1127 lines
35 KiB
TeX
Executable File
1127 lines
35 KiB
TeX
Executable File
% use [nosolutions] flag to hide solutions.
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% use [solutions] flag to show solutions.
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\documentclass[
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solutions
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]{../../resources/ormc_handout}
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\usepackage{../../resources/macros}
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\usepackage{tkz-graph}
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\uptitlel{Intermediate 2}
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\uptitler{\smallurl{}}
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\title{Graph Theory and Instant Insanity}
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\subtitle{
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Prepared by Mark on \today \\
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Based on a handout by Oleg Gleizer
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}
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\begin{document}
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\maketitle
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\section{Instant Insanity}
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The puzzle you have in front of you is called {\it Instant Insanity}.
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It consists of four cubes, with faces colored with four colors:
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red, blue, green, and white. The objective is to put the cubes in a row
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so that each side, front, back, upper, and lower,
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of the row shows each of the four colors. \\
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\begin{center}
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\includegraphics[width=2.2in]
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{II.jpg}
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\end{center}
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There are 41,472 different arrangements
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of the cubes. Only one is a solution.
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Finding it by trial and error is quite difficult,
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but we have witnessed a few students
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do just that.
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However, that rarely happens.
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We'd like to solve this puzzle today, and
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to do that, we'll need a few tools.
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\section{Cubic Nets}
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A {\it cubic net} is a 2D picture
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simultaneously showing all the six sides
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(a.k.a.~faces) of a 3D cube,
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please take a look at the examples below. \\
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\begin{center}
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\begin{tikzpicture} [scale = .3]
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\draw [line width = 1.5pt] (0,0) -- (12,0) --
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(12,3) -- (0,3) -- (0,0);
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\draw [line width = 1.5pt] (6,-3) --
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(9,-3) -- (9,6) -- (6,6) -- (6,-3);
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\draw [line width = 1.5pt] (3,0) -- (3,3);
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\draw [line width = 1.5pt] (21,0) -- (33,0) --
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(33,3) -- (21,3) -- (21,0);
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\draw [line width = 1.5pt] (24,0) --
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(24,6) -- (27,6) -- (27,0);
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\draw [line width = 1.5pt] (27,0) --
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(27,-3) -- (30,-3) -- (30,3);
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\end{tikzpicture}
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\end{center}
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\problem{}
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Draw a cubic net different from the two above.
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\vfill
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\pagebreak
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\problem{}
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An ant wants to crawl from point $A$ of a cubic room
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to the opposite point $B$, as in the picture below.
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\begin{center} \begin{small}
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\begin{tikzpicture}
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\draw (1,1) -- (4,1) -- (4,4) -- (1,4) -- (1,1);
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\draw (0,0) -- (3,0) -- (3,3) -- (0,3) -- (0,0);
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\draw (0,0) -- (1,1);
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\draw (3,0) -- (4,1);
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\draw (0,3) -- (1,4);
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\draw (3,3) -- (4,4);
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\filldraw (0,0) circle (3pt);
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\filldraw (4,4) circle (3pt);
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\coordinate [label=below left:{A}] (a) at (0,0);
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\coordinate [label=above right:{B}] (b) at (4,4);
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\end{tikzpicture}
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\end{small} \end{center}
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The insect can crawl on any surface,
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a floor, ceiling, or wall, but cannot fly through the air.
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Find at least two different shortest paths for the ant
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(there is more than one).
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Let's look at the nets of the puzzle's cubes. \\
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\begin{center} \begin{small}
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\begin{tikzpicture} [scale = .3] \label{pic:ii_cubes}
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\filldraw [red] (0,0) -- (9,0) -- (9,3) -- (0,3) -- (0,0);
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\filldraw [blue] (9,0) -- (12,0) -- (12,3) -- (9,3) -- (9,0);
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\filldraw [green] (6,3) -- (9,3) -- (9,6) -- (6,6) -- (6,3);
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\draw [line width = 1.5pt] (0,0) -- (12,0) --
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(12,3) -- (0,3) -- (0,0);
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\draw [line width = 1.5pt] (6,-3) --
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(9,-3) -- (9,6) -- (6,6) -- (6,-3);
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\draw [line width = 1.5pt] (3,0) -- (3,3);
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\coordinate [label=below:{Cube 1}] (c1) at (7.6,-3.5);
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\node[text = white] at (0 + 1.5, 0 + 1.5) {\textbf{Red}};
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\node[text = white] at (0 + 4.5, 0 + 1.5) {\textbf{Red}};
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\node[text = white] at (0 + 7.5, 0 + 1.5) {\textbf{Red}};
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\node[text = white] at (0 + 10.5, 0 + 1.5) {\textbf{Blue}};
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\node[text = black] at (0 + 7.5, 0 + 4.5) {\textbf{Grn}};
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\node[text = black] at (0 + 7.5, 0 - 1.5) {\textbf{Wht}}; % spell:disable-line
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\filldraw [red] (21,0) -- (27,0) -- (27,3) --
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(21,3) -- (21,0);
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\filldraw [green] (27,3) -- (27,0) -- (30,0) --
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(30,3) -- (27,3);
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\filldraw [blue] (27,0) -- (27,-3) -- (30,-3) --
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(30,0) -- (27,0);
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\draw [line width = 1.5pt] (21,0) -- (33,0) --
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(33,3) -- (21,3) -- (21,0);
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\draw [line width = 1.5pt] (27,-3) --
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(30,-3) -- (30,6) -- (27,6) -- (27,-3);
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\draw [line width = 1.5pt] (24,0) -- (24,3);
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\coordinate [label=below:{Cube 2}] (c2) at (28.6,-3.5);
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\node[text = white] at (21 + 1.5, 0 + 1.5) {\textbf{Red}};
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\node[text = white] at (21 + 4.5, 0 + 1.5) {\textbf{Red}};
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\node[text = black] at (21 + 7.5, 0 + 1.5) {\textbf{Grn}};
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\node[text = black] at (21 + 10.5, 0 + 1.5) {\textbf{Wht}}; % spell:disable-line
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\node[text = black] at (21 + 7.5, 0 + 4.5) {\textbf{Wht}}; % spell:disable-line
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\node[text = white] at (21 + 7.5, 0 - 1.5) {\textbf{Blue}};
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\filldraw [red] (0,-15) -- (3,-15) -- (3,-12)
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-- (0,-12) -- (0,-15);
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\filldraw [green] (3,-15) -- (6,-15) -- (6,-12)
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-- (3,-12) -- (3,-15);
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\filldraw [blue] (6,-18) -- (9,-18) -- (9,-12)
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-- (6,-12) -- (6,-15);
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\draw [line width = 1.5pt] (0,-15) -- (12,-15) --
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(12,-12) -- (0,-12) -- (0,-15);
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\draw [line width = 1.5pt] (6,-18) --
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(9,-18) -- (9,-9) -- (6,-9) -- (6,-18);
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\draw [line width = 1.5pt] (3,-15) -- (3,-12);
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\coordinate [label=below:{Cube 3}] (c3) at (7.6,-18.5);
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\node[text = white] at (0 + 1.5, -15 + 1.5) {\textbf{Red}};
