Add diamond coloring conjecture, parity-separation reformulation, and counterexample search
Extends paper with: a notation section for color-class preimages; the plane diamond coloring definition (4-coloring whose two classes lift to a 2-coloring of some BFS-rooted diamond scaffold); a connectedness lemma for the scaffold; a proposition reformulating the property as parity- separation of two color classes by BFS layers; a remark noting this is strictly stronger than 4CT; and the conjecture that every maximal planar graph admits such a coloring. Adds plane_diamond_coloring.py with get_plane_diamond_scaffold and a counterexample search that reduces the per-root check to 4-colorability of an auxiliary graph forcing two colors onto opposite parity layers. Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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\maketitle
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\section*{Notation}
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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 $c$.
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\section{Definitions}
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\begin{definition}
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@@ -90,6 +96,16 @@ Equivalently, $L_i = \{v \in V(G) : d(v, u) = i\}$, where $d(v, u)$ denotes the
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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 $i$.
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\end{definition}
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\begin{definition}
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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
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\[
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C^\diamond(v) = c_a \quad \text{for every } v \in C^{-1}(c_a),
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\]
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\[
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C^\diamond(v) = c_b \quad \text{for every } v \in C^{-1}(c_b).
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\]
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\end{definition}
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\section{Results}
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\begin{theorem}
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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.
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\end{proof}
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\begin{lemma}
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The diamond scaffold $G^\diamond$ of a maximal planar graph $G$ relative to $u$ is connected.
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\end{lemma}
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\begin{proof}
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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.
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\end{proof}
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\begin{proposition}
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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 $u$,
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\[
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C^{-1}(c_a) \subseteq \bigcup_{i \text{ even}} L_i \qquad \text{and} \qquad C^{-1}(c_b) \subseteq \bigcup_{i \text{ odd}} L_i.
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\]
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\end{proposition}
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\begin{proof}
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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 $u$.
|
||||
\end{remark}
|
||||
|
||||
\begin{conjecture}
|
||||
Every maximal planar graph $G$ has a plane diamond coloring.
|
||||
\end{conjecture}
|
||||
|
||||
\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.
|
||||
|
||||
\end{thebibliography}
|
||||
|
||||
\end{document}
|
||||
|
||||
%-----------------------------------------------------------------------
|
||||
|
||||
@@ -0,0 +1,74 @@
|
||||
"""Plane diamond coloring on maximal planar graphs."""
|
||||
from typing import Any
|
||||
from sage.all import Graph, graphs # type: ignore[attr-defined] # pylint: disable=no-name-in-module
|
||||
from lib.colored_graphs import canonize_and_save_graph
|
||||
|
||||
|
||||
def get_plane_diamond_scaffold(g: Graph, v: Any) -> Graph:
|
||||
"""
|
||||
Return the diamond scaffold of g relative to v.
|
||||
|
||||
The diamond scaffold is the spanning subgraph of g obtained by removing
|
||||
every edge whose endpoints lie in the same level of the distance
|
||||
partition of g from v.
|
||||
"""
|
||||
distances = dict(g.breadth_first_search(v, report_distance=True))
|
||||
scaffold = g.copy()
|
||||
scaffold.delete_edges([(x, y) for x, y in g.edges(labels=False) if distances[x] == distances[y]])
|
||||
return scaffold
|
||||
|
||||
|
||||
def has_plane_diamond_coloring_at_root(g: Graph, u: Any) -> bool:
|
||||
"""
|
||||
Return True iff g admits a proper 4-coloring with two color classes
|
||||
parity-separated by the BFS distance partition from u.
|
||||
|
||||
Equivalent to 4-colorability of the auxiliary graph H_u obtained by
|
||||
adjoining vertices v_a, v_b to g, with v_a adjacent to every odd-parity
|
||||
layer vertex, v_b adjacent to every even-parity layer vertex, and a
|
||||
v_a v_b edge.
|
||||
"""
|
||||
distances = dict(g.breadth_first_search(u, report_distance=True))
|
||||
odd_vertices = [v for v in g.vertices() if distances[v] % 2 == 1]
|
||||
even_vertices = [v for v in g.vertices() if distances[v] % 2 == 0]
|
||||
h = g.copy()
|
||||
v_a = max(g.vertices()) + 1
|
||||
v_b = v_a + 1
|
||||
h.add_vertex(v_a)
|
||||
h.add_vertex(v_b)
|
||||
h.add_edges([(v_a, w) for w in odd_vertices])
|
||||
h.add_edges([(v_b, w) for w in even_vertices])
|
||||
h.add_edge(v_a, v_b)
|
||||
return h.chromatic_number() <= 4
|
||||
|
||||
|
||||
def has_plane_diamond_coloring(g: Graph) -> bool:
|
||||
"""Return True iff some root vertex of g witnesses a plane diamond coloring."""
|
||||
return any(has_plane_diamond_coloring_at_root(g, u) for u in g.vertices())
|
||||
|
||||
|
||||
def search_counterexample(n: int, num_trials: int) -> Graph | None:
|
||||
"""
|
||||
Sample random maximal planar graphs of order n and return the first one
|
||||
with no plane diamond coloring, or None if none is found.
|
||||
"""
|
||||
for trial in range(num_trials):
|
||||
g = graphs.RandomTriangulation(n)
|
||||
if not has_plane_diamond_coloring(g):
|
||||
print(f"Counterexample found on trial {trial + 1}")
|
||||
return g
|
||||
if (trial + 1) % 10 == 0:
|
||||
print(f"Checked {trial + 1}/{num_trials} graphs of order {n}, no counterexample yet")
|
||||
return None
|
||||
|
||||
|
||||
if __name__ == "__main__":
|
||||
import sys
|
||||
n = int(sys.argv[1]) if len(sys.argv) > 1 else 12
|
||||
num_trials = int(sys.argv[2]) if len(sys.argv) > 2 else 100
|
||||
counterexample = search_counterexample(n, num_trials)
|
||||
if counterexample is None:
|
||||
print(f"No counterexample found in {num_trials} random triangulations of order {n}")
|
||||
else:
|
||||
canonical, graph_dir = canonize_and_save_graph(counterexample)
|
||||
print(f"Counterexample saved to {graph_dir}")
|
||||
Reference in New Issue
Block a user