\include{preamble}
\setcounter{try}{51}


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\begin{document}
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\centerline{\bf Math 365 -- Wed.\ 4/16/19 -- 10.4 \& 10.5 Connectivity, Euler trails, and Hamilton paths}


\bigskip
\begin{try}~
\begin{enumerate}[(a)]
\item For each of the following graphs, compute the vertex and edge connectivity. Be sure to justify your answers.\\\smallskip
\centerline{$G_1 = \begin{matrix}\begin{tikzpicture}
\foreach \x/\y in {0/a, 1/b, 2/c, 3/d} {\node[bV, label=above:{$\y$}] (\y) at (\x,1){};}
\foreach \x/\y in {0/g, 1/f, 2/e} {\node[bV, label=below:{$\y$}] (\y) at (\x,0){};}
\draw (a) to (b)  to (f) to (e) to (c) to (d) to (e) to (b) to (g)to (f) to (a);
\end{tikzpicture}\end{matrix} \qquad \qquad 
G_2 = \begin{matrix}\begin{tikzpicture}
\foreach \x/\y in {0/a, 1/b, 2/c, 3/d} {\node[bV, label=above:{$\y$}] (\y) at (\x,1){};}
\foreach \x/\y in {0/h, 1/g, 2/f, 3/e} {\node[bV, label=below:{$\y$}] (\y) at (\x,0){};}
\draw (a) to (b)  to (h) to (g) to (c) to (b) (d) to (e)  (b) to (f) to (g) to (a) (f) to (c);
\end{tikzpicture}\end{matrix}$}
\centerline{$G_3 = \begin{matrix}\begin{tikzpicture}[scale=1.5]
\foreach \x/\y in {0/a, 1/b, 2/c} {\node[bV, label=above:{$\y$}] (\y) at (\x,1){};}
\foreach \x/\y in {0/d, 1/e, 2/f} {\node[bV, label=below right:{$\y$}] (\y) at (\x,0){};}
\foreach \x/\y in {0/g, 1/h, 2/i} {\node[bV, label=below:{$\y$}] (\y) at (\x,-1){};}
\draw (a) to (c) to (i) to (g) to (a) (d) to (f) (b) to (h) (d) to (b) (h) to (f);
\end{tikzpicture}\end{matrix}
\qquad \qquad 
G_4 = \begin{matrix}\begin{tikzpicture}[scale=1.5]
\foreach \x/\y in {0/a, 1/b, 2/c, 3/d} {\node[bV, label=above:{$\y$}] (\y) at (\x,1){};}
\foreach \x/\y in {0/i, 1/h, 2/g, 4/f, 3/e} {\node[bV, label=below:{$\y$}] (\y) at (\x,0){};}
\draw (c) to [bend right=60] (a) to (h) to (b) to (i) to (c) to (g) to (i) to (a) to (d) to (e) to (f) 
	(g) to (d) to (h) to [bend right=60] (e) (g) to [bend right=60] (f);
\end{tikzpicture}\end{matrix}$}


\item Recall that $\kappa(G)$ is the vertex connectivity of $G$ and $\lambda(G)$ is the edge connectivity of $G$. Give examples of graphs for which each of the following are satisfied.
\begin{enumerate}[(i)]
\item $\kappa(G) = \lambda(G)<\min_{v \in V} \deg(v)$ 
\item  $\kappa(G) < \lambda(G) = \min_{v \in V} \deg(v)$ 
\item  $\kappa(G) < \lambda(G) < \min_{v \in V} \deg(v)$
\item $\kappa(G) = \lambda(G) = \min_{v \in V} \deg(v)$
\end{enumerate}\vfill
\item For which values of $m$ and $n$ does the complete bipartite graph $K_{m, n}$ have a cut vertex? A cut edge? What is are $\kappa(K_{m,n})$ and $\lambda(K_{m,n})$?\vfill
\item For which values of $n$ does the  wheel $W_n$ have a cut vertex? A cut edge? What is are $\kappa(W_n)$ and $\lambda(W_n)$?
\vfill\vfill

\end{enumerate}
\end{try}


\begin{try}(Euler trails and circuits)

\begin{enumerate}[(a)]
\item Decide which of the following graphs have Eulerian circuits, which have Eulerian trails, and which have neither. For those that have an Eulerian circuits or trails, give an example.

