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\setcounter{try}{54}


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\centerline{\bf Math 365 -- Monday 4/29/19}



\begin{try}Let $A$, $B$, $C$, $D$, $G$, and $H$ be the graphs
\\ \smallskip
\centerline{$\displaystyle
A = \begin{matrix}\begin{tikzpicture}[scale = 1.5]
\node[bV, label=above:{$a$}] (a) at (0,1){};
\node[bV, label=above:{$b$}] (b) at (1,1){};
\node[bV, label=below:{$c$}] (c) at (0,0){};
\node[bV, label=below:{$d$}] (d) at (1,0){};
\draw (a) to (b) to (c) to (a) (b) to (d);
\end{tikzpicture}\end{matrix}\qquad \qquad 
B= \begin{matrix}\begin{tikzpicture}
\node[bV, label=above:{$a$}] (a) at (0,0){};
\node[bV, label=above:{$b$}] (b) at (1,1){};
\node[bV, label=below:{$f$}] (f) at (1,-1){};
\node[bV, label=above:{$g$}] (g) at (2,0){};
\node[bV, label=above:{$c$}] (c) at (3,1){};
\node[bV, label=below:{$e$}] (e) at (3,-1){};
\node[bV, label=above:{$d$}] (d) at (4,0){};
\draw (a) to (b) to (g) to (f) to (a) to (g) to (c) to (d) to (e) to (g) to (d) (b) to (c);
\end{tikzpicture}\end{matrix}\qquad \qquad 
C = \begin{matrix}\begin{tikzpicture}
\node[bV, label=above:{$a$}] (a) at (1,1){};
\node[bV, label=left:{$b$}] (b) at (0,0){};
\node[bV, label=right:{$c$}] (c) at (2,0){};
\node[bV, label=below:{$d$}] (d) at (1,-1){};
\draw (a) to (b) to (d) to (c) to (a) to (d);
\end{tikzpicture}\end{matrix}$}

\smallskip

\centerline{$\displaystyle
D = \begin{matrix}\begin{tikzpicture}
\node[bV, label=left:{$a$}] (a) at (0,2){};
\node[bV, label=right:{$b$}] (b) at (3,2){};
\node[bV, label=below left:{$c$}] (c) at (0,0){};
\node[bV, label=below:{$d$}] (d) at (3,0){};
\node[bV, label=above:{$e$}] (e) at (.75,1){};
\node[bV, label=below:{$f$}] (f) at (2.25,1){};
\draw (a) to (b) to (d) to (c) to (a) to (e) to (c) to (f) to (e) to (b) to (f) to (d) .. controls (5,3) and (3,3) .. (a);
\end{tikzpicture}\end{matrix}~  
G= \begin{matrix}\begin{tikzpicture}[scale=.75]
\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); 
\node[above] at (O0) {\small $a$}; 
\node[above] at (O1) {\small $b$}; 
\node[left] at (O2) {\small $c$}; 
\node[right] at (O3) {\small $d$}; 
\node[above] at (O4) {\small $e$}; 
\node[above left, inner sep = 2pt] at (I0) {\small $f$}; 
\node[above] at (I1) {\small $g$}; 
\node[left] at (I2) {\small $h$}; 
\node[right] at (I3) {\small $i$}; 
\node[above] at (I4) {\small $j$}; 
\end{tikzpicture}\end{matrix}\qquad \quad 
H= \begin{matrix}\begin{tikzpicture}[scale=1.75]
\foreach \x [count=\y from 0] in {a, b, ..., i}{ 
	\node [bV] (\x) at (90-40*\y :1cm){}; 
	\node[anchor=-40*\y-90] at (\x) {$\x$};
	}
\draw (a) to (b) to (c) to (d) to (e) to (f) to (g) to (h) to (i) to (a) (b) to (i) to (c) to (h) to (a) to (g) to (d) to (i) to (e) (b) to (h)
	(g) to (c) to (f);
\end{tikzpicture}\end{matrix}\qquad\qquad  $}
\begin{enumerate}[(a)]
\item Calculate the chromatic numbers for  $A$, $B$, $C$, $D$, $G$, and $H$. For each, give an example of a vertex coloring of the corresponding graph using exactly $\chi$ colors. 



\item Which of $A$, $B$, $C$, $D$, $G$, and $H$ have the property that removing a single vertex will reduce the chromatic number?




\item Classify all graphs with chromatic number (i) 1, and (ii) 2.  



\item What are the chromatic numbers of 

\centerline{(i) $K_n$, \qquad (ii) $K_{m,n}$, \qquad (iii) $C_n$, \qquad (iv) $W_n$, \qquad (v) $Q_n$?}
\end{enumerate}
\end{try}


\begin{try}
\begin{enumerate}[(a)]
\item What are the clique and independence numbers of $A$, $B$, $C$, $D$, $G$, and $H$ from the previous problem? How do $\omega$ and $|V|/\alpha$ compare to $\chi$ for each graph?



\item What are the clique and independence numbers of 

\centerline{(i) $K_n$, \qquad (ii) $K_{m,n}$, \qquad (iii) $C_n$, \qquad (iv) $W_n$, \qquad (v) $Q_n$?}
How do $\omega$ and $|V|/\alpha$ compare to $\chi$ for each graph? (You may need to break into cases.)



\item Explain why the clique number of the complement of a bipartite is no smaller than the number of vertices in each part. (Recall that the \emph{parts} of a bipartite graph are the two collections of pairwise non-adjacent vertices.)



\item Notice that the graph 
$$G = \begin{matrix}\begin{tikzpicture}[scale=1.5]
\foreach \x in {1,3,5,7,9}{\node[bV, label=above:{\small$x_{\x}$}] (\x) at (\x/2 + .5, 1){};}
\foreach \x in {2,4,6,8,10}{\node[bV, label=below:{\small$x_{\x}$}] (\x) at (\x/2, 0){};}
\foreach \x in {4,6,8,10}{\draw (1) to (\x);}
\foreach \x in {2,6,8,10}{\draw (3) to (\x);}
\foreach \x in {2,4,8,10}{\draw (5) to (\x);}
\foreach \x in {2,4,6,10}{\draw (7) to (\x);}
\foreach \x in {2,4,6,8}{\draw (9) to (\x);}
\end{tikzpicture}\end{matrix}$$
is bipartite, so should have chromatic number $2$. Now color this graph using the so-called ``greedy algorithm'': name your colors ``color 1, color 2, \dots''. First color $x_1$ with color 1; then color $x_2$ with the lowest color possible (i.e.\ color 1 if you can, but color 2 if you can't); then color $x_3$ with the lowest color possible; and so on. How many colors did you need? What is a better way to color $G$?
\end{enumerate}

\end{try}
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