Graph theory partial cleanup

Intermediate/An Introduction to Graph Theory/main.tex

@@ -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}

Intermediate/An Introduction to Graph Theory/parts/0 intro.tex

@@ -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

Intermediate/An Introduction to Graph Theory/parts/1 paths.tex

@@ -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}[
+\begin{tikzpicture}[
-		node distance={20mm},
+	node distance={20mm},
-		thick,
+	thick,
-		main/.style = {draw, circle}
+	main/.style = {draw, circle}
-	]
+]
 
 		\node[main] (1) {$x_1$};
 		\node[main] (2) [above right of=1] {$x_2$};

Intermediate/An Introduction to Graph Theory/parts/2 planar.tex

@@ -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

Intermediate/An Introduction to Graph Theory/parts/3 counting.tex

@@ -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}
+
+
+

Intermediate/An Introduction to Graph Theory/tikxset.tex

@@ -38,5 +38,8 @@
 	},
 	every path/.style = {
 		line width = 0.3mm
-	}
+	},
+	node distance={20mm},
+	thick,
+	main/.style = {draw, circle}
 }