\centerline{$\begin{matrix}\begin{tikzpicture}
\foreach \x/\y in {0/a, 1/b, 2/c, 3/d} {\node[bV, label=above:{$\y$}] (\y) at (\x,1){};}
\foreach \x/\y in {0/g, 1/f, 2/e} {\node[bV, label=below:{$\y$}] (\y) at (\x,0){};}
\draw (a) to (b)  to (f) to (e) to (c) to (d) to (e) to (b) to (g)to (f) to (a);
\end{tikzpicture}\end{matrix} \qquad \qquad 
\begin{matrix}\begin{tikzpicture}
\foreach \x/\y in {0/a, 1/b, 2/c, 3/d} {\node[bV, label=above:{$\y$}] (\y) at (\x,1){};}
\foreach \x/\y in {0/h, 1/g, 2/f, 3/e} {\node[bV, label=below:{$\y$}] (\y) at (\x,0){};}
\draw (a) to (b)  to (h) to (g) to (c) to (d) to (e) to (f) to (g) to (a) (f) to (c) (a) to (h);
\end{tikzpicture}\end{matrix}$}

\centerline{$\begin{matrix}\begin{tikzpicture}
\foreach \x/\y in {0/a, 1/b, 2/c, 3/d} {\node[bV, label=above:{$\y$}] (\y) at (\x,1){};}
\foreach \x/\y in {0/h, 1/g, 2/f, 3/e} {\node[bV, label=below:{$\y$}] (\y) at (\x,0){};}
\draw (a) to (b)  to (h) to (g) to (c) to (d) to (e) to (f) to (g) to (a) (f) to (c);
\end{tikzpicture}\end{matrix}
\qquad \qquad 
\begin{matrix}\begin{tikzpicture}[scale=1.5]
\foreach \x/\y in {0/a, 1/b, 2/c, 3/d} {\node[bV, label=above:{$\y$}] (\y) at (\x,1){};}
\foreach \x/\y in {0/i, 1/h, 2/g, 4/f, 3/e} {\node[bV, label=below:{$\y$}] (\y) at (\x,0){};}
\draw (c) to [bend right=60] (a) to (h) to (b) to (i) to (c) to (g) to (i) to (a) to (d) to (e) to (f) 
	(g) to (d) to (h) to [bend right=60] (e) (g) to [bend right=60] (f);
\end{tikzpicture}\end{matrix}$}



\item For which values of $m$, $n$ do the following have an Euler trail? an Euler circuit?
$$\text{(i)}~ K_n  \qquad \text{(ii)}~ Q_n \qquad \text{(iii)}~ K_{m,n}$$


\end{enumerate}
\end{try}


\pagebreak
\begin{try}(Hamilton paths and cycles)
\begin{enumerate}[(a)]
\item Decide which of the following has a Hamilton path, a Hamilton cycle, or neither. If a cycle of path exists, give an example. If not, give an argument as to why not. 
$$
\text{(i)}~\begin{matrix}\begin{tikzpicture}
\node[bV, label=above:{$a$}] (a) at (0,1){};
\node[bV, label=below:{$b$}] (b) at (0,-1){};
\node[bV, label=left:{$c$}] (c) at (1,0){};
\node[bV, label=right:{$d$}] (d) at (2,0){};
\node[bV, label=above:{$e$}] (e) at (3,1){};
\node[bV, label=below:{$f$}] (f) at (3,-1){};
\draw (c) to (b) to (a) to (c) to (d) to (e) to (f) to (d);
\end{tikzpicture}\end{matrix}
\qquad\qquad 
\text{(ii)}~
\begin{matrix}\begin{tikzpicture}[scale=1.5]
\node[bV, label=above:{$a$}] (a) at (0,1){};
\node[bV, label=below:{$b$}] (b) at (0,0){};
\node[bV, label=below left:{$c$}] (c) at (.5,.5){};
\node[bV, label=above:{$d$}] (d) at (1,1){};
\node[bV, label=below:{$e$}] (e) at (1,0){};
\node[bV, label=below:{$f$}] (f) at (2,0){};
\draw (a) to (b) to (e) to (c) to (a) to (d) to (e) to (f) (c) to (d);
\end{tikzpicture}\end{matrix}
\qquad\qquad 
\text{(iii)}~
\begin{matrix}\begin{tikzpicture}[scale=1.5]
\node[bV, label=above:{$a$}] (a) at (0,1){};
\node[bV, label=below:{$b$}] (b) at (0,0){};
\node[bV, label=below left:{$c$}] (c) at (.5,.5){};
\node[bV, label=above:{$d$}] (d) at (1,1){};
\node[bV, label=below:{$e$}] (e) at (1,0){};
\node[bV, label=below:{$f$}] (f) at (2,0){};
\node[bV, label=above:{$g$}] (g) at (2,1){};
\draw (a) to (b) to (e) to (c) to (a) to (d) to (e) to (f) (c) to (d) to (g) to (f);
\end{tikzpicture}\end{matrix}
$$
%(iii) 
% \includegraphics{G9}
%
%(iv) 
% \includegraphics{G10}

