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\begin{document}

\title{Plane Diamond Coloring}

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\author{Eric Bauerfeld}
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\begin{abstract}
\end{abstract}

\maketitle

\section*{Notation}

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}

\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 $u$:
\[
    L_0 = \{u\}, \qquad L_{i+1} = \{v \in V(G) \setminus (L_0 \cup \cdots \cup L_i) : v \text{ is adjacent to some } w \in L_i\}.
\]
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}

\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 $u$,
\[
    C^{-1}(c_a) \subseteq \bigcup_{i \text{ even}} L_i \qquad \text{and} \qquad C^{-1}(c_b) \subseteq \bigcup_{i \text{ odd}} L_i.
\]
\end{definition}

\section{Results}

\begin{remark}
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}

\begin{conjecture}
Every maximal planar graph $G$ has a plane diamond coloring.
\end{conjecture}

\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 $13$.
\end{theorem}

\begin{proof}
Let $G$ be the maximal planar graph on $13$ vertices with graph6 string\footnote{We use the standard graph6 encoding of McKay; see \cite{mckaygraph6}.}
\[
    \texttt{LQhO?SFIwnTYTi}
\]
shown in Figure~\ref{fig:counterexample}. Equivalently, $G$ has edge set
\begin{align*}
&\{0,2\}, \{0,4\}, \{0,11\}, \{1,3\}, \{1,5\}, \{1,12\}, \{2,4\}, \{2,9\}, \{2,11\}, \\
&\{3,5\}, \{3,10\}, \{3,12\}, \{4,7\}, \{4,9\}, \{4,11\}, \{5,8\}, \{5,10\}, \{5,12\}, \\
&\{6,7\}, \{6,8\}, \{6,9\}, \{6,10\}, \{6,11\}, \{6,12\}, \{7,8\}, \{7,9\}, \{7,10\}, \\
&\{7,11\}, \{8,9\}, \{8,10\}, \{8,12\}, \{9,11\}, \{10,12\}.
\end{align*}
We have $|V(G)| = 13$ and $|E(G)| = 33 = 3 \cdot 13 - 6$, so $G$ is a triangulation.

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 $H_u$.

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.

For minimality and uniqueness, we exhaustively enumerated every maximal planar graph of order at most $13$ using \texttt{Sage}'s \texttt{graphs.planar\_graphs} generator (with \texttt{minimum\_connectivity=3} and \texttt{maximum\_face\_size=3}). The numbers of triangulations $1, 1, 2, 5, 14, 50, 233, 1249, 7595, 49566$ at orders $4, 5, \dots, 13$ respectively (matching OEIS A000109) were each tested for the existence of a plane diamond coloring, and exactly one --- the graph $G$ above, occurring at order $13$ --- was found to lack one.
\end{proof}

\begin{figure}[htbp]
    \centering
    \includegraphics[width=0.45\textwidth]{counterexample.png}
    \caption{The unique smallest maximal planar graph with no plane diamond coloring; it has $13$ vertices and degree sequence $(6,6,6,6,6,6,6,5,5,4,4,3,3)$.}
    \label{fig:counterexample}
\end{figure}

\begin{conjecture} \label{conj:mindeg5}
Every maximal planar graph $G$ of minimum degree at least $5$ has a plane diamond coloring.
\end{conjecture}

\begin{theorem} \label{thm:mindeg5counterexample}
Conjecture~\ref{conj:mindeg5} is false. The smallest counterexamples have order $28$, and every maximal planar graph of minimum degree at least $5$ and order at most $27$ admits a plane diamond coloring.
\end{theorem}

\begin{proof}
By exhaustive enumeration via \texttt{Sage}'s \texttt{graphs.planar\_graphs} generator (with \texttt{minimum\_connectivity=3}, \texttt{maximum\_face\_size=3}, and \texttt{minimum\_degree=5}) and the auxiliary-graph reduction described in the proof of Theorem~\ref{thm:counterexample}, every maximal planar graph of minimum degree at least $5$ and order in $\{12, 13, \dots, 27\}$ admits a plane diamond coloring. The numbers of such triangulations at orders $12, 13, \dots, 27$ are
\[
    1,\ 0,\ 1,\ 1,\ 3,\ 4,\ 12,\ 23,\ 73,\ 192,\ 651,\ 2070,\ 7290,\ 25381,\ 91441,\ 329824,
\]
totalling $456{,}967$ graphs, none of which is a counterexample.

At order $28$, however, counterexamples do exist. Of the $1{,}204{,}737$ maximal planar graphs of minimum degree at least $5$ and order $28$, exactly four lack a plane diamond coloring. The graph in Figure~\ref{fig:mindeg5counterexample} is one such, with canonical graph6 string
\[
    \verb+[??DAaGP@OA_AI@DCPOaI_gh@PO?????C??B???|C?CIG?GIA?iD@?TPC?VQG_Bi+.
\]
It has $|V| = 28$, $|E| = 78 = 3 \cdot 28 - 6$, minimum degree $5$, and chromatic number $4$. The remaining three counterexamples at order $28$ have canonical graph6 strings
\[
    \verb+[?`???I@PCAG????@COGaGA_OD?DD?Aa_AII?PPV???Y??@ii?ATT?@T?T@agAgX+,
\]
\[
    \verb+[??DAaGP@OA_AI@DCPOaI_gh@PO?????C??BIA??gG?PC?IPC?Ig_?tIG?TO??F~+,
\]
\[
    \verb+[C_OQ?_?O@?a?aOOCC??A??GCCOCCO?gg?II?SSO@PPI_I_I_I_}??@yi_?LTS?B+ .
\]
Direct computation (using \texttt{Sage}'s \texttt{chromatic\_number}) verifies $\chi(H_u) > 4$ for every $u$ in each of these graphs.
\end{proof}

\begin{figure}[htbp]
    \centering
    \includegraphics[width=0.45\textwidth]{min_degree_5_counterexample.png}
    \caption{One of four smallest counterexamples to Conjecture~\ref{conj:mindeg5}: a maximal planar graph on $28$ vertices with minimum degree $5$ admitting no plane diamond coloring.}
    \label{fig:mindeg5counterexample}
\end{figure}

\begin{thebibliography}{9}

\bibitem{appel1977every}
K.~Appel and W.~Haken,
\emph{Every planar map is four colorable},
Illinois Journal of Mathematics, vol.~21, no.~3, pp.~429--567, 1977.

\bibitem{robertson1997four}
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}

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