From 236a3f828805b228d372a6766d940f85e24d3eb9 Mon Sep 17 00:00:00 2001 From: didericis Date: Mon, 1 Jun 2026 01:41:11 -0400 Subject: [PATCH] Add level-cycle coloring conjecture --- papers/coloring_nested_tire_graphs/paper.tex | 70 +++++++++++++------- 1 file changed, 46 insertions(+), 24 deletions(-) diff --git a/papers/coloring_nested_tire_graphs/paper.tex b/papers/coloring_nested_tire_graphs/paper.tex index f2d6f64..26110f0 100644 --- a/papers/coloring_nested_tire_graphs/paper.tex +++ b/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)