237 lines
5.5 KiB
TeX
237 lines
5.5 KiB
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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