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\node[text = black] at (0 + 4.5, -15 + 1.5) {\textbf{Grn}};
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\node[text = white] at (0 + 7.5, -15 + 1.5) {\textbf{Blue}};
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\node[text = black] at (0 + 10.5, -15 + 1.5) {\textbf{Wht}}; % spell:disable-line
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\node[text = black] at (0 + 7.5, -15 + 4.5) {\textbf{Wht}}; % spell:disable-line
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\node[text = white] at (0 + 7.5, -15 - 1.5) {\textbf{Blue}};
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\filldraw [red] (21,-15) -- (24,-15) --
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(24,-12) -- (21,-12) -- (21,-15);
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\filldraw [blue] (24,-15) -- (27,-15) --
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(27,-12) -- (24,-12) -- (24,-15);
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\filldraw [green] (27,-18) -- (30,-18) --
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(30,-12) -- (27,-12) -- (27,-18);
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\filldraw [blue] (30,-15) -- (33,-15) --
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(33,-12) -- (30,-12) -- (30,-15);
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\draw [line width = 1.5pt] (21,-15) -- (33,-15) --
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(33,-12) -- (21,-12) -- (21,-15);
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\draw [line width = 1.5pt] (27,-18) --
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(30,-18) -- (30,-9) -- (27,-9) -- (27,-18);
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\draw [line width = 1.5pt] (24,-15) -- (24,-12);
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\coordinate [label=below:{Cube 4}] (c4) at (28.6,-18.5);
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\node[text = white] at (21 + 1.5, -15 + 1.5) {\textbf{Red}};
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\node[text = white] at (21 + 4.5, -15 + 1.5) {\textbf{Blue}};
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\node[text = black] at (21 + 7.5, -15 + 1.5) {\textbf{Grn}};
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\node[text = white] at (21 + 10.5, -15 + 1.5) {\textbf{Blue}};
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\node[text = black] at (21 + 7.5, -15 + 4.5) {\textbf{Wht}}; % spell:disable-line
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\node[text = black] at (21 + 7.5, -15 - 1.5) {\textbf{Grn}};
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\end{tikzpicture}
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\end{small} \end{center}
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\medskip
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Note that each cube is different.
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\vfill
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\pagebreak
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\section{Graphs}
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\begin{tcolorbox}[
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colback=white,
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colframe=gray!75!black,
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title={Last week's lesson}
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]
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A \textit{graph} is a collection of nodes (vertices) and connections between them (edges). If an edge $e$ connects the vertices $v_i$ and $v_j$, then we write $e = {v_i, v_j}$. An example is below.
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\begin{center}
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\begin{tikzpicture} [scale = .6] \label{pic:1}
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\SetGraphUnit{5}
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\Vertex{B}
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\WE(B){A}
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\EA(B){C}
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\Edge(B)(A)
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\Edge(C)(B)
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\tikzset{EdgeStyle/.append style = {bend left = 50}}
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\Edge(A)(C)
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\Edge(C)(A)
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\coordinate [label=above:{$e_1$}] (e1) at (-2.1,.0);
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\coordinate [label=above:{$e_2$}] (e2) at (0,2.45);
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\coordinate [label=below:{$e_3$}] (e3) at (0,-2.5);
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\coordinate [label=above:{$e_4$}] (e4) at (2.1,.0);
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\end{tikzpicture}
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\end{center}
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More formally, a graph is defined by a set of vertices $\{v_1, v_2, ...\}$, and a set of edges $\{\ \{v_1, v_2\}, \{v_1, v_3\}, ...\ \}$.
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\medskip
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If the order of the vertices in an edge does not matter,
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a graph is called {\it undirected}. A graph is called
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a {\it directed graph} if the order of the vertices does matter.
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For example, the (undirected) graph above
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has three vertices, $A$, $B$, and $C$, and four edges,
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$e_1 =\{A,B\}$, $e_2 = \{A,C\}$, $e_3 = \{A,C\}$,
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and $e_4 = \{B,C\}$.
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\end{tcolorbox}
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Let's represent Cube 1 by a graph. \\
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The vertices will be the face colors: Blue, Green, Red, and White, so $V = \{B,G,R,W\}$. \\
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Two vertices are be connected by an edge if and only if the corresponding faces are opposing each other on the cube. \\
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Cube 1 has the following edges: $e_1 = \{B,R\}$, $e_2 = \{G,W\}$, and the loop $e_3 = \{R,R\}$. To emphasize that all the three edges represent the first cube, let us mark them with the number $1$. \\
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\begin{center} \begin{small}
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\begin{tikzpicture}
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\filldraw [blue] (0,5) -- (1,5) -- (1,6) --
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(0,6) -- (0,5);
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\draw [line width = 1.5pt] (0,5) --
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(1,5) -- (1,6) -- (0,6) -- (0,5);
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\filldraw [green] (5,5) -- (6,5) -- (6,6) --
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(5,6) -- (5,5);
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\draw [line width = 1.5pt] (5,5) --
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(6,5) -- (6,6) -- (5,6) -- (5,5);
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\filldraw [red] (0,0) -- (1,0) -- (1,1) --
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(0,1) -- (0,0);
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\draw [line width = 1.5pt] (0,0) --
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(1,0) -- (1,1) -- (0,1) -- (0,0);
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\draw [line width = 1.5pt] (5,0) --
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(6,0) -- (6,1) -- (5,1) -- (5,0);
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\draw (-.5,-.5) circle (.3) node {1};
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\draw [line width = 1.5pt] (0,.5) ..
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controls (- .9,.4) and (-.65,-.2) .. (-.7,-.3);
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\draw [line width = 1.5pt] (.5,0) ..
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controls (.4,-.9) and (-.2,-.65) .. (-.3,-.7);
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\draw (.5,2) circle (.3) node {1};
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\draw [line width = 1.5pt] (.5,1) -- (.5,1.7);
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\draw [line width = 1.5pt] (.5,2.3) -- (.5,5);
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\draw (5.5,4) circle (.3) node {1};
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\draw [line width = 1.5pt] (5.5,5) -- (5.5,4.3);
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\draw [line width = 1.5pt] (5.5,3.7) -- (5.5,1);
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\node[text = white] at (0.5, 5.5) {\textbf{Blue}};
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\node[text = white] at (0.5, 0.5) {\textbf{Red}};
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\node[text = black] at (5.5, 0.5) {\textbf{Wht}}; % spell:disable-line
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\node[text = black] at (5.5, 5.5) {\textbf{Grn}};
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\end{tikzpicture}
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\end{small} \end{center}
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\bigskip
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Cube 2 has the following pairs
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of opposing faces, $\{B,W\}$, $\{G,R\}$,
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and $\{R,W\}$. Let us add them to the graph
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as the edges $e_4$, $e_5$, and $e_6$. \\
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\begin{center} \begin{small}
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\begin{tikzpicture}
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\filldraw [blue] (0,5) -- (1,5) -- (1,6) --
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(0,6) -- (0,5);
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\draw [line width = 1.5pt] (0,5) --
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(1,5) -- (1,6) -- (0,6) -- (0,5);
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\filldraw [green] (5,5) -- (6,5) -- (6,6) --
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(5,6) -- (5,5);
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\draw [line width = 1.5pt] (5,5) --
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(6,5) -- (6,6) -- (5,6) -- (5,5);
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\filldraw [red] (0,0) -- (1,0) -- (1,1) --
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(0,1) -- (0,0);
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\draw [line width = 1.5pt] (0,0) --
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(1,0) -- (1,1) -- (0,1) -- (0,0);
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\draw [line width = 1.5pt] (5,0) --
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(6,0) -- (6,1) -- (5,1) -- (5,0);
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\draw (-.5,-.5) circle (.3) node {1};
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\draw [line width = 1.5pt] (0,.5) ..
