Improved graph theory handout
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127
Intermediate/An Introduction to Graph Theory/parts/0 intro.tex
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127
Intermediate/An Introduction to Graph Theory/parts/0 intro.tex
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\section{Graphs}
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\definition{}
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A \textit{set} is an unordered collection of objects. \par
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This means that the sets $\{1, 2, 3\}$ and $\{3, 2, 1\}$ are identical.
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\definition{}
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A \textit{graph} $G = (N, E)$ consists of two sets: a set of \textit{vertices} $V$, and a set of \textit{edges} $E$. \par
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Verticies are simply named \say{points,} and edges are connections between pairs of vertices. \par
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In the graph below, $V = \{a, b, c, d\}$ and $E = \{~ (a,b),~ (a,c),~ (a,d),~ (c,d) ~\}$.
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\begin{center}
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\begin{tikzpicture}
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\begin{scope}[layer = nodes]
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\node[main] (a) at (0, 0) {$a$};
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\node[main] (b) at (0, -1) {$b$};
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\node[main] (c) at (2, -1) {$c$};
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\node[main] (d) at (4, 0) {$d$};
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\end{scope}
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\draw[-]
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(a) edge (b)
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(a) edge (c)
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(a) edge (d)
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(c) edge (d)
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;
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\end{tikzpicture}
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\end{center}
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Vertices are also sometimes called \textit{nodes}. You'll see both terms in this handout. \par
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\problem{}
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Draw the graph defined by the following vertex and edge sets: \par
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$V = \{A,B,C,D,E\}$ \par
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$E = \{~ (A,B),~ (A,C),~ (A,D),~ (A,E),~ (B,C),~ (C,D),~ (D,E) ~\}$\par
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\vfill
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We can use graphs to solve many different kinds of problems. \par
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Most situations that involve some kind of \say{relation} between elements can be represented by a graph.
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\pagebreak
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Graphs are fully defined by their vertices and edges. The exact position of each vertex and edge doesn't matter---only which nodes are connected to each other. The same graph can be drawn in many different ways.
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\problem{}
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Show that the graphs below are equivalent by comparing the sets of their vertices and edges.
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\begin{center}
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\adjustbox{valign=c}{
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\begin{tikzpicture}
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\begin{scope}[layer = nodes]
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\node[main] (a) at (0, 0) {$a$};
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\node[main] (b) at (2, 0) {$b$};
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\node[main] (c) at (2, -2) {$c$};
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\node[main] (d) at (0, -2) {$d$};
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\end{scope}
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\draw[-]
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(a) edge (b)
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(b) edge (c)
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(c) edge (d)
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(d) edge (a)
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(a) edge (c)
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(b) edge (d)
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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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\begin{scope}[layer = nodes]
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\node[main] (a) at (0, 0) {$a$};
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\node[main] (b) at (-2, -2) {$b$};
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\node[main] (c) at (0, -2) {$c$};
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\node[main] (d) at (2, -2) {$d$};
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\end{scope}
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\draw[-]
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(a) edge (b)
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(b) edge (c)
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(c) edge (d)
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(d) edge (a)
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(a) edge (c)
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(b) edge[out=270, in=270, looseness=1] (d)
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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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\definition{}
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The degree $D(v)$ of a vertex $v$ of a graph
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is the number of the edges of the graph
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connected to that vertex.
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\theorem{Handshake Lemma}<handshake>
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In any graph, the sum of the degrees of its vertices equals twice the number of the edges.
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\problem{}
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Prove \ref{handshake}.
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\vfill
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\problem{}
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Show that all graphs have an even number number of vertices with odd degree.
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\vfill
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\problem{}
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One girl tells another, \say{There are 25 kids
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in my class. Isn't it funny that each of them
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has 5 friends in the class?} \say{This cannot be true,} immediately replies the other girl.
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How did she know?
