MA0301/exercise10/main.tex

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2021-05-03 01:06:04 +02:00
\documentclass[12pt]{article}
\usepackage{ntnu}
\usepackage{ntnu-math}
\author{Øystein Tveit}
\title{MA0301 Exercise 10}
\usepackage{amsthm}
\usepackage{mathabx}
\usetikzlibrary{arrows.meta}
\begin{document}
\ntnuTitle{}
\break{}
Because there are no exercise where there are multiple edges between two vertices, I will use strings of vertex names to represent a walk.
\begin{excs}
\exc{}
\begin{subexcs}
\subexc{}
\[ bcbcd \]
\subexc{}
\[ bacbed \]
\subexc{}
\[ bcd \]
\subexc{}
\[ bcb \]
\subexc{}
\[ befgedcb \]
\subexc{}
\[ bacb \]
\end{subexcs}
\exc{}
\begin{figure}[H]
\center
\scalebox{2}{
\input{diagrams/ex2.tex}
}
\end{figure}
\exc{}
By using trial and error, starting with the nodes that had a higher degree, I managed to bring it down to three nodes.
\includeDiagram[scale=2, width=12cm]{diagrams/ex3.tex}
The red vertices represent the guards
\exc{}
\begin{subexcs}
\subexc{}
The graphs are not isomorphic because the shortest cycle between the vertices with a degree of 3 has a different length.
\subexc{}
The graphs are isomorphic
\end{subexcs}
\exc{}
\begin{subexcs}
\subexc{}
\includeDiagram[scale=2, width=12cm]{diagrams/ex5_a.tex}
\[ adhijkgcbgjfbefiedba \]
\subexc{}
Because $deg(e) = deg(f) = 3$ is now odd, they have to be the starting vertex and ending vertex.
\includeDiagram[scale=2, width=12cm]{diagrams/ex5_b.tex}
\[ dabdhijkgcbgjfbefie \]
\end{subexcs}
\exc{}
\begin{subexcs}
\subexc{}
$G_1$ is not an induced subgraph if it's missing an edge $e_1$ between $v_1, v_2 \in G_1$ where $e_1 \in G$
\subexc{}
\includeDiagram[scale=0.8, width=6cm, pdf=true]{diagrams/ex6_b.pdf}
$G_1$ contains the vertices $c$ and $d$ while it is missing the edge $cd$ even though $cd$ was present in $G$. Therefore, it is not an induced subgraph
\end{subexcs}
\exc{}
\begin{align*}
\sum_{deg(v) \in V} = 2 |E| \\
3|V| \leq 2 |E| \\
|V| \leq \frac{2 |E|}{3} \\
|V| \leq \frac{2 \cdot 17}{3} \\
|V| \leq \frac{34}{3} \\
|V| \leq 11.33 \\
\end{align*}
The max amount of vertices in $G$ has to be $11$
\end{excs}
\end{document}