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controls (-.9,.4) and (-.65,-.2) .. (-.7,-.3);
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\draw [line width = 1.5pt] (.5,0) ..
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controls (.4,-.9) and (-.2,-.65) .. (-.3,-.7);
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\draw (.5,2) circle (.3) node {1};
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\draw [line width = 1.5pt] (.5,1) -- (.5,1.7);
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\draw [line width = 1.5pt] (.5,2.3) -- (.5,5);
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\draw (5.5,4) circle (.3) node {1};
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\draw [line width = 1.5pt] (5.5,5) -- (5.5,4.3);
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\draw [line width = 1.5pt] (5.5,3.7) -- (5.5,1);
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\draw (2,4) circle (.3) node {2};
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\draw [line width = 1.5pt] (1,5) -- (1.8,4.2);
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\draw [line width = 1.5pt] (5,1) -- (2.2,3.8);
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\draw (4,4) circle (.3) node {2};
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\draw [line width = 1.5pt] (5,5) -- (4.2,4.2);
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\draw [line width = 1.5pt] (1,1) -- (3.8,3.8);
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\draw (4,.5) circle (.3) node {2};
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\draw [line width = 1.5pt] (1,.5) -- (3.7,.5);
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\draw [line width = 1.5pt] (5,.5) -- (4.3,.5);
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\node[text = white] at (0.5, 5.5) {\textbf{Blue}};
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\node[text = white] at (0.5, 0.5) {\textbf{Red}};
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\node[text = black] at (5.5, 0.5) {\textbf{Wht}}; % spell:disable-line
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\node[text = black] at (5.5, 5.5) {\textbf{Grn}};
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\end{tikzpicture}
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\end{small} \end{center}
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\bigskip
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Let us now make the graph
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representing all four cubes. \\
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\begin{center} \begin{small}
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\begin{tikzpicture} \label{pic:II_comfiguration}
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\filldraw [blue] (0,5) -- (1,5) -- (1,6) --
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(0,6) -- (0,5);
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\draw [line width = 1.5pt] (0,5) --
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(1,5) -- (1,6) -- (0,6) -- (0,5);
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\filldraw [green] (5,5) -- (6,5) -- (6,6) --
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(5,6) -- (5,5);
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\draw [line width = 1.5pt] (5,5) --
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(6,5) -- (6,6) -- (5,6) -- (5,5);
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\filldraw [red] (0,0) -- (1,0) -- (1,1) --
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(0,1) -- (0,0);
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\draw [line width = 1.5pt] (0,0) --
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(1,0) -- (1,1) -- (0,1) -- (0,0);
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\draw [line width = 1.5pt] (5,0) --
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(6,0) -- (6,1) -- (5,1) -- (5,0);
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\draw (-.5,-.5) circle (.3) node {1};
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\draw [line width = 1.5pt] (0,.5) ..
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controls (-.9,.4) and (-.65,-.2) .. (-.7,-.3);
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\draw [line width = 1.5pt] (.5,0) ..
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controls (.4,-.9) and (-.2,-.65) .. (-.3,-.7);
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\draw (.5,2) circle (.3) node {1};
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\draw [line width = 1.5pt] (.5,1) -- (.5,1.7);
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\draw [line width = 1.5pt] (.5,2.3) -- (.5,5);
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\draw (5.5,4) circle (.3) node {1};
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\draw [line width = 1.5pt] (5.5,5) -- (5.5,4.3);
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\draw [line width = 1.5pt] (5.5,3.7) -- (5.5,1);
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\draw (1.5,3.5) circle (.3) node {2};
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\draw [line width = 1.5pt] (.8,5) ..
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controls (.8,4.6) and (1,4.2) .. (1.3,3.7);
|
|
\draw [line width = 1.5pt] (5,.8) ..
|
|
controls (2.5,1.5) and (2.1,2.5) .. (1.6,3.22);
|
|
\draw (4,4) circle (.3) node {2};
|
|
\draw [line width = 1.5pt] (5,5) -- (4.2,4.2);
|
|
\draw [line width = 1.5pt] (1,1) -- (3.8,3.8);
|
|
\draw (4,.5) circle (.3) node {2};
|
|
\draw [line width = 1.5pt] (1,.5) -- (3.7,.5);
|
|
\draw [line width = 1.5pt] (5,.5) -- (4.3,.5);
|
|
\draw (-1,3) circle (.3) node {3};
|
|
\draw [line width = 1.5pt] (0,5) ..
|
|
controls (-.7,4.3) and (-.9,3.6) .. (-1,3.3);
|
|
\draw [line width = 1.5pt] (0,1) ..
|
|
controls (-.7,1.7) and (-.9,2.3) .. (-1,2.7);
|
|
\draw (2.5,4) circle (.3) node {3};
|
|
\draw [line width = 1.5pt] (1,5.2) ..
|
|
controls (1.4,5.2) and (2,4.5) .. (2.3,4.2);
|
|
\draw [line width = 1.5pt] (5.2,1) ..
|
|
controls (4.7,1.8) and (3.5,3) .. (2.7,3.8);
|
|
\draw (7,3) circle (.3) node {3};
|
|
\draw [line width = 1.5pt] (6,5) ..
|
|
controls (6.7,4.3) and (6.9,3.6) .. (7,3.3);
|
|
\draw [line width = 1.5pt] (6,1) ..
|
|
controls (6.7,1.7) and (6.9,2.3) .. (7,2.7);
|
|
\draw (-.5,6.5) circle (.3) node {4};
|
|
\draw [line width = 1.5pt] (0,5.5) ..
|
|
controls (-.9,5.4) and (-.65,6.2) .. (-.7,6.26);
|
|
\draw [line width = 1.5pt] (-.24,6.65) ..
|
|
controls (.1,6.7) and (.45,6.7) .. (.5,6);
|
|
\draw (8.5,3) circle (.3) node {4};
|
|
\draw [line width = 1.5pt] (6,5.5) ..
|
|
controls (7.3,5) and (8.2,4) .. (8.5,3.3);
|
|
\draw [line width = 1.5pt] (6,.8) ..
|
|
controls (7.3,1) and (8.2,2) .. (8.5,2.7);
|
|
\draw (2.3,1.2) circle (.3) node {4};
|
|
\draw [line width = 1.5pt] (1,.8) -- (2.04,1.1);
|
|
\draw [line width = 1.5pt] (2.6,1.3) ..
