Add level-cycle coloring conjecture

papers/coloring_nested_tire_graphs/paper.tex

@@ -159,13 +159,13 @@ planar region these faces cover.
 \begin{definition}[Tire graph]
 \label{def:tire-graph}
 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
-outerplanar graph} $O \subseteq T$ with $V(B_{\mathrm{out}}) \cap V(O)
-= \emptyset$, where
+\emph{outer boundary} $B_{\mathrm{out}} \subseteq T$ and a
+\emph{connected inner outerplanar graph} $O \subseteq T$ with
+$V(B_{\mathrm{out}}) \cap V(O) = \emptyset$, where
 \begin{itemize}
   \item $B_{\mathrm{out}}$ is either a simple cycle of length $\geq 3$
         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
         boundary of $O$'s outer face in the inherited embedding,
         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
 vertex); the inner boundary $B_{\mathrm{in}}$ is then a non-simple
 closed walk that visits the cut-vertex multiple times.  The relaxed
-definition accommodates outerplanar inner graphs with bridges,
-cut-vertices, or multiple connected components.  When either
+definition accommodates outerplanar inner graphs with bridges or
+cut-vertices, while $O$ itself remains connected.  When either
 boundary is degenerate, the tread is a closed disk with that vertex
 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
 $B_{\mathrm{out}}$ 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
-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
 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
@@ -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
 twice and visiting cut-vertices multiple times.  This walk is the
 inner boundary $B_{\mathrm{in}}$.  No further restriction on $O$'s
-internal structure is needed: when $R_{C'}$ has more than two
-boundary components in the surface-classification sense (i.e.\
-several disjoint simple cycles on $L_{d+1}$), these correspond
-precisely to the multiple connected components or bridge crossings of
-$O$, and the outer-face boundary closed walk of $O$ captures them
-collectively.
+internal structure is needed: when the inner side has several lobes
+meeting through cut-vertices or bridges of $O$, the outer-face
+boundary closed walk of the connected graph $O$ captures them by
+revisiting those vertices or traversing those bridges twice.
 
 \emph{Tire structure.}  The triangular faces of $T_{C'}$ inside the closed
 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
 arc.  No additional hypothesis is needed.
 
-\emph{Multi-hole topology of $R_{C'}$.}  Even when $R_{C'}$ encloses
-several disjoint depth-$>d$ sub-regions, the inner outerplanar graph
-$O$ captures the multi-hole structure as a disconnected or
-non-$2$-connected outerplanar graph (with bridges or multiple
-components), and its outer-face boundary closed walk serves as
-$B_{\mathrm{in}}$ traversing bridges twice and visiting cut-vertices
-multiple times.
+\emph{Non-$2$-connected inner topology.}  Even when the inner side of
+$R_{C'}$ has several lobes joined at cut-vertices or across bridges,
+the connected inner outerplanar graph $O$ captures this structure as
+a non-$2$-connected outerplanar graph, and its outer-face boundary
+closed walk serves as $B_{\mathrm{in}}$ by traversing bridges twice
+and visiting cut-vertices multiple times.
 
 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
@@ -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
 non-trivial interiors (= containing depth-$\ge d+2$ vertices of
 $G$).  This happens, for instance, when $O^{(T_p)}$ has chords or
-cut-vertices that subdivide its inner region, or when $O^{(T_p)}$
-has multiple connected components in $G[L_{d+1}] \cap V_{C'_p}$.
+cut-vertices that subdivide its inner region.
 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
 interior, $T_p$ has exactly one child.
@@ -1160,6 +1156,32 @@ This is the structural setup underlying the chain-pigeonhole
 program for tire treads.
 \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]
 \label{def:seam}
 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
 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
-$d - 1$.  The inner dual of an outerplanar graph is a tree (a forest,
-if the outerplanar graph is disconnected), so each edge of
+$d - 1$.  The inner dual of a connected outerplanar graph is a tree,
+so each edge of
 $O^{(T_p)}$ lies on at most two of its bounded face cycles.  Hence
 $e$ lies on at most one other bounded face cycle of $O^{(T_p)}$,
 corresponding (Theorem~\ref{thm:tread-tree}, child--face bijection)