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Binary files /dev/null and b/papers/plane_diamond_coloring/paper.pdf differ diff --git a/papers/plane_diamond_coloring/paper.tex b/papers/plane_diamond_coloring/paper.tex new file mode 100644 index 0000000..573dd4f --- /dev/null +++ b/papers/plane_diamond_coloring/paper.tex @@ -0,0 +1,111 @@ +%% filename: amsart-template.tex +%% version: 1.1 +%% date: 2014/07/24 +%% +%% American Mathematical Society +%% Technical Support +%% Publications Technical Group +%% 201 Charles Street +%% Providence, RI 02904 +%% USA +%% tel: (401) 455-4080 +%% (800) 321-4267 (USA and Canada only) +%% fax: (401) 331-3842 +%% email: tech-support@ams.org +%% +%% Copyright 2008-2010, 2014 American Mathematical Society. +%% +%% 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. +%% The latest version of this license is in +%% http://www.latex-project.org/lppl.txt +%% and version 1.3c or later is part of all distributions of LaTeX +%% version 2005/12/01 or later. +%% +%% This work has the LPPL maintenance status `maintained'. +%% +%% The Current Maintainer of this work is the American Mathematical +%% Society. +%% +%% ==================================================================== + +% AMS-LaTeX v.2 template for use with amsart +% +% Remove any commented or uncommented macros you do not use. + +\documentclass{amsart} + +\newtheorem{theorem}{Theorem}[section] +\newtheorem{lemma}[theorem]{Lemma} + +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\newtheorem{example}[theorem]{Example} +\newtheorem{xca}[theorem]{Exercise} + +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} + +\numberwithin{equation}{section} + +\begin{document} + +\title{Plane Diamond Coloring} + +% Remove any unused author tags. + +% author one information +\author{Eric Bauerfeld} +\address{} +\curraddr{} +\email{} +\thanks{} + + +\subjclass[2010]{Primary } + +\keywords{} + +\date{} + +\dedicatory{} + +\begin{abstract} +\end{abstract} + +\maketitle + +\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} + +\end{document} + +%----------------------------------------------------------------------- +% End of amsart-template.tex +%-----------------------------------------------------------------------