|
|
controls (4,2) and (5,4) .. (5.2,5);
|
|
|
|
\node[text = white] at (0.5, 5.5) {\textbf{Blue}};
|
|
\node[text = white] at (0.5, 0.5) {\textbf{Red}};
|
|
\node[text = black] at (5.5, 0.5) {\textbf{Wht}}; % spell:disable-line
|
|
\node[text = black] at (5.5, 5.5) {\textbf{Grn}};
|
|
\end{tikzpicture}
|
|
\end{small} \end{center}
|
|
|
|
\problem{}
|
|
Check if the above representation
|
|
is correct for Cubes 3 and 4.
|
|
\vfill
|
|
\pagebreak
|
|
|
|
With the help of the above graph,
|
|
solving the puzzle becomes as easy
|
|
as a walk in the park, literally.
|
|
Imagine that the vertices of the above graph
|
|
are the clearings and the edges are the paths.
|
|
An edge marked by the number $i$ represents
|
|
two opposing faces of the $i$-th cube.
|
|
Let us try to find a closed walk, a.k.a.~a cycle,
|
|
in the graph that visits each clearing once
|
|
and uses the paths marked by the different numbers,
|
|
$i=1,2,3,4$. If we order the front and rear sides
|
|
of the cubes accordingly,
|
|
then the front and rear of the stack
|
|
will show all the four different colors
|
|
in the order prescribed by our walk. \\
|
|
|
|
For example, here is such an (oriented) cycle,
|
|
represented by the magenta arrows
|
|
on the picture below. \\
|
|
|
|
\begin{center} \begin{small}
|
|
\begin{tikzpicture}
|
|
\draw [line width = 3pt, color = magenta, <-]
|
|
(.5,1) -- (.5,1.7);
|
|
\draw [line width = 3pt, color = magenta, <-]
|
|
(.5,2.3) -- (.5,5);
|
|
\draw [line width = 3pt, color = magenta, <-]
|
|
(5,5) -- (4.2,4.2);
|
|
\draw [line width = 3pt, color = magenta, ->]
|
|
(1,1) -- (3.8,3.8);
|
|
\draw [line width = 3pt, color = magenta, ->]
|
|
(6,5.5) .. controls (7.3,5) and (8.2,4) .. (8.5,3.3);
|
|
\draw [line width = 3pt, color = magenta, <-]
|
|
(6,.8) .. controls (7.3,1) and (8.2,2) .. (8.5,2.7);
|
|
\draw [line width = 3pt, color = magenta, <-]
|
|
(1,5.2) .. controls (1.4,5.2) and (2,4.5) .. (2.3,4.2);
|
|
\draw [line width = 3pt, color = magenta, ->]
|
|
(5.2,1) .. controls (4.7,1.8) and (3.5,3) .. (2.7,3.8);
|
|
|
|
|
|
\filldraw [blue] (0,5) -- (1,5) -- (1,6) --
|
|
(0,6) -- (0,5);
|
|
\draw [line width = 1.5pt] (0,5) --
|
|
(1,5) -- (1,6) -- (0,6) -- (0,5);
|
|
\filldraw [green] (5,5) -- (6,5) -- (6,6) --
|
|
(5,6) -- (5,5);
|
|
\draw [line width = 1.5pt] (5,5) --
|
|
(6,5) -- (6,6) -- (5,6) -- (5,5);
|
|
\filldraw [red] (0,0) -- (1,0) -- (1,1) --
|
|
(0,1) -- (0,0);
|
|
\draw [line width = 1.5pt] (0,0) --
|
|
(1,0) -- (1,1) -- (0,1) -- (0,0);
|
|
\draw [line width = 1.5pt] (5,0) --
|
|
(6,0) -- (6,1) -- (5,1) -- (5,0);
|
|
\draw (-.5,-.5) circle (.3) node {1};
|
|
\draw [line width = 1.5pt] (0,.5) ..
|
|
controls (-.9,.4) and (-.65,-.2) .. (-.7,-.3);
|
|
\draw [line width = 1.5pt] (.5,0) ..
|
|
controls (.4,-.9) and (-.2,-.65) .. (-.3,-.7);
|
|
\draw (.5,2) circle (.3) node {1};
|
|
\draw (5.5,4) circle (.3) node {1};
|
|
\draw [line width = 1.5pt] (5.5,5) -- (5.5,4.3);
|
|
\draw [line width = 1.5pt] (5.5,3.7) -- (5.5,1);
|
|
\draw (1.5,3.5) circle (.3) node {2};
|
|
\draw [line width = 1.5pt] (.8,5) ..
|
|
controls (.8,4.6) and (1,4.2) .. (1.3,3.7);
|
|
\draw [line width = 1.5pt] (5,.8) ..
|
|
controls (2.5,1.5) and (2.1,2.5) .. (1.6,3.22);
|
|
\draw (4,4) circle (.3) node {2};
|
|
\draw (4,.5) circle (.3) node {2};
|
|
\draw [line width = 1.5pt] (1,.5) -- (3.7,.5);
|
|
\draw [line width = 1.5pt] (5,.5) -- (4.3,.5);
|
|
\draw (-1,3) circle (.3) node {3};
|
|
\draw [line width = 1.5pt] (0,5) ..
|
|
controls (-.7,4.3) and (-.9,3.6) .. (-1,3.3);
|
|
\draw [line width = 1.5pt] (0,1) ..
|
|
controls (-.7,1.7) and (-.9,2.3) .. (-1,2.7);
|
|
\draw (2.5,4) circle (.3) node {3};
|
|
\draw (7,3) circle (.3) node {3};
|
|
\draw [line width = 1.5pt] (6,5) ..
|
|
controls (6.7,4.3) and (6.9,3.6) .. (7,3.3);
|
|
\draw [line width = 1.5pt] (6,1) ..
|
|
controls (6.7,1.7) and (6.9,2.3) .. (7,2.7);
|
|
\draw (-.5,6.5) circle (.3) node {4};
|
|
\draw [line width = 1.5pt] (0,5.5) ..
|
|
controls (-.9,5.4) and (-.65,6.2) .. (-.7,6.26);
|
|
\draw [line width = 1.5pt] (-.24,6.65) ..
|
|
controls (.1,6.7) and (.45,6.7) .. (.5,6);
|
|
\draw (8.5,3) circle (.3) node {4};
|
|
\draw (2.3,1.2) circle (.3) node {4};
|
|
\draw [line width = 1.5pt] (1,.8) -- (2.04,1.1);
|
|
\draw [line width = 1.5pt] (2.6,1.3) ..
|
|
controls (4,2) and (5,4) .. (5.2,5);
|
|
|
|
|
|
\node[text = white] at (0.5, 5.5) {\textbf{Blue}};
|
|
\node[text = white] at (0.5, 0.5) {\textbf{Red}};
|
|
\node[text = black] at (5.5, 0.5) {\textbf{Wht}}; % spell:disable-line
|
|
\node[text = black] at (5.5, 5.5) {\textbf{Grn}};
|
|
\end{tikzpicture}
|
|
\end{small} \end{center}
|
|
\bigskip
|
|
|
|
The first leg of the walk tells us
|
|
to take Cube 1 and to make sure
|
|
that its blue side is facing forward.
