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Add Motivation section and restore diamond scaffold definition

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Frames the paper around the scaffold-first 4-coloring program: 2-color
the BFS-layered bipartite spanning subgraph (the diamond scaffold),
then promote select vertices with two new colors to absorb the
discarded same-layer edges. Reintroduces the diamond scaffold
definition (removed in b5a9030 along with the equivalence machinery)
since it now plays a motivational rather than definitional role.
Replaces hardcoded definition/theorem/conjecture numbers with stable
\ref{}-based cross-references.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
didericis
2026-05-09 13:50:43 -04:00
parent ad88a2a4c7
commit a0fc10f2fa
4 changed files with 113 additions and 52 deletions
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For a coloring $C : V(G) \to S$ and a color $c \in S$, we write $C^{-1}(c) = \{v \in V(G) : C(v) = c\}$ for the preimage of $c$ under $C$, i.e., the color class of $c$. For a coloring $C : V(G) \to S$ and a color $c \in S$, we write $C^{-1}(c) = \{v \in V(G) : C(v) = c\}$ for the preimage of $c$ under $C$, i.e., the color class of $c$.
\section{Motivation}
Let $G$ be a maximal planar graph. By the Four Color Theorem \cite{appel1977every,robertson1997four}, $G$ admits a proper $4$-coloring; the question that motivates this paper is whether such a coloring can always be exhibited via a particular structural construction, rather than as the output of an ad hoc case analysis.
The construction we have in mind proceeds as follows. Fix a root vertex $u \in V(G)$ and consider the BFS layering $\{L_0, L_1, L_2, \dots\}$ of $G$ from $u$ (Definition~\ref{def:distance-partition}). Remove from $G$ every edge whose two endpoints lie in the same layer $L_i$, producing a spanning subgraph $G^\diamond \subseteq G$ which we call the \emph{diamond scaffold} of $G$ relative to $u$ (Definition~\ref{def:scaffold}). For any edge $\{x, y\} \in E(G)$ the BFS depths of $x$ and $y$ differ by at most $1$, so every edge surviving in $G^\diamond$ joins some level $L_i$ to $L_{i+1}$; in particular $G^\diamond$ is bipartite, and the parity of the layer index supplies a canonical proper $2$-coloring of $G^\diamond$ using two colors, which we denote $c_a$ and $c_b$.
To extend this $2$-coloring of the scaffold to a proper $4$-coloring of the original graph $G$, we must dispose of the edges discarded in passing from $G$ to $G^\diamond$ --- namely the edges whose endpoints share a BFS layer. The natural strategy is to recolor a chosen subset of vertices using two new colors $c_c, c_d$, just enough to resolve the discarded same-layer conflicts while preserving the canonical $\{c_a, c_b\}$-coloring on the remaining vertices. A \emph{plane diamond coloring} of $G$ (Definition~\ref{def:diamond}) is precisely a proper $4$-coloring of $G$ obtained in this way: two of its color classes, $C^{-1}(c_a)$ and $C^{-1}(c_b)$, are confined to even-indexed and odd-indexed BFS layers respectively, so that they extend the canonical $2$-coloring of the diamond scaffold $G^\diamond$ relative to some root $u$.
This paper investigates which maximal planar graphs admit such a coloring. The Four Color Theorem guarantees a proper $4$-coloring; the diamond coloring asks for one obeying the additional structural constraint above. Theorem~\ref{thm:counterexample} shows the constraint genuinely fails on some triangulations, with a unique smallest obstruction of order $13$; Conjecture~\ref{conj:mindeg5} asserts that the obstructions disappear once $\delta(G) \geq 5$.
\section{Definitions} \section{Definitions}
\begin{definition} \begin{definition} \label{def:distance-partition}
Let $G$ be a graph and let $u \in V(G)$. The \emph{distance partition} of $G$ from $u$ is the partition $\{L_0, L_1, L_2, \dots\}$ of $V(G)$ obtained by breadth-first search from $u$: Let $G$ be a graph and let $u \in V(G)$. The \emph{distance partition} of $G$ from $u$ is the partition $\{L_0, L_1, L_2, \dots\}$ of $V(G)$ obtained by breadth-first search from $u$:
\[ \[
L_0 = \{u\}, \qquad L_{i+1} = \{v \in V(G) \setminus (L_0 \cup \cdots \cup L_i) : v \text{ is adjacent to some } w \in L_i\}. L_0 = \{u\}, \qquad L_{i+1} = \{v \in V(G) \setminus (L_0 \cup \cdots \cup L_i) : v \text{ is adjacent to some } w \in L_i\}.
