\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}