|
|
Then the red side, opposite to the blue one,
|
|
will face the rear.
|
|
|
|
\begin{center}
|
|
\begin{tikzpicture}
|
|
\coordinate [label=center:{Front:}] (f) at (0,.5);
|
|
\filldraw [blue] (1,0) -- (2,0) -- (2,1) -- (1,1) -- (1,0);
|
|
\draw [line width = 1.5pt] (1,0) -- (2,0) --
|
|
(2,1) -- (1,1) -- (1,0);
|
|
|
|
\coordinate [label=center:{Rear:}] (f) at (0,-1);
|
|
\filldraw [red] (1,-1.5) -- (2,-1.5) -- (2,-.5) --
|
|
(1,-.5) -- (1,-1.5);
|
|
\draw [line width = 1.5pt] (1,-1.5) -- (2,-1.5) -- (2,-.5) --
|
|
(1,-.5) -- (1,-1.5);
|
|
|
|
\node[text = white] at (1.5, 0.5) {\textbf{Blue}};
|
|
\node[text = white] at (1.5, -1) {\textbf{Red}};
|
|
\end{tikzpicture}
|
|
\end{center}
|
|
\bigskip
|
|
|
|
The next leg of the walk tells us
|
|
to take Cube 2 and to place it in such a way
|
|
that its red side faces
|
|
us while the opposing green side faces the rear.
|
|
Since we go in a cycle that visits all the colors
|
|
one-by-one, neither color repeats the ones
|
|
already used on their sides of the stack. \\
|
|
|
|
\begin{center}
|
|
\begin{tikzpicture}
|
|
\coordinate [label=center:{Front:}] (f) at (0,.5);
|
|
\filldraw [blue] (1,0) -- (2,0) -- (2,1) -- (1,1) -- (1,0);
|
|
\filldraw [red] (2,0) -- (3,0) -- (3,1) -- (2,1) -- (2,0);
|
|
\draw [line width = 1.5pt] (1,0) -- (2,0) --
|
|
(2,1) -- (1,1) -- (1,0);
|
|
\draw [line width = 1.5pt] (2,0) -- (3,0) --
|
|
(3,1) -- (2,1);
|
|
|
|
\coordinate [label=center:{Rear:}] (f) at (0,-1);
|
|
\filldraw [red] (1,-1.5) -- (2,-1.5) -- (2,-.5) --
|
|
(1,-.5) -- (1,-1.5);
|
|
\filldraw [green] (2,-1.5) -- (3,-1.5) -- (3,-.5) --
|
|
(2,-.5) -- (2,-1.5);
|
|
\draw [line width = 1.5pt] (1,-1.5) -- (2,-1.5) -- (2,-.5) --
|
|
(1,-.5) -- (1,-1.5);
|
|
\draw [line width = 1.5pt] (2,-1.5) -- (3,-1.5) --
|
|
(3,-.5) -- (2,-.5);
|
|
|
|
\node[text = white] at (1.5, 0.5) {\textbf{Blue}};
|
|
\node[text = black] at (2.5, 0.5) {\textbf{Red}};
|
|
\node[text = white] at (1.5, -1) {\textbf{Red}};
|
|
\node[text = black] at (2.5, -1) {\textbf{Grn}};
|
|
|
|
\end{tikzpicture}
|
|
\end{center}
|
|
\bigskip
|
|
|
|
The third leg of the walk tells us
|
|
to take Cube 4, not Cube 3, and to place it
|
|
green side forward, white side facing the rear. \\
|
|
|
|
\begin{center}
|
|
\begin{tikzpicture}
|
|
\coordinate [label=center:{Front:}] (f) at (0,.5);
|
|
\filldraw [blue] (1,0) -- (2,0) -- (2,1) -- (1,1) -- (1,0);
|
|
\filldraw [red] (2,0) -- (3,0) -- (3,1) -- (2,1) -- (2,0);
|
|
\filldraw [green] (4,0) -- (5,0) -- (5,1) -- (4,1) -- (4,0);
|
|
\draw [line width = 1.5pt] (1,0) -- (2,0) --
|
|
(2,1) -- (1,1) -- (1,0);
|
|
\draw [line width = 1.5pt] (2,0) -- (3,0) --
|
|
(3,1) -- (2,1);
|
|
\draw [line width = 1.5pt] (4,0) -- (5,0) --
|
|
(5,1) -- (4,1) -- (4,0);
|
|
|
|
\coordinate [label=center:{Rear:}] (f) at (0,-1);
|
|
\filldraw [red] (1,-1.5) -- (2,-1.5) -- (2,-.5) --
|
|
(1,-.5) -- (1,-1.5);
|
|
\filldraw [green] (2,-1.5) -- (3,-1.5) -- (3,-.5) --
|
|
(2,-.5) -- (2,-1.5);
|
|
\draw [line width = 1.5pt] (1,-1.5) -- (2,-1.5) -- (2,-.5) --
|
|
(1,-.5) -- (1,-1.5);
|
|
\draw [line width = 1.5pt] (2,-1.5) -- (3,-1.5) --
|
|
(3,-.5) -- (2,-.5);
|
|
\draw [line width = 1.5pt] (4,-1.5) -- (5,-1.5) --
|
|
(5,-.5) -- (4,-.5) -- (4,-1.5);
|
|
|
|
\node[text = white] at (1.5, 0.5) {\textbf{Blue}};
|
|
\node[text = black] at (2.5, 0.5) {\textbf{Red}};
|
|
\node[text = white] at (1.5, -1) {\textbf{Red}};
|
|
\node[text = black] at (2.5, -1) {\textbf{Grn}};
|
|
|
|
\node[text = black] at (4.5, 0.5) {\textbf{Grn}};
|
|
\node[text = black] at (4.5, -1) {\textbf{Wht}}; % spell:disable-line
|
|
|
|
\end{tikzpicture}
|
|
\end{center}
|
|
\bigskip
|
|
|
|
Finally, the last leg of the walk
|
|
tells us to take Cube 3 and to place it
|
|
the white side facing forward,
|
|
the opposite blue side facing the rear. \\
|
|
|
|
\begin{center}
|
|
\begin{tikzpicture}
|
|
\coordinate [label=center:{Front:}] (f) at (0,.5);
|
|
\filldraw [blue] (1,0) -- (2,0) -- (2,1) -- (1,1) -- (1,0);
|
|
\filldraw [red] (2,0) -- (3,0) -- (3,1) -- (2,1) -- (2,0);
|
|
\filldraw [green] (4,0) -- (5,0) -- (5,1) -- (4,1) -- (4,0);
|
|
\draw [line width = 1.5pt] (1,0) -- (2,0) --
|
|
(2,1) -- (1,1) -- (1,0);
|
|
\draw [line width = 1.5pt] (2,0) -- (3,0) --
|
|
(3,1) -- (2,1);
|
|
\draw [line width = 1.5pt] (4,0) -- (5,0) --
|
|
(5,1) -- (4,1) -- (4,0);
|
|
\draw [line width = 1.5pt] (3,0) -- (4,0);
|
|
\draw [line width = 1.5pt] (3,1) -- (4,1);
|
|
|
|
\coordinate [label=center:{Rear:}] (f) at (0,-1);
|
|
\filldraw [red] (1,-1.5) -- (2,-1.5) -- (2,-.5) --
|
|
(1,-.5) -- (1,-1.5);
|
|
\filldraw [green] (2,-1.5) -- (3,-1.5) -- (3,-.5) --
|
|
(2,-.5) -- (2,-1.5);
|
|
\filldraw [blue] (3,-1.5) -- (4,-1.5) -- (4,-.5) --
|
|
(3,-.5) -- (3,-1.5);
|
|
\draw [line width = 1.5pt] (1,-1.5) -- (2,-1.5) -- (2,-.5) --
|
|
(1,-.5) -- (1,-1.5);
|
|
\draw [line width = 1.5pt] (2,-1.5) -- (3,-1.5) --
|
|
(3,-.5) -- (2,-.5);
|
|
\draw [line width = 1.5pt] (4,-1.5) -- (5,-1.5) --
|
|
(5,-.5) -- (4,-.5) -- (4,-1.5);
|
|
\draw [line width = 1.5pt] (3,-1.5) -- (4,-1.5);
|
|
