158 lines
3.2 KiB
TeX
158 lines
3.2 KiB
TeX
\section{Counting Graphs}
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\definition{}
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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.
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\problem{}
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Draw a bipartite graph with 5 vertices.
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\vfill
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\problem{}
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Is the following graph bipartite? \par
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\hint{Be careful.}
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\begin{center}
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\begin{tikzpicture}
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% Nodes
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\begin{scope}
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\node[main] (A) at (0mm, 0mm) {$A$};
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\node[main] (B) at (0mm, -10mm) {$B$};
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\node[main] (C) at (0mm, -20mm) {$C$};
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\node[main] (D) at (20mm, 0mm) {$D$};
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\node[main] (E) at (20mm, -10mm) {$E$};
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\node[main] (F) at (20mm, -20mm) {$F$};
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\end{scope}
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% Edges
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\draw
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(A) edge (D)
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(A) edge (E)
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(B) edge (F)
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(C) edge (E)
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(C) edge (D)
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(E) edge (F)
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;
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\end{tikzpicture}
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\end{center}
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\vfill
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\definition{}
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A \textit{subgraph} is a graph inside another graph. \par
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In the next problem, the left graph contains the left graph. \par
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The triangle is a subgraph of the larger graph.
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\problem{}
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Find two subgraphs of the triangle in the larger graph.
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\begin{center}
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\adjustbox{valign=c}{
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\begin{tikzpicture}
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% Nodes
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\begin{scope}
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\node[main] (1) {1};
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\node[main] (2) [right of=1] {2};
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\node[main] (3) [below of=1] {3};
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\end{scope}
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% Edges
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\draw
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(1) edge (2)
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(2) edge (3)
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(3) edge (1)
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;
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\end{tikzpicture}
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}
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\hspace{20mm}
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\adjustbox{valign=c}{
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\begin{tikzpicture}
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% Nodes
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\begin{scope}
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\node[main] (1) {1};
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\node[main] (4) [below of=1] {4};
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\node[main] (3) [left of=4] {3};
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\node[main] (5) [right of=4] {5};
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\node[main] (6) [right of=5] {6};
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\node[main] (2) [above of=6] {2};
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\node[main] (7) [below of=4] {7};
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\end{scope}
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% Edges
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\draw
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(1) edge (4)
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(2) edge (5)
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(2) edge (6)
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(3) edge (4)
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(4) edge (5)
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(4) edge (7)
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(5) edge (6)
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(3) edge (7)
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;
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\end{tikzpicture}
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}
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\end{center}
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\vfill
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\pagebreak
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A few special graphs have names. Here are a few you should know before we begin:
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\definition{The path graph}
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The \textit{path graph} on $n$ vertices (written $P_n$) is a straight line of vertices connected by edges. \par
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$P_5$ is shown below.
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\begin{center}
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\begin{tikzpicture}
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\node[main] (1) {1};
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\node[main] (2) [right of=1] {2};
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\node[main] (3) [right of=2] {3};
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\node[main] (4) [right of=3] {4};
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\node[main] (5) [right of=4] {5};
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\draw[-] (1) -- (2);
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\draw[-] (2) -- (3);
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\draw[-] (3) -- (4);
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\draw[-] (4) -- (5);
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\end{tikzpicture}
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\end{center}
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\definition{The complete graph}
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The \textit{complete graph} on $n$ vertices (written $K_n$) is the graph that has $n$ nodes, all of which share an edge.
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$K_4$ is shown below.
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\begin{center}
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\begin{tikzpicture}
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\node[main] (1) {A};
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\node[main] (2) [above right of=1] {B};
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\node[main] (3) [below right of=1] {C};
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\node[main] (4) [above right of=3] {D};
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\draw[-] (1) -- (2);
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\draw[-] (1) -- (3);
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\draw[-] (1) -- (4);
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\draw[-] (2) -- (3);
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\draw[-] (2) -- (4);
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\draw[-] (3) -- (4);
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\end{tikzpicture}
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\end{center}
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\problem{}
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\begin{enumerate}
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\item How many times does $P_4$ appear in $K_9$?
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\item How many times does $C_4$ appear in $K_9$?
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\item How many times does $K_{4,4}$ appear in $K_9$?
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\item How many times does $C_5$ appear in $K_8$?
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\item How many times does $K_{3,3}$ appear in $K_{12}$?
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\item How many times does $K_{3,3}$ appear in $K_{6,6}$?
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\end{enumerate}
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