@@ -95,7 +105,11 @@ Let $G$ be a graph and let $u \in V(G)$. The \emph{distance partition} of $G$ fr
Equivalently, $L_i = \{v \in V(G) : d(v, u) = i\}$, where $d(v, u)$ denotes the graph distance between $v$ and $u$ in $G$. We call each $L_i$ the \emph{$i$-th level} of the partition. Equivalently, $L_i = \{v \in V(G) : d(v, u) = i\}$, where $d(v, u)$ denotes the graph distance between $v$ and $u$ in $G$. We call each $L_i$ the \emph{$i$-th level} of the partition.
\end{definition} \end{definition}
\begin{definition} \begin{definition} \label{def:scaffold}
Let $G$ be a maximal planar graph and let $u \in V(G)$, with distance partition $\{L_0, L_1, L_2, \dots\}$ from $u$. The \emph{diamond scaffold} of $G$ relative to $u$ is the spanning subgraph $G^\diamond \subseteq G$ obtained from $G$ by removing every edge $\{x, y\} \in E(G)$ such that $x, y \in L_i$ for some $i$.
\end{definition}
\begin{definition} \label{def:diamond}
Let $G$ be a maximal planar graph. A \emph{plane diamond coloring} of $G$ is a proper $4$-coloring $C$ of $G$ for which there exist a vertex $u \in V(G)$ and two distinct colors $c_a, c_b$ such that, with respect to the distance partition $\{L_0, L_1, L_2, \dots\}$ of $G$ from $u$, Let $G$ be a maximal planar graph. A \emph{plane diamond coloring} of $G$ is a proper $4$-coloring $C$ of $G$ for which there exist a vertex $u \in V(G)$ and two distinct colors $c_a, c_b$ such that, with respect to the distance partition $\{L_0, L_1, L_2, \dots\}$ of $G$ from $u$,
\[ \[
C^{-1}(c_a) \subseteq \bigcup_{i \text{ even}} L_i \qquad \text{and} \qquad C^{-1}(c_b) \subseteq \bigcup_{i \text{ odd}} L_i. C^{-1}(c_a) \subseteq \bigcup_{i \text{ even}} L_i \qquad \text{and} \qquad C^{-1}(c_b) \subseteq \bigcup_{i \text{ odd}} L_i.
@@ -105,14 +119,14 @@ Let $G$ be a maximal planar graph. A \emph{plane diamond coloring} of $G$ is a p
\section{Results} \section{Results}
\begin{remark} \begin{remark}
Definition 1.2 imposes a structural condition on $4$-colorings of maximal planar graphs strictly stronger than the conclusion of the Four Color Theorem \cite{appel1977every,robertson1997four}: it requires not merely the existence of a proper $4$-coloring, but the existence of a proper $4$-coloring together with a root $u$ such that two of the four color classes are separated by the parity of the BFS layering from $u$. Definition~\ref{def:diamond} imposes a structural condition on $4$-colorings of maximal planar graphs strictly stronger than the conclusion of the Four Color Theorem \cite{appel1977every,robertson1997four}: it requires not merely the existence of a proper $4$-coloring, but the existence of a proper $4$-coloring together with a root $u$ such that two of the four color classes are separated by the parity of the BFS layering from $u$.
\end{remark} \end{remark}
\begin{conjecture} \begin{conjecture}
Every maximal planar graph $G$ has a plane diamond coloring. Every maximal planar graph $G$ has a plane diamond coloring.
\end{conjecture} \end{conjecture}
\begin{theorem} \begin{theorem} \label{thm:counterexample}
The preceding conjecture is false. Moreover, the smallest counterexample has order $13$, and is unique up to isomorphism among triangulations of order at most $13$. The preceding conjecture is false. Moreover, the smallest counterexample has order $13$, and is unique up to isomorphism among triangulations of order at most $13$.