\draw [line width = 1.5pt] (3,-.5) -- (4,-.5);
|
|
|
|
\node[text = white] at (1.5, 0.5) {\textbf{Blue}};
|
|
\node[text = black] at (2.5, 0.5) {\textbf{Red}};
|
|
\node[text = white] at (1.5, -1) {\textbf{Red}};
|
|
\node[text = black] at (2.5, -1) {\textbf{Grn}};
|
|
\node[text = black] at (3.5, 0.5) {\textbf{Wht}}; % spell:disable-line
|
|
\node[text = white] at (3.5, -1) {\textbf{Blue}};
|
|
\node[text = black] at (4.5, 0.5) {\textbf{Grn}};
|
|
\node[text = black] at (4.5, -1) {\textbf{Wht}}; % spell:disable-line
|
|
\end{tikzpicture}
|
|
\end{center}
|
|
\bigskip
|
|
|
|
Now the front and rear of the stack are done.
|
|
If we manage to find a second oriented cycle
|
|
in the original graph that has all the properties
|
|
of the first cycle, but uses none of its edges,
|
|
we would be able to do the upper and lower sides
|
|
of the stack and to complete the puzzle.
|
|
Using the edges we have already traversed
|
|
during our first walk will mess up
|
|
the front-rear configuration, but there are still
|
|
a plenty of the edges left!
|
|
|
|
\problem{}
|
|
Complete the puzzle.
|
|
|
|
\vfill
|
|
\pagebreak
|
|
|
|
|
|
\section{Traveling salesman problem}
|
|
|
|
\problem{} \label{pr:tsp}
|
|
A salesman with the home office
|
|
in Albuquerque has to fly to Boston,
|
|
Chicago, and Denver, visiting each city once,
|
|
and then to come back to the home office.
|
|
The airfare prices, shown on the graph below,
|
|
do not depend on the direction of the travel.
|
|
Find the cheapest way. \\
|
|
\note{This was on last week's handout, but not everyone had the chance to solve it.}
|
|
|
|
\begin{center}
|
|
\begin{normalsize}
|
|
\begin{tikzpicture}
|
|
\tikzset{EdgeStyle/.append style = {-}}
|
|
\SetGraphUnit{3}
|
|
\Vertex{A}
|
|
\SOWE(A){B}
|
|
\SOEA(A){C}
|
|
\SO(A){D}
|
|
\Edge(A)(B)
|
|
\Edge(A)(C)
|
|
\Edge(A)(D)
|
|
\Edge(B)(D)
|
|
\Edge(C)(D)
|
|
\tikzset{EdgeStyle/.append style = {bend right = 70}}
|
|
\Edge(B)(C)
|
|
\coordinate [label=left:{\$1400}] (ab) at (-1.8,-1.75);
|
|
\coordinate [label=right:{\$1000}] (ac) at (1.8,-1.75);
|
|
\coordinate [label=right:{\$400}] (ad) at (-.1,-1.75);
|
|
\coordinate [label=below:{\$800}] (bc) at (0,-4.8);
|
|
\coordinate [label=below:{\$1200}] (bd) at (-1.5,-3);
|
|
\coordinate [label=below:{\$900}] (cd) at (1.5,-3);
|
|
\end{tikzpicture}
|
|
\end{normalsize}
|
|
\end{center}
|
|
|
|
\vfill
|
|
\pagebreak
|
|
|
|
\section{Planar graphs}
|
|
\label{sec:PG}
|
|
|
|
A graph is called {\it planar},
|
|
if it can be drawn in the plane in such a way
|
|
that no edges cross one another.
|
|
\problem{}
|
|
Show that the following graph is planar.
|
|
|
|
\begin{center}
|
|
\tikzset{EdgeStyle/.append style = {-}}
|
|
\begin{tikzpicture} [scale = .8]
|
|
\SetGraphUnit{3.5}
|
|
\draw [color = white] (0,0) -- (0,-5);
|
|
\Vertex{A}
|
|
\EA(A){B}
|
|
\SO(B){C}
|
|
\SO(A){D}
|
|
\Edge(A)(B)
|
|
\Edge(A)(C)
|
|
\Edge(A)(D)
|
|
\Edge(B)(C)
|
|
\Edge(B)(D)
|
|
\Edge(C)(D)
|
|
\end{tikzpicture}
|
|
\end{center}
|
|
\vfill
|
|
|
|
\problem{}
|
|
Is it possible to connect three houses,
|
|
A, B, and C, to three utility sources,
|
|
water (W), gas (G), and electricity (E),
|
|
without using the third dimension,
|
|
either on the plane or sphere,
|
|
so that the utility lines do not intersect? \\
|
|
|
|
\begin{center}
|
|
\begin{tikzpicture} [scale = .8]
|
|
\draw (0,0) circle (.5);
|
|
\coordinate [label=center:{W}] (w) at (-.05,-.03);
|
|
\draw (4,0) circle (.5);
|
|
\coordinate [label=center:{G}] (g) at (3.95,-.03);
|
|
\draw (8,0) circle (.5);
|
|
\coordinate [label=center:{E}] (e) at (7.95,-.03);
|
|
\draw (0,3) circle (.5);
|
|
\coordinate [label=center:{A}] (a) at (-.05,2.97);
|
|
\draw (4,3) circle (.5);
|
|
\coordinate [label=center:{B}] (b) at (3.95,2.97);
|
|
\draw (8,3) circle (.5);
|
|
\coordinate [label=center:{C}] (c) at (7.95,2.97);
|
|
\end{tikzpicture}
|
|
\end{center}
|
|
|
|
\vfill
|
|
\pagebreak
|
|
|
|
A {\it subdivision} of a graph $G$
|
|
is a graph resulting from the subdivision
|
|
of the edges of $G$. The subdivision
|
|
of an edge $e = (v_1,v_2)$ is a graph
|
|
containing one new vertex $v_3$,
|
|
with the edges $e_1 = (v_1,v_3)$ and
|
|
$e_2 = (v_3,v_2)$ replacing the edge $e$. \\
|
|
|
|
\begin{center}
|
|
\begin{tikzpicture} [scale = .8]
|
|
\draw (0,0) circle (.5);
|
|
\coordinate [label=center:{$v_1$}] (v12) at (0,-.03);
|
|
\draw (4,0) circle (.5);
|
|
\coordinate [label=center:{$v_3$}] (v3) at (4,-.03);
|
|
\draw (8,0) circle (.5);
|
|
\draw (.5,0) -- (3.5,0);
|
|
\draw (4.5,0) -- (7.5,0);
|
|
\draw (.5,3) -- (7.5,3);
|
|
\coordinate [label=center:{$v_2$}] (v22) at (8,-.03);
|
|
\draw (0,3) circle (.5);
|
|
\coordinate [label=center:{$v_1$}] (v11) at (0,2.97);
|
|
\draw (8,3) circle (.5);
|
|
\coordinate [label=center:{$v_2$}] (v21) at (8,2.97);
|
|
\coordinate [label=above:{$e$}] (e) at (4,3);
|
|
\coordinate [label=above:{$e_1$}] (e1) at (2,0);
|
|
\coordinate [label=above:{$e_2$}] (e2) at (6,0);
|
|
\end{tikzpicture}
|
|
\end{center}
|
|
|
|
\problem{}
|
|
What is the degree of a subdivision vertex?
