% use [nosolutions] flag to hide solutions.
% use [solutions] flag to show solutions.
\documentclass[
solutions
]{../../resources/ormc_handout}
\usepackage{tkz-graph}
\begin{document}
\maketitle
<Intermediate 2>
<ORMC Summer Sessions>
{Graph Theory and Instant Insanity}
{
Prepared by Mark on \today \\
Based on a handout by Oleg Gleizer
}
\section{Instant Insanity}
The puzzle you have in front of you is called {\it Instant Insanity}.
It consists of four cubes, with faces colored with four colors:
red, blue, green, and white. The objective is to put the cubes in a row
so that each side, front, back, upper, and lower,
of the row shows each of the four colors. \\
\begin{center}
\includegraphics[width=2.2in]
{II.jpg}
\end{center}
There are 41,472 different arrangements
of the cubes. Only one is a solution.
Finding it by trial and error is quite difficult,
but we have witnessed a few students
do just that.
However, that rarely happens.
We'd like to solve this puzzle today, and
to do that, we'll need a few tools.
\section{Cubic Nets}
A {\it cubic net} is a 2D picture
simultaneously showing all the six sides
(a.k.a.~faces) of a 3D cube,
please take a look at the examples below. \\
\begin{center}
\begin{tikzpicture} [scale = .3]
\draw [line width = 1.5pt] (0,0) -- (12,0) --
(12,3) -- (0,3) -- (0,0);
\draw [line width = 1.5pt] (6,-3) --
(9,-3) -- (9,6) -- (6,6) -- (6,-3);
\draw [line width = 1.5pt] (3,0) -- (3,3);
\draw [line width = 1.5pt] (21,0) -- (33,0) --
(33,3) -- (21,3) -- (21,0);
\draw [line width = 1.5pt] (24,0) --
(24,6) -- (27,6) -- (27,0);
\draw [line width = 1.5pt] (27,0) --
(27,-3) -- (30,-3) -- (30,3);
\end{tikzpicture}
\end{center}
\problem{}
Draw a cubic net different from the two above.
\vfill
\pagebreak
\problem{}
An ant wants to crawl from point $A$ of a cubic room
to the opposite point $B$, as in the picture below.
\begin{center} \begin{small}
\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 (4,4) circle (3pt);
\coordinate [label=below left:{A}] (a) at (0,0);
\coordinate [label=above right:{B}] (b) at (4,4);
\end{tikzpicture}
\end{small} \end{center}
The insect can crawl on any surface,
a floor, ceiling, or wall, but cannot fly through the air.
Find at least two different shortest paths for the ant
(there is more than one).
Let's look at the nets of the puzzle's cubes. \\
\begin{center} \begin{small}
\begin{tikzpicture} [scale = .3] \label{pic:ii_cubes}
\filldraw [red] (0,0) -- (9,0) -- (9,3) -- (0,3) -- (0,0);
\filldraw [blue] (9,0) -- (12,0) -- (12,3) -- (9,3) -- (9,0);
\filldraw [green] (6,3) -- (9,3) -- (9,6) -- (6,6) -- (6,3);
\draw [line width = 1.5pt] (0,0) -- (12,0) --
(12,3) -- (0,3) -- (0,0);
\draw [line width = 1.5pt] (6,-3) --
(9,-3) -- (9,6) -- (6,6) -- (6,-3);
\draw [line width = 1.5pt] (3,0) -- (3,3);
\coordinate [label=below:{Cube 1}] (c1) at (7.6,-3.5);
\node[text = white] at (0 + 1.5, 0 + 1.5) {\textbf{Red}};
\node[text = white] at (0 + 4.5, 0 + 1.5) {\textbf{Red}};
\node[text = white] at (0 + 7.5, 0 + 1.5) {\textbf{Red}};
\node[text = white] at (0 + 10.5, 0 + 1.5) {\textbf{Blue}};
\node[text = black] at (0 + 7.5, 0 + 4.5) {\textbf{Grn}};
\node[text = black] at (0 + 7.5, 0 - 1.5) {\textbf{Wht}};
\filldraw [red] (21,0) -- (27,0) -- (27,3) --
(21,3) -- (21,0);
\filldraw [green] (27,3) -- (27,0) -- (30,0) --
(30,3) -- (27,3);
\filldraw [blue] (27,0) -- (27,-3) -- (30,-3) --
(30,0) -- (27,0);
\draw [line width = 1.5pt] (21,0) -- (33,0) --
(33,3) -- (21,3) -- (21,0);
\draw [line width = 1.5pt] (27,-3) --
(30,-3) -- (30,6) -- (27,6) -- (27,-3);
\draw [line width = 1.5pt] (24,0) -- (24,3);
\coordinate [label=below:{Cube 2}] (c2) at (28.6,-3.5);
\node[text = white] at (21 + 1.5, 0 + 1.5) {\textbf{Red}};
\node[text = white] at (21 + 4.5, 0 + 1.5) {\textbf{Red}};
\node[text = black] at (21 + 7.5, 0 + 1.5) {\textbf{Grn}};
\node[text = black] at (21 + 10.5, 0 + 1.5) {\textbf{Wht}};
\node[text = black] at (21 + 7.5, 0 + 4.5) {\textbf{Wht}};
\node[text = white] at (21 + 7.5, 0 - 1.5) {\textbf{Blue}};
\filldraw [red] (0,-15) -- (3,-15) -- (3,-12)
-- (0,-12) -- (0,-15);
\filldraw [green] (3,-15) -- (6,-15) -- (6,-12)
-- (3,-12) -- (3,-15);
\filldraw [blue] (6,-18) -- (9,-18) -- (9,-12)
-- (6,-12) -- (6,-15);
\draw [line width = 1.5pt] (0,-15) -- (12,-15) --
(12,-12) -- (0,-12) -- (0,-15);
\draw [line width = 1.5pt] (6,-18) --
(9,-18) -- (9,-9) -- (6,-9) -- (6,-18);
\draw [line width = 1.5pt] (3,-15) -- (3,-12);