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\vfill
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\pagebreak
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237
Intermediate/An Introduction to Graph Theory/parts/1 paths.tex
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237
Intermediate/An Introduction to Graph Theory/parts/1 paths.tex
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\section{Paths and cycles}
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A \textit{path} in a graph is, intuitively, a sequence of edges: $(x_1, x_2, x_4, ... )$. \par
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I've highlighted one possible path in the graph below.
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\begin{center}
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\begin{tikzpicture}[
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node distance={15mm},
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thick,
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main/.style = {draw, circle}
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]
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\node[main] (1) {$x_1$};
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\node[main] (2) [above right of=1] {$x_2$};
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\node[main] (3) [below right of=1] {$x_3$};
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\node[main] (4) [above right of=3] {$x_4$};
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\node[main] (5) [above right of=4] {$x_5$};
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\node[main] (6) [below right of=4] {$x_6$};
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\node[main] (7) [below right of=5] {$x_7$};
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\draw[-] (1) -- (2);
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\draw[-] (1) -- (3);
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\draw[-] (2) -- (5);
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\draw[-] (2) -- (4);
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\draw[-] (3) -- (6);
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\draw[-] (3) -- (4);
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\draw[-] (4) -- (5);
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\draw[-] (5) -- (7);
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\draw[-] (6) -- (7);
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\draw [
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line width=2mm,
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draw=black,
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opacity=0.4
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] (1) -- (2) -- (4) -- (3) -- (6);
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\end{tikzpicture}
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\end{center}
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A \textit{cycle} is a path that starts and ends on the same vertex:
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\begin{center}
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\begin{tikzpicture}[
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node distance={15mm},
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thick,
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main/.style = {draw, circle}
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]
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\node[main] (1) {$x_1$};
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\node[main] (2) [above right of=1] {$x_2$};
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\node[main] (3) [below right of=1] {$x_3$};
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\node[main] (4) [above right of=3] {$x_4$};
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\node[main] (5) [above right of=4] {$x_5$};
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\node[main] (6) [below right of=4] {$x_6$};
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\node[main] (7) [below right of=5] {$x_7$};
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\draw[-] (1) -- (2);
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\draw[-] (1) -- (3);
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\draw[-] (2) -- (5);
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\draw[-] (2) -- (4);
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\draw[-] (3) -- (6);
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\draw[-] (3) -- (4);
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\draw[-] (4) -- (5);
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\draw[-] (5) -- (7);
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\draw[-] (6) -- (7);
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\draw[
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line width=2mm,
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draw=black,
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opacity=0.4
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] (2) -- (4) -- (3) -- (6) -- (7) -- (5) -- (2);
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\end{tikzpicture}
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\end{center}
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A \textit{Eulerian\footnotemark} path is a path that traverses each edge exactly once. \par
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A Eulerian cycle is a cycle that does the same.
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\footnotetext{Pronounced ``oiler-ian''. These terms are named after a Swiss mathematician, Leonhard Euler (1707-1783), who is usually considered the founder of graph theory.}
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\vspace{2mm}
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Similarly, a {\it Hamiltonian} path is a path in a graph that visits each vertex exactly once, \par
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and a Hamiltonian cycle is a closed Hamiltonian path.
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\medskip
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An example of a Hamiltonian path is below.
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\begin{center}
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\begin{tikzpicture}[
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node distance={15mm},
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thick,
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main/.style = {draw, circle}
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]
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\node[main] (1) {$x_1$};
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\node[main] (2) [above right of=1] {$x_2$};
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\node[main] (3) [below right of=1] {$x_3$};
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\node[main] (4) [above right of=3] {$x_4$};
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\node[main] (5) [above right of=4] {$x_5$};
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\node[main] (6) [below right of=4] {$x_6$};
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\node[main] (7) [below right of=5] {$x_7$};
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\draw[-] (1) -- (2);
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\draw[-] (1) -- (3);
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\draw[-] (2) -- (5);
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\draw[-] (2) -- (4);
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\draw[-] (3) -- (6);
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\draw[-] (3) -- (4);
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\draw[-] (4) -- (5);
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\draw[-] (5) -- (7);
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\draw[-] (6) -- (7);
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\draw [
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line width=2mm,
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draw=black,
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opacity=0.4
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] (1) -- (2) -- (4) -- (3) -- (6) -- (7) -- (5);
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\end{tikzpicture}
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\end{center}
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\vfill
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\pagebreak
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\definition{}
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We say a graph is \textit{connected} if there is a path between every pair of vertices. A graph is called \textit{disconnected} otherwise.