|
|
|
|
\vfill
|
|
|
|
A graph $H$ is called a {\it subgraph}
|
|
of a graph $G$ if the sets of vertices
|
|
and edges of $H$ are subsets of the sets
|
|
of vertices and edges of $G$. \\
|
|
|
|
The following graphs are known as $K_{3,3}$
|
|
and $K_5$. \\
|
|
|
|
\begin{center}
|
|
\begin{tikzpicture} %[scale = .8]
|
|
\filldraw (0,0) circle (3pt);
|
|
\filldraw (2,0) circle (3pt);
|
|
\filldraw (4,0) circle (3pt);
|
|
\filldraw (0,2) circle (3pt);
|
|
\filldraw (2,2) circle (3pt);
|
|
\filldraw (4,2) circle (3pt);
|
|
\draw (0,0) -- (0,2);
|
|
\draw (0,0) -- (2,2);
|
|
\draw (0,0) -- (4,2);
|
|
\draw (2,0) -- (0,2);
|
|
\draw (2,0) -- (2,2);
|
|
\draw (2,0) -- (4,2);
|
|
\draw (4,0) -- (0,2);
|
|
\draw (4,0) -- (2,2);
|
|
\draw (4,0) -- (4,2);
|
|
\coordinate [label=below:{$K_{3,3}$}] (l) at (2,-.2);
|
|
\end{tikzpicture} \hspace{60pt}
|
|
\begin{tikzpicture} %[scale = .8]
|
|
\filldraw (90:2) circle (3pt);
|
|
\filldraw (162:2) circle (3pt);
|
|
\filldraw (234:2) circle (3pt);
|
|
\filldraw (306:2) circle (3pt);
|
|
\filldraw (18:2) circle (3pt);
|
|
\draw (90:2) -- (162:2);
|
|
\draw (90:2) -- (234:2);
|
|
\draw (90:2) -- (306:2);
|
|
\draw (90:2) -- (18:2);
|
|
\draw (162:2) -- (234:2);
|
|
\draw (162:2) -- (306:2);
|
|
\draw (162:2) -- (18:2);
|
|
\draw (234:2) -- (306:2);
|
|
\draw (234:2) -- (18:2);
|
|
\draw (306:2) -- (18:2);
|
|
\coordinate [label=below:{$K_5$}] (l) at (0,-1.7);
|
|
\end{tikzpicture}
|
|
\end{center}
|
|
\vspace{15pt}
|
|
|
|
Let $H$ be a graph that is a subdivision
|
|
of either $K_{3,3}$ or $K_5$. If $H$ is
|
|
a subgraph of a graph $G$, then $H$ is called
|
|
a {\it Kuratowski subgraph},
|
|
after a famous Polish mathematician
|
|
Kazimierz Kuratowski (1896-1980). \\
|
|
|
|
\pagebreak
|
|
|
|
\theorem{}
|
|
A graph is planar if and only if
|
|
it has no Kuratowski subgraph.
|
|
|
|
\problem{}
|
|
Is the following graph planar?
|
|
Why or why not? \\
|
|
|
|
\begin{center}
|
|
\begin{tikzpicture} [scale = .8]
|
|
\filldraw (90:2) circle (3pt);
|
|
\filldraw (150:2) circle (3pt);
|
|
\filldraw (210:2) circle (3pt);
|
|
\filldraw (270:2) circle (3pt);
|
|
\filldraw (330:2) circle (3pt);
|
|
\filldraw (30:2) circle (3pt);
|
|
\draw (90:2) -- (150:2);
|
|
\draw (90:2) -- (210:2);
|
|
\draw (90:2) -- (270:2);
|
|
\draw (90:2) -- (330:2);
|
|
\draw (90:2) -- (30:2);
|
|
\draw (150:2) -- (210:2);
|
|
\draw (150:2) -- (270:2);
|
|
\draw (150:2) -- (330:2);
|
|
\draw (150:2) -- (30:2);
|
|
\draw (210:2) -- (270:2);
|
|
\draw (210:2) -- (330:2);
|
|
\draw (210:2) -- (330:2);
|
|
\draw (210:2) -- (30:2);
|
|
\draw (270:2) -- (330:2);
|
|
\draw (270:2) -- (30:2);
|
|
\draw (330:2) -- (30:2);
|
|
\end{tikzpicture}
|
|
\end{center}
|
|
|
|
\vfill
|
|
|
|
\problem{}
|
|
Is the following graph planar?
|
|
Why or why not?
|
|
|
|
\begin{center}
|
|
\includegraphics[width=2.2in]{4Dcube2.jpg}
|
|
\end{center}
|
|
|
|
\vfill
|
|
\pagebreak
|
|
|
|
|
|
\section{Euler characteristic}
|
|
|
|
Let $G$ be a planar graph,
|
|
drawn with no edge intersections.
|
|
The edges of $G$ divide the plane into regions,
|
|
called {\it faces}. The regions
|
|
enclosed by the graph are called
|
|
the {\it interior faces}.
|
|
The region surrounding the graph is called
|
|
the {\it exterior (or infinite) face}.
|
|
The faces of $G$ include both the interior faces
|
|
and the exterior one. For example,
|
|
the following graph has two interior faces,
|
|
$F_1$, bounded by the edges $e_1$, $e_2$, $e_4$;
|
|
and $F_2$, bounded by the edges $e_1$, $e_3$, $e_4$.