\coordinate [label=below:{Cube 3}] (c3) at (7.6,-18.5);
\node[text = white] at (0 + 1.5, -15 + 1.5) {\textbf{Red}};
\node[text = black] at (0 + 4.5, -15 + 1.5) {\textbf{Grn}};
\node[text = white] at (0 + 7.5, -15 + 1.5) {\textbf{Blue}};
\node[text = black] at (0 + 10.5, -15 + 1.5) {\textbf{Wht}};
\node[text = black] at (0 + 7.5, -15 + 4.5) {\textbf{Wht}};
\node[text = white] at (0 + 7.5, -15 - 1.5) {\textbf{Blue}};
\filldraw [red] (21,-15) -- (24,-15) --
(24,-12) -- (21,-12) -- (21,-15);
\filldraw [blue] (24,-15) -- (27,-15) --
(27,-12) -- (24,-12) -- (24,-15);
\filldraw [green] (27,-18) -- (30,-18) --
(30,-12) -- (27,-12) -- (27,-18);
\filldraw [blue] (30,-15) -- (33,-15) --
(33,-12) -- (30,-12) -- (30,-15);
\draw [line width = 1.5pt] (21,-15) -- (33,-15) --
(33,-12) -- (21,-12) -- (21,-15);
\draw [line width = 1.5pt] (27,-18) --
(30,-18) -- (30,-9) -- (27,-9) -- (27,-18);
\draw [line width = 1.5pt] (24,-15) -- (24,-12);
\coordinate [label=below:{Cube 4}] (c4) at (28.6,-18.5);
\node[text = white] at (21 + 1.5, -15 + 1.5) {\textbf{Red}};
\node[text = white] at (21 + 4.5, -15 + 1.5) {\textbf{Blue}};
\node[text = black] at (21 + 7.5, -15 + 1.5) {\textbf{Grn}};
\node[text = white] at (21 + 10.5, -15 + 1.5) {\textbf{Blue}};
\node[text = black] at (21 + 7.5, -15 + 4.5) {\textbf{Wht}};
\node[text = black] at (21 + 7.5, -15 - 1.5) {\textbf{Grn}};
\end{tikzpicture}
\end{small} \end{center}
\medskip
Note that each cube is different.
\vfill
\pagebreak
\section{Graphs}
\begin{tcolorbox}[
colback=white,
colframe=gray!75!black,
title={Last week's lesson}
]
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.
\begin{center}
\begin{tikzpicture} [scale = .6] \label{pic:1}
\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}
More formally, a graph is defined by a set of vertices $\{v_1, v_2, ...\}$, and a set of edges $\{\ \{v_1, v_2\}, \{v_1, v_3\}, ...\ \}$.
\medskip
If the order of the vertices in an edge does not matter,
a graph is called {\it undirected}. A graph is called
a {\it directed graph} if the order of the vertices does matter.
For example, the (undirected) graph above
has three vertices, $A$, $B$, and $C$, and four edges,
$e_1 =\{A,B\}$, $e_2 = \{A,C\}$, $e_3 = \{A,C\}$,
and $e_4 = \{B,C\}$.
\end{tcolorbox}
Let's represent Cube 1 by a graph. \\
The vertices will be the face colors: Blue, Green, Red, and White, so $V = \{B,G,R,W\}$. \\
Two vertices are be connected by an edge if and only if the corresponding faces are opposing each other on the cube. \\
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 $1$. \\
\begin{center} \begin{small}
\begin{tikzpicture}
\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 [line width = 1.5pt] (.5,1) -- (.5,1.7);
\draw [line width = 1.5pt] (.5,2.3) -- (.5,5);
\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);
\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}};
\node[text = black] at (5.5, 5.5) {\textbf{Grn}};
\end{tikzpicture}
\end{small} \end{center}
\bigskip
Cube 2 has the following pairs
of opposing faces, $\{B,W\}$, $\{G,R\}$,
and $\{R,W\}$. Let us add them to the graph
as the edges $e_4$, $e_5$, and $e_6$. \\
\begin{center} \begin{small}
\begin{tikzpicture}
\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 [line width = 1.5pt] (.5,1) -- (.5,1.7);
\draw [line width = 1.5pt] (.5,2.3) -- (.5,5);
\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 (2,4) circle (.3) node {2};
\draw [line width = 1.5pt] (1,5) -- (1.8,4.2);
\draw [line width = 1.5pt] (5,1) -- (2.2,3.8);
\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);
\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}};
\node[text = black] at (5.5, 5.5) {\textbf{Grn}};
\end{tikzpicture}
\end{small} \end{center}
\bigskip
Let us now make the graph
representing all four cubes. \\
\begin{center} \begin{small}
\begin{tikzpicture} \label{pic:II_comfiguration}
\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 [line width = 1.5pt] (.5,1) -- (.5,1.7);
\draw [line width = 1.5pt] (.5,2.3) -- (.5,5);
\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 [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}};
\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}};
\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}};
\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}};
\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}};
\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}