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V.1  Graph coloring


$\chi(G)$ = (vertex) chromatic number of $G = min\{k ~|~$ there exists $f: V(G) \rightarrow \{1, ..., k\}$ such that if $v_1$ and $v_2$ are adjacent, then $f(v_1) \not= f(v_2)\}$


$\chi(G) \geq \omega(G) = $ clique number of $G = max\{n ~|~ K_n \subset G\}$


$c^{-1}(j)$ is a set of independent vertices called a color class.

$\chi(G) \geq 2$ iff $G$ contains an edge