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Graph theory partial cleanup

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Mark 2023-07-09 21:37:57 -07:00
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GPG Key ID: AD62BB059C2AAEE4
6 changed files with 181 additions and 7 deletions
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3
Intermediate/An Introduction to Graph Theory/main.tex
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@ -24,6 +24,7 @@
\input{parts/0 intro.tex}
\input{parts/1 paths.tex}
\input{parts/2 planar.tex}
%\input{parts/3 counting.tex}
\end{document}

6
Intermediate/An Introduction to Graph Theory/parts/0 intro.tex
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@ -123,5 +123,11 @@ in my class. Isn't it funny that each of them
has 5 friends in the class?} \say{This cannot be true,} immediately replies the other girl.
How did she know?
\vfill
\problem{}
Say $G$ is a graph with nine vertices. Show that $G$ has at least five vertices of degree six or at least six vertices of degree 5.
\vfill
\pagebreak

10
Intermediate/An Introduction to Graph Theory/parts/1 paths.tex
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@ -201,11 +201,11 @@ Is there an Eulerian path in the following graph? \par
Is there an Eulerian path in the following graph? \par
\begin{center}
\begin{tikzpicture}[
node distance={20mm},
thick,
main/.style = {draw, circle}
]
\begin{tikzpicture}[
node distance={20mm},
thick,
main/.style = {draw, circle}
]
\node[main] (1) {$x_1$};
\node[main] (2) [above right of=1] {$x_2$};

7
Intermediate/An Introduction to Graph Theory/parts/2 planar.tex Normal file
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@ -0,0 +1,7 @@
\section{Planar Graphs}
\textbf{TODO.} Will feature planar graphs, euler's formula, utility problem, utility problem on a torus
\vfill
\pagebreak

157
Intermediate/An Introduction to Graph Theory/parts/3 counting.tex Normal file
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@ -0,0 +1,157 @@
\section{Counting Graphs}
\definition{}
A graph is \textit{bipartite} if its nodes can be split into two groups, where no two nodes in the same group share an edge. One such graph is shown below.
\problem{}
Draw a bipartite graph with 5 vertices.
\vfill
\problem{}
Is the following graph bipartite? \par
\hint{Be careful.}
\begin{center}
\begin{tikzpicture}
% Nodes
\begin{scope}
\node[main] (A) at (0mm, 0mm) {$A$};
\node[main] (B) at (0mm, -10mm) {$B$};
\node[main] (C) at (0mm, -20mm) {$C$};
\node[main] (D) at (20mm, 0mm) {$D$};
\node[main] (E) at (20mm, -10mm) {$E$};
\node[main] (F) at (20mm, -20mm) {$F$};
\end{scope}
% Edges
\draw
(A) edge (D)
(A) edge (E)
(B) edge (F)
(C) edge (E)
(C) edge (D)
(E) edge (F)
;
\end{tikzpicture}
\end{center}
\vfill
\definition{}
A \textit{subgraph} is a graph inside another graph. \par
In the next problem, the left graph contains the left graph. \par
The triangle is a subgraph of the larger graph.
\problem{}
Find two subgraphs of the triangle in the larger graph.
\begin{center}
\adjustbox{valign=c}{
\begin{tikzpicture}
% Nodes
\begin{scope}
\node[main] (1) {1};
\node[main] (2) [right of=1] {2};
\node[main] (3) [below of=1] {3};
\end{scope}
% Edges
\draw
(1) edge (2)
(2) edge (3)
(3) edge (1)
;
\end{tikzpicture}
}
\hspace{20mm}
\adjustbox{valign=c}{
\begin{tikzpicture}
% Nodes
\begin{scope}
\node[main] (1) {1};
\node[main] (4) [below of=1] {4};
\node[main] (3) [left of=4] {3};
\node[main] (5) [right of=4] {5};
\node[main] (6) [right of=5] {6};
\node[main] (2) [above of=6] {2};
\node[main] (7) [below of=4] {7};
\end{scope}
% Edges
\draw
(1) edge (4)
(2) edge (5)
(2) edge (6)
(3) edge (4)
(4) edge (5)
(4) edge (7)
(5) edge (6)
(3) edge (7)
;
\end{tikzpicture}
}
\end{center}
\vfill
\pagebreak
A few special graphs have names. Here are a few you should know before we begin:
\definition{The path graph}
The \textit{path graph} on $n$ vertices (written $P_n$) is a straight line of vertices connected by edges. \par
$P_5$ is shown below.
\begin{center}
\begin{tikzpicture}
\node[main] (1) {1};
\node[main] (2) [right of=1] {2};
\node[main] (3) [right of=2] {3};
\node[main] (4) [right of=3] {4};
\node[main] (5) [right of=4] {5};
\draw[-] (1) -- (2);
\draw[-] (2) -- (3);
\draw[-] (3) -- (4);
\draw[-] (4) -- (5);
\end{tikzpicture}
\end{center}
\definition{The complete graph}
The \textit{complete graph} on $n$ vertices (written $K_n$) is the graph that has $n$ nodes, all of which share an edge.
$K_4$ is shown below.
\begin{center}
\begin{tikzpicture}
\node[main] (1) {A};
\node[main] (2) [above right of=1] {B};
\node[main] (3) [below right of=1] {C};
\node[main] (4) [above right of=3] {D};
\draw[-] (1) -- (2);
\draw[-] (1) -- (3);
\draw[-] (1) -- (4);
\draw[-] (2) -- (3);
\draw[-] (2) -- (4);
\draw[-] (3) -- (4);
\end{tikzpicture}
\end{center}
\problem{}
\begin{enumerate}
\item How many times does $P_4$ appear in $K_9$?
\item How many times does $C_4$ appear in $K_9$?
\item How many times does $K_{4,4}$ appear in $K_9$?
\item How many times does $C_5$ appear in $K_8$?
\item How many times does $K_{3,3}$ appear in $K_{12}$?
\item How many times does $K_{3,3}$ appear in $K_{6,6}$?
\end{enumerate}

5
Intermediate/An Introduction to Graph Theory/tikxset.tex
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@ -38,5 +38,8 @@
},
every path/.style = {
line width = 0.3mm
}
},
node distance={20mm},
thick,
main/.style = {draw, circle}
}