$ has multiple bounded faces with
non-trivial interiors (= containing depth-$\ge d+2$ vertices of non-trivial interiors (= containing depth-$\ge d+2$ vertices of
$G$). This happens, for instance, when $O^{(T_p)}$ has chords or $G$). This happens, for instance, when $O^{(T_p)}$ has chords or
cut-vertices that subdivide its inner region, or when $O^{(T_p)}$ cut-vertices that subdivide its inner region.
has multiple connected components in $G[L_{d+1}] \cap V_{C'_p}$.
By contrast, if $O^{(T_p)}$ is a simple cycle (the spoke-only case By contrast, if $O^{(T_p)}$ is a simple cycle (the spoke-only case
of Remark~\ref{rem:hamilton-cycle-spoke-only}) with a non-empty of Remark~\ref{rem:hamilton-cycle-spoke-only}) with a non-empty
interior, $T_p$ has exactly one child. interior, $T_p$ has exactly one child.
@@ -1160,6 +1156,32 @@ This is the structural setup underlying the chain-pigeonhole
program for tire treads. program for tire treads.
\end{remark} \end{remark}
\begin{definition}[Level-cycle three-colour restriction]
\label{def:level-cycle-three-colour-restriction}
Let $G$ be a maximal planar graph, let $S \subseteq V(G)$ be a level
source, and let $c \colon V(G) \to \{1,2,3,4\}$ be a proper
$4$-vertex-colouring of $G$. We say that $c$ has the
\emph{level-cycle three-colour restriction} with respect to $S$ if,
for every level $d \geq 0$ and every simple cycle
$C \subseteq G[L_d]$, the colour set used on $C$ has size at most
three:
\[
|c(V(C))| \leq 3.
\]
Equivalently, every simple cycle contained in a single level omits at
least one of the four colours. The omitted colour may depend on the
cycle; in particular, distinct cycles in the same level, the same tire
tread, or the same inner outerplanar component are not required to
omit the same colour.
\end{definition}
\begin{conjecture}[Level-cycle three-colour conjecture]
\label{conj:level-cycle-three-colour}
Let $G$ be a maximal planar graph and let $S \subseteq V(G)$ be any
level source. Then $G$ admits a proper $4$-vertex-colouring with the
level-cycle three-colour restriction with respect to $S$.
\end{conjecture}
\begin{definition}[Seam] \begin{definition}[Seam]
\label{def:seam} \label{def:seam}
A \emph{seam} of a maximal planar graph $G$ is a simple cycle A \emph{seam} of a maximal planar graph $G$ is a simple cycle
@@ -1223,8 +1245,8 @@ $e \in C_{T'}$.
By Theorem~\ref{thm:tread-tree}, $C_T$ is the boundary cycle of a By Theorem~\ref{thm:tread-tree}, $C_T$ is the boundary cycle of a
bounded face of the parent's inner outerplanar graph $O^{(T_p)}$, bounded face of the parent's inner outerplanar graph $O^{(T_p)}$,
where $T_p \in \mathcal{T}(G, S)$ is the parent of $T$ at depth where $T_p \in \mathcal{T}(G, S)$ is the parent of $T$ at depth
$d - 1$. The inner dual of an outerplanar graph is a tree (a forest, $d - 1$. The inner dual of a connected outerplanar graph is a tree,
if the outerplanar graph is disconnected), so each edge of so each edge of
$O^{(T_p)}$ lies on at most two of its bounded face cycles. Hence $O^{(T_p)}$ lies on at most two of its bounded face cycles. Hence
$e$ lies on at most one other bounded face cycle of $O^{(T_p)}$, $e$ lies on at most one other bounded face cycle of $O^{(T_p)}$,
corresponding (Theorem~\ref{thm:tread-tree}, child--face bijection) corresponding (Theorem~\ref{thm:tread-tree}, child--face bijection)