\end{theorem} \end{theorem}
@@ -130,7 +144,7 @@ shown in Figure~\ref{fig:counterexample}. Equivalently, $G$ has edge set
\end{align*} \end{align*}
We have $|V(G)| = 13$ and $|E(G)| = 33 = 3 \cdot 13 - 6$, so $G$ is a triangulation. We have $|V(G)| = 13$ and $|E(G)| = 33 = 3 \cdot 13 - 6$, so $G$ is a triangulation.
By Definition 1.2, it suffices to show that for every root $u \in V(G)$, no proper $4$-coloring $C$ of $G$ admits two distinct colors $c_a, c_b$ with $C^{-1}(c_a)$ contained in the union of even-indexed levels and $C^{-1}(c_b)$ contained in the union of odd-indexed levels of the distance partition from $u$. By Definition~\ref{def:diamond}, it suffices to show that for every root $u \in V(G)$, no proper $4$-coloring $C$ of $G$ admits two distinct colors $c_a, c_b$ with $C^{-1}(c_a)$ contained in the union of even-indexed levels and $C^{-1}(c_b)$ contained in the union of odd-indexed levels of the distance partition from $u$.
For a fixed root $u$, the existence of such a triple $(C, c_a, c_b)$ is equivalent to $4$-colorability of the auxiliary graph $H_u$ obtained from $G$ by adjoining two new vertices $\alpha, \beta$, joining $\alpha$ to every vertex in odd-indexed levels, joining $\beta$ to every vertex in even-indexed levels, and adding the edge $\{\alpha, \beta\}$. Indeed, in any proper $4$-coloring of $H_u$ the colors of $\alpha$ and $\beta$ are distinct and absent from the odd-parity and even-parity layers of $G$ respectively, yielding $c_a := C(\alpha)$ and $c_b := C(\beta)$. Conversely, given a $4$-coloring satisfying the parity-separation condition, setting $C(\alpha) := c_a$ and $C(\beta) := c_b$ extends it to a proper $4$-coloring of $H_u$. For a fixed root $u$, the existence of such a triple $(C, c_a, c_b)$ is equivalent to $4$-colorability of the auxiliary graph $H_u$ obtained from $G$ by adjoining two new vertices $\alpha, \beta$, joining $\alpha$ to every vertex in odd-indexed levels, joining $\beta$ to every vertex in even-indexed levels, and adding the edge $\{\alpha, \beta\}$. Indeed, in any proper $4$-coloring of $H_u$ the colors of $\alpha$ and $\beta$ are distinct and absent from the odd-parity and even-parity layers of $G$ respectively, yielding $c_a := C(\alpha)$ and $c_b := C(\beta)$. Conversely, given a $4$-coloring satisfying the parity-separation condition, setting $C(\alpha) := c_a$ and $C(\beta) := c_b$ extends it to a proper $4$-coloring of $H_u$.
@@ -146,12 +160,12 @@ For minimality and uniqueness, we exhaustively enumerated every maximal planar g
\label{fig:counterexample} \label{fig:counterexample}
\end{figure} \end{figure}
\begin{conjecture} \begin{conjecture} \label{conj:mindeg5}
Every maximal planar graph $G$ of minimum degree at least $5$ has a plane diamond coloring. Every maximal planar graph $G$ of minimum degree at least $5$ has a plane diamond coloring.
\end{conjecture} \end{conjecture}
\begin{remark} \begin{remark}
We have verified Conjecture 2.4 computationally for all maximal planar graphs of minimum degree at least $5$ and order at most $N$, by exhaustive enumeration via \texttt{Sage}'s \texttt{graphs.planar\_graphs} generator and the auxiliary-graph reduction described in the proof of Theorem 2.3. No counterexample has been found. We have verified Conjecture~\ref{conj:mindeg5} computationally for all maximal planar graphs of minimum degree at least $5$ and order at most $N$, by exhaustive enumeration via \texttt{Sage}'s \texttt{graphs.planar\_graphs} generator and the auxiliary-graph reduction described in the proof of Theorem~\ref{thm:counterexample}. No counterexample has been found.
\end{remark} \end{remark}
\begin{thebibliography}{9} \begin{thebibliography}{9}