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Equivalently, $G$ has edge set +\begin{align*} +&\{0,8\}, \{0,10\}, \{0,14\}, \{1,7\}, \{1,10\}, \{1,12\}, \{2,9\}, \{2,11\}, \{2,13\}, \\ +&\{3,9\}, \{3,11\}, \{3,15\}, \{4,8\}, \{4,12\}, \{4,14\}, \{4,15\}, \{5,6\}, \{5,13\}, \\ +&\{5,14\}, \{5,15\}, \{6,11\}, \{6,13\}, \{6,15\}, \{7,10\}, \{7,12\}, \{7,14\}, \{8,10\}, \\ +&\{8,12\}, \{8,14\}, \{9,11\}, \{9,13\}, \{9,15\}, \{10,12\}, \{10,14\}, \{11,13\}, \\ +&\{11,15\}, \{12,13\}, \{12,14\}, \{12,15\}, \{13,14\}, \{13,15\}, \{14,15\}. +\end{align*} +We have $|V(G)| = 16$ and $|E(G)| = 42 = 3 \cdot 16 - 6$, 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 $u$. + +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 $H_u$. + +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. +\end{proof} + +\begin{figure}[htbp] + \centering + \includegraphics[width=0.45\textwidth]{counterexample.png} + \caption{A maximal planar graph on $16$ vertices with no plane diamond coloring.} + \label{fig:counterexample} +\end{figure} + \begin{thebibliography}{9} \bibitem{appel1977every} @@ -159,6 +195,11 @@ N.~Robertson, D.~Sanders, P.~Seymour, and R.~Thomas, \emph{The four-colour theorem}, Journal of Combinatorial Theory, Series B, vol.~70, no.~1, pp.~2--44, 1997. +\bibitem{mckaygraph6} +B.~D.~McKay, +\emph{Description of graph6, sparse6 and digraph6 encodings}, +\url{https://users.cecs.anu.edu.au/~bdm/data/formats.txt}. + \end{thebibliography} \end{document}