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List coloring
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Thm 12:  Let $G$ be a near triangulation with outer cycle $C = x_1x_2...x_k$ such that $L(x_1) = \{1\}$, $L(x_2) = \{2\}$, $|L(x_i)| \geq 3$, and $|L(x)| \geq 5$ for all $x \in V(G-C)$.

Cor:   Every planar graph is 5-colorable.
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Perfect graphs
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Recall:  $\chi(G) \geq \omega(G) = $ clique number of $G = max\{n ~|~ K_n \subset G\}$


$G$ is perfect if $\chi(H) = \omega(H)$ for every induced subgraph $H \subset G$.

Ex:  If $G$ is bipartite, $G$ is perfect.

Ex:  A triangle free graph containing an odd cycle is not perfect.

If $G = (V, E)$, the complement of $G = \overline{G} = (V, V^{(2)} - E)$.

Note:  $\omega(\overline{G}) = \alpha(G)$ independence number of $G$ = maximal size of an independent set of vertices.

Thm 15:  The complement of a bipartite graph is perfect.

Recall Cor 3.10:  If $G$ bipartite, then $\alpha(G) =$ minimal number of edges and vertices covering all vertices (where $e = xy$ covers $x$ and $x$ covers $x$).

%%Thm 16:  If $G$ bipartite, then the line graph $H = L(G)$ and its complement %%$\overline{H}$ are perfect.

%%Thm 17:  Comparability graphs and their complements are perfect.

Thm 18:  A graph $G$ is perfect iff for every induced subgraph $H \subset G$, there is an independent set of vertices $I$ such that $\omega(H - I) < \omega(I)$

I.e., $G$ is perfect iff every induced subgraph $H$, there exists an independent set of vertices $I$ such that $I \cap D \not= \emptyset$ for every clique $D$ of maximal order $\omega(G)$. 

(recall, a clique = maximal complete subgraph).

Thm 19:  A graph, $H$, obtained from a perfect graph, $G$ by replacing its vertices with perfect graphs, $G_i$ is perfect  ($v(H) = \cup V(G_i)$, $E(H) = [\cup E(G_i)] \cup \{v_iv_j ~|~ v_i \in G_i, v_j \in G_j$ such that the vertices in $G$ corresponding to $G_i, G_j$ are adjacent $\}$).

Thm 20:  The complement of a perfect graph is perfect.

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