Add plane diamond coloring paper with 2-colorability theorem
Defines distance partition (BFS layers from a chosen vertex) and the diamond scaffold of a maximal planar graph (G with all same-level edges removed), then proves the diamond scaffold is 2-colorable by parity of level. Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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%% filename: amsart-template.tex
|
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%% version: 1.1
|
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%% date: 2014/07/24
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%%
|
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%% American Mathematical Society
|
||||
%% Technical Support
|
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%% Publications Technical Group
|
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%% 201 Charles Street
|
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%% Providence, RI 02904
|
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%% USA
|
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%% tel: (401) 455-4080
|
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%% (800) 321-4267 (USA and Canada only)
|
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%% fax: (401) 331-3842
|
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%% email: tech-support@ams.org
|
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%%
|
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%% Copyright 2008-2010, 2014 American Mathematical Society.
|
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%%
|
||||
%% This work may be distributed and/or modified under the
|
||||
%% conditions of the LaTeX Project Public License, either version 1.3c
|
||||
%% of this license or (at your option) any later version.
|
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%% The latest version of this license is in
|
||||
%% http://www.latex-project.org/lppl.txt
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%% and version 1.3c or later is part of all distributions of LaTeX
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%% version 2005/12/01 or later.
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%%
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%% This work has the LPPL maintenance status `maintained'.
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%%
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%% The Current Maintainer of this work is the American Mathematical
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%% Society.
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%%
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%% ====================================================================
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% AMS-LaTeX v.2 template for use with amsart
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%
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% Remove any commented or uncommented macros you do not use.
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\documentclass{amsart}
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\newtheorem{theorem}{Theorem}[section]
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\newtheorem{lemma}[theorem]{Lemma}
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\theoremstyle{definition}
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\numberwithin{equation}{section}
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\begin{document}
|
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\title{Plane Diamond Coloring}
|
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|
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% Remove any unused author tags.
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% author one information
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\author{Eric Bauerfeld}
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\address{}
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\curraddr{}
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\email{}
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\thanks{}
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\keywords{}
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\date{}
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\dedicatory{}
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\begin{abstract}
|
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\end{abstract}
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\maketitle
|
||||
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||||
\section{Definitions}
|
||||
|
||||
\begin{definition}
|
||||
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}
|
||||
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 $i$.
|
||||
\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}
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\end{document}
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%-----------------------------------------------------------------------
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% End of amsart-template.tex
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%-----------------------------------------------------------------------
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Reference in New Issue
Block a user