|
|
Its exterior face, $F_3$, is bounded by the edges
|
|
$e_2$, $e_3$.
|
|
|
|
\begin{center} \label{pic:nsgrap}
|
|
\begin{tikzpicture} [scale = .6]
|
|
\SetGraphUnit{5}
|
|
\Vertex{B}
|
|
\WE(B){A}
|
|
\EA(B){C}
|
|
\Edge(B)(A)
|
|
\Edge(C)(B)
|
|
\tikzset{EdgeStyle/.append style = {bend left = 50}}
|
|
\Edge(A)(C)
|
|
\Edge(C)(A)
|
|
\coordinate [label=above:{$e_1$}] (e1) at (-2.1,.0);
|
|
\coordinate [label=above:{$e_2$}] (e2) at (0,2.45);
|
|
\coordinate [label=below:{$e_3$}] (e3) at (0,-2.5);
|
|
\coordinate [label=above:{$e_4$}] (e4) at (2.1,.0);
|
|
\end{tikzpicture}
|
|
\end{center}
|
|
|
|
The {\it Euler characteristic} of a graph
|
|
is the number of the graph's vertices minus
|
|
the number of the edges plus the number of the faces.
|
|
\begin{equation}
|
|
\chi = V - E + F
|
|
\end{equation}
|
|
|
|
\problem{}
|
|
Compute the Euler characteristic
|
|
of the graph above.
|
|
\vfill
|
|
|
|
\problem{}
|
|
Compute the Euler characteristic
|
|
of the following graph.
|
|
|
|
|
|
\begin{center}
|
|
\begin{tikzpicture} [scale = .8]
|
|
\draw (.5,3) -- (7.5,3);
|
|
\draw (0,3) circle (.5);
|
|
\coordinate [label=center:{$v_1$}] (v11) at (0,2.97);
|
|
\draw (8,3) circle (.5);
|
|
\coordinate [label=center:{$v_2$}] (v21) at (8,2.97);
|
|
\coordinate [label=above:{$e$}] (e) at (4,3);
|
|
\end{tikzpicture}
|
|
\end{center}
|
|
|
|
\vfill
|
|
\pagebreak
|
|
|
|
\problem{}<3Dcube>
|
|
Is the following graph planar?
|
|
If you think it is, please re-draw
|
|
the graph so that it has no intersecting edges.
|
|
If you think the graph is not planar,
|
|
please explain why. \\
|
|
|
|
\begin{center}
|
|
\begin{tikzpicture}
|
|
\draw (1,1) -- (4,1) -- (4,4) -- (1,4) -- (1,1);
|
|
\draw (0,0) -- (3,0) -- (3,3) -- (0,3) -- (0,0);
|
|
\draw (0,0) -- (1,1);
|
|
\draw (3,0) -- (4,1);
|
|
\draw (0,3) -- (1,4);
|
|
\draw (3,3) -- (4,4);
|
|
\filldraw (0,0) circle (3pt);
|
|
\filldraw (3,0) circle (3pt);
|
|
\filldraw (0,3) circle (3pt);
|
|
\filldraw (3,3) circle (3pt);
|
|
\filldraw (4,1) circle (3pt);
|
|
\filldraw (1,1) circle (3pt);
|
|
\filldraw (1,4) circle (3pt);
|
|
\filldraw (4,4) circle (3pt);
|
|
\end{tikzpicture}
|
|
\end{center}
|
|
|
|
\vfill
|
|
|
|
\problem{}
|
|
Compute the Euler characteristic of the graph
|
|
from Problem \ref{3Dcube}.
|
|
|
|
\vfill
|
|
\pagebreak
|
|
|
|
Let us consider the below picture
|
|
of a {\it regular dodecahedron} as a graph,
|
|
the vertices representing those of the graph,
|
|
and the edges, both solid and dashed,
|
|
representing the edges of the graph.
|
|
|
|
\problem{}<dodec>
|
|
Is the graph planar?
|
|
If you think it is planar,
|
|
please re-draw the graph so that it has
|
|
no intersecting edges. If you think the graph
|
|
is not planar, please explain why. \\
|
|
|
|
\begin{center}
|
|
\includegraphics[width=2.5in]
|
|
{dodecahedron.jpg}
|
|
\end{center}
|
|
|
|
\vfill
|
|
|
|
\problem{}
|
|
Compute the Euler characteristic of the graph
|
|
from Problem \ref{dodec}.
|
|
Can you conjecture what the Euler characteristic
|
|
of every planar graph is equal to?
|
|
|
|
\vfill
|
|
\pagebreak
|
|
|
|
A graph is called a {\it tree}
|
|
if it is connected and has no cycles.
|
|
Here is an example. \\
|
|
|
|
\begin{center}
|
|
\begin{tikzpicture} [scale = .6]
|
|
\SetGraphUnit{5}
|
|
\Vertex{A}
|
|
\SOWE(A){B}
|
|
\SO(A){C}
|
|
\SOEA(A){D}
|
|
\SOWE(B){E}
|
|
\SO(B){F}
|
|
\SOEA(D){G}
|
|
\Edge(A)(B)
|
|
\Edge(A)(C)
|
|
\Edge(A)(D)
|
|
\Edge(B)(E)
|
|
\Edge(B)(F)
|
|
\Edge(D)(G)
|
|
%\tikzset{EdgeStyle/.append style = {bend left = 50}}
|
|
\end{tikzpicture}
|
|
\end{center}
|
|
\bigskip
|
|
|
|
A path is called {\it simple}
|
|
if it does not include any of its edges
|
|
more than once.
|
|
|
|
\problem{}
|
|
Prove that a graph in which any two vertices are connected by one
|
|
and only one simple path is a tree.
|
|
|
|
\problem{}
|
|
What is the Euler characteristic
|
|
of a finite tree?
|
|
|
|
\theorem{}<eu_char>
|
|
Let a finite connected planar graph have
|
|
$V$ vertices, $E$ edges, and $F$ faces.
|
|
Then $V - E + F = 2$.
|
|
|
|
\problem{}
|
|
Prove Theorem \ref{eu_char}.
|
|
Hint: removing an edge from a cycle
|
|
does not change the number of vertices
|
|
and reduces the number of edges and faces
|
|
by one.
|
|
|
|
\problem{}
|
|
There are three ponds in a botanical garden,
|
|
connected by ten non-intersecting brooks
|
|
so that the ducks can sweem from any pond to any other.
|
|
How many islands are there in the garden?
|
|
|
|
\problem{}
|
|
All the vertices of a finite graph
|
|
have degree three.
|
|
Prove that the graph has a cycle.
|
|
|
|
\problem{}
|
|
Draw an infinite tree with every vertex
|
|
of degree three.
|
|
|
|
\problem{}
|
|
Prove that a connected finite graph is a tree
|
|
if and only if $V = E + 1$.
|
|
|
|
\problem{}
|
|
Give an example of a finite graph
|
|
that is not a tree,
|
|
but satisfies the relation $V = E + 1$.
|
|
|
|
\end{document} |