Define plane diamond coloring directly via parity-separation

Replaces the scaffold-mediated definition with the equivalent direct condition (two color classes contained in opposite-parity BFS layers from some root) and removes the scaffold definition, 2-colorability theorem, connectedness lemma, and equivalence proposition that existed solely to translate between the two formulations. Updates the refutation proof to invoke the new definition directly. Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
2026-05-09 13:04:46 -04:00
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@@ -96,54 +96,16 @@ Equivalently, $L_i = \{v \in V(G) : d(v, u) = i\}$, where $d(v, u)$ denotes the
 \end{definition}
 \begin{definition}
-Let $G$ be a maximal planar graph with a plane embedding, and let $\{L_0, L_1, L_2, \dots\}$ be the distance partition of $G$ from some $u \in V(G)$. The \emph{diamond scaffold} of $G$ relative to $u$ is the spanning subgraph $G^\diamond \subseteq G$ obtained by removing every edge $\{x, y\} \in E(G)$ such that $x, y \in L_i$ for some $i$.
+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 $u$,
 \end{definition}
 \begin{definition}
 Let $G$ be a maximal planar graph. A \emph{plane diamond coloring} of $G$ is a proper $4$-coloring $C$ of $G$ such that there exist two colors $c_a, c_b$ and a diamond scaffold $G^\diamond$ of $G$ with a proper $2$-coloring $C^\diamond : V(G) \to \{c_a, c_b\}$ satisfying
 \[
-    C^\diamond(v) = c_a \quad \text{for every } v \in C^{-1}(c_a),
+    C^{-1}(c_a) \subseteq \bigcup_{i \text{ even}} L_i \qquad \text{and} \qquad C^{-1}(c_b) \subseteq \bigcup_{i \text{ odd}} L_i.
 \]
 \[
    C^\diamond(v) = c_b \quad \text{for every } v \in C^{-1}(c_b).
 \]
 \end{definition}
 \section{Results}
 \begin{theorem}
 The diamond scaffold of any maximal planar graph $G$ is $2$-colorable.
 \end{theorem}
 \begin{proof}
 Let $\{L_0, L_1, L_2, \dots\}$ be the distance partition of $G$ from the chosen vertex $u$, and let $G^\diamond$ be the resulting diamond scaffold. We show $G^\diamond$ is bipartite by exhibiting a proper $2$-coloring.
 For any edge $\{x, y\} \in E(G)$, the depths of $x$ and $y$ differ by at most $1$: if $x \in L_i$, then prepending the edge $\{y, x\}$ to a shortest path from $x$ to $u$ gives a walk of length $i + 1$ from $y$ to $u$, so $y \in L_j$ for some $j \leq i + 1$, and symmetrically $i \leq j + 1$. Hence $|i - j| \leq 1$.
 By construction, $G^\diamond$ contains no edge with both endpoints in the same level $L_i$. Combined with the bound above, every edge of $G^\diamond$ joins some $L_i$ to $L_{i+1}$. Color each vertex $v \in L_i$ by the parity of $i$. Every edge of $G^\diamond$ connects vertices of opposite parity, so this is a proper $2$-coloring.
 \end{proof}
 \begin{lemma}
 The diamond scaffold $G^\diamond$ of a maximal planar graph $G$ relative to $u$ is connected.
 \end{lemma}
 \begin{proof}
 Let $\{L_0, L_1, L_2, \dots\}$ be the distance partition of $G$ from $u$. We show by induction on $i$ that every vertex of $L_i$ is connected to $u$ in $G^\diamond$. The base case $i = 0$ is immediate, since $L_0 = \{u\}$. For $i \geq 1$, let $v \in L_i$. By definition of $L_i$, there is a shortest path from $v$ to $u$ of length $i$ in $G$, whose penultimate vertex $w$ lies in $L_{i-1}$. The edge $\{v, w\}$ joins $L_i$ to $L_{i-1}$, hence is not a level edge, hence belongs to $G^\diamond$. By the inductive hypothesis $w$ is connected to $u$ in $G^\diamond$, so $v$ is as well.
 \end{proof}
 \begin{proposition}
 A maximal planar graph $G$ has a plane diamond coloring if and only if there exist a proper $4$-coloring $C$ of $G$, 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 $u$,
 \[
    C^{-1}(c_a) \subseteq \bigcup_{i \text{ even}} L_i \qquad \text{and} \qquad C^{-1}(c_b) \subseteq \bigcup_{i \text{ odd}} L_i.
 \]
 \end{proposition}
 \begin{proof}
 Since $G^\diamond$ is connected and bipartite (Theorem 2.1 and Lemma 2.2), its proper $2$-coloring is unique up to swapping the two colors, and is given by the parity of level. Hence a proper $2$-coloring $C^\diamond : V(G) \to \{c_a, c_b\}$ of $G^\diamond$ exists with $C^\diamond(v) = c_a$ on the even-parity layers and $C^\diamond(v) = c_b$ on the odd-parity layers (or vice versa). The agreement condition $C(v) = C^\diamond(v)$ on $C^{-1}(c_a) \cup C^{-1}(c_b)$ is then equivalent to the stated containment.
 \end{proof}
 \begin{remark}
-The conjecture below asserts a structural property of $4$-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 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 $u$.
 \end{remark}
 \begin{conjecture}
@@ -151,7 +113,7 @@ Every maximal planar graph $G$ has a plane diamond coloring.
 \end{conjecture}
 \begin{theorem}
-Conjecture 2.5 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}
 \begin{proof}
@@ -168,9 +130,9 @@ shown in Figure~\ref{fig:counterexample}. Equivalently, $G$ has edge set
 \end{align*}
 We have $|V(G)| = 13$ and $|E(G)| = 33 = 3 \cdot 13 - 6$, so $G$ is a triangulation.
-By Proposition 2.4, it suffices to verify that for every choice of root $u \in V(G)$ and every pair of distinct colors $c_a, c_b$, no proper $4$-coloring $C$ of $G$ satisfies $C^{-1}(c_a) \subseteq \bigcup_{i \text{ even}} L_i^{(u)}$ and $C^{-1}(c_b) \subseteq \bigcup_{i \text{ odd}} L_i^{(u)}$, where $\{L_i^{(u)}\}$ denotes the distance partition from $u$.
+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 $u$.
-For a fixed root $u$, the existence of such a $4$-coloring $C$ together with colors $c_a, c_b$ 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 $\bigcup_{i \text{ odd}} L_i^{(u)}$, joining $\beta$ to every vertex in $\bigcup_{i \text{ even}} L_i^{(u)}$, 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 the required colors $c_a := C(\alpha)$ and $c_b := C(\beta)$. Conversely, given $C, c_a, c_b$ as in the proposition, setting $C(\alpha) := c_a$ and $C(\beta) := c_b$ extends $C$ 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 $H_u$.
 A direct computation (using \texttt{Sage}'s \texttt{chromatic\_number}) verifies that $\chi(H_u) > 4$ for every $u \in V(G)$, so $G$ admits no plane diamond coloring.