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\problem{}
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Draw a disconnected graph with four vertices. \par
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Then, draw a graph with four vertices, all of degree one.
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\vfill
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\problem{}
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Find a Hamiltonian cycle in the following graph.
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\begin{center}
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\begin{tikzpicture}[
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node distance={20mm},
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thick,
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main/.style = {draw, circle}
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]
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\node[main] (1) {$x_1$};
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\node[main] (2) [above right of=1] {$x_2$};
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\node[main] (3) [below right of=1] {$x_3$};
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\node[main] (4) [above right of=3] {$x_4$};
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\node[main] (5) [above right of=4] {$x_5$};
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\node[main] (6) [below right of=4] {$x_6$};
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\node[main] (7) [below right of=5] {$x_7$};
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\draw[-] (1) -- (2);
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\draw[-] (1) -- (3);
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\draw[-] (2) -- (5);
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\draw[-] (2) -- (4);
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\draw[-] (3) -- (6);
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\draw[-] (3) -- (4);
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\draw[-] (4) -- (5);
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\draw[-] (5) -- (7);
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\draw[-] (6) -- (7);
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\end{tikzpicture}
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\end{center}
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\vfill
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\pagebreak
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\problem{}
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Is there an Eulerian path in the following graph? \par
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\begin{center}
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\begin{tikzpicture}[
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node distance={20mm},
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thick,
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main/.style = {draw, circle}
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]
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\node[main] (1) {$x_1$};
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\node[main] (2) [above right of=1] {$x_2$};
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\node[main] (3) [below right of=1] {$x_3$};
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\node[main] (4) [above right of=3] {$x_4$};
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\node[main] (5) [above right of=4] {$x_5$};
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\node[main] (6) [below right of=4] {$x_6$};
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\node[main] (7) [below right of=5] {$x_7$};
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\draw[-] (1) -- (2);
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\draw[-] (1) -- (3);
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\draw[-] (2) -- (5);
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\draw[-] (2) -- (4);
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\draw[-] (3) -- (6);
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\draw[-] (3) -- (4);
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\draw[-] (4) -- (5);
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\draw[-] (5) -- (7);
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\draw[-] (6) -- (7);
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\end{tikzpicture}
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\end{center}
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\vfill
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\problem{}
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Is there an Eulerian path in the following graph? \par
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\begin{center}
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\begin{tikzpicture}[
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node distance={20mm},
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thick,
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main/.style = {draw, circle}
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]
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\node[main] (1) {$x_1$};
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\node[main] (2) [above right of=1] {$x_2$};
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\node[main] (3) [below right of=1] {$x_3$};
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\node[main] (4) [above right of=3] {$x_4$};
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\node[main] (5) [above right of=4] {$x_5$};
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\node[main] (6) [below right of=4] {$x_6$};
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\node[main] (7) [below right of=5] {$x_7$};
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\draw[-] (1) -- (2);
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\draw[-] (1) -- (3);
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\draw[-] (2) -- (4);
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\draw[-] (3) -- (6);
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\draw[-] (3) -- (4);
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\draw[-] (4) -- (5);
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\draw[-] (5) -- (7);
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\draw[-] (6) -- (7);
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\end{tikzpicture}
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\end{center}
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\vfill
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\problem{}
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When does an Eulerian path exist? \par
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\hint{Look at the degree of each node.}
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\vfill
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\pagebreak
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