\item For which values of $m$, $n$ do the following have an Hamilton path? a Hamilton cycle?
$$\text{(i)}~ K_n  \qquad \text{(ii)}~ Q_n \qquad \text{(iii)}~ K_{m,n}$$

\item The \emph{Petersen graph} is 
$$P= 
\begin{matrix}\begin{tikzpicture}
\foreach \x in {0,1,2,3,4} { \node[bV] (I\x) at (\x*360/5 + 90:1){}; \node[bV] (O\x) at (\x*360/5 + 90:2){};}
\draw (O0) to (O1) to (O2) to (O3) to (O4) to (O0)
	(O0) to (I0) (O1) to (I1) (O2) to (I2) (O3) to (I3) (O4) to (I4)
	(I0) to (I2) to (I4) to (I1) to (I3) to (I0); 
\end{tikzpicture}\end{matrix}
$$
Argue that it does not have a Hamilton cycle, but the induced subgraph $G - v$, for any $v \in V$, does.

\item For each of the following, decide whether Dirac's theorem applies, whether Ore's theorem applies, and whether the graph has a Hamilton cycle.

$$
\text{(i)}~\begin{matrix}\begin{tikzpicture}[scale=1.5]
\node[bV] (a) at (0,0){};
\node[bV] (b) at (1,0){};
\node[bV] (c) at (1,1){};
\node[bV] (d) at (.5,1.5){};
\node[bV] (e) at (0,1){};
\draw (a) to (b) to (c) to (d) to (e) to (a) (c) to (e);
\end{tikzpicture}\end{matrix}
\qquad \qquad 
\text{(ii)}~
\begin{matrix}\begin{tikzpicture}[scale=1.5]
\node[bV] (a) at (0,0){};
\node[bV] (b) at (1,0){};
\node[bV] (c) at (1,1){};
\node[bV] (d) at (.5,1.5){};
\node[bV] (e) at (0,1){};
\draw (a) to (b) to (c) to (d) to (e) to (a) (c) to (e) to (b);
\end{tikzpicture}\end{matrix}
\qquad\qquad
\text{(iii)}~\begin{matrix}\begin{tikzpicture}[scale=1.5]
\node[bV] (a) at (0,0){};
\node[bV] (b) at (1,0){};
\node[bV] (c) at (1,1){};
\node[bV] (e) at (.5,.5){};
\node[bV] (d) at (0,1){};
\draw (a) to (b) to (c) to (d) to (a) to (c) (d) to (b);
\end{tikzpicture}\end{matrix}
\qquad \qquad 
\text{(iv)}~
\begin{matrix}\begin{tikzpicture}[scale=1.5]
\node[bV] (a) at (0,0){};
\node[bV] (b) at (1,0){};
\node[bV] (c) at (1,1){};
\node[bV] (e) at (.5,.75){};
\node[bV] (f) at (.5,.25){};
\node[bV] (d) at (0,1){};
\draw (a) to (b) to (c) to (d) to (a) to (f) to (b) (c) to (e) to (d) (e) to (f);
\end{tikzpicture}\end{matrix}
$$
\item Give an example of a simple graph with at least 3 vertices that satisfies (1) the degree of every vertex is at least $(|V|-1)/2$, but (2) does not have a Hamilton circuit. (This shows that  that Dirac's  bound is ``sharp''.)
\end{enumerate}
\end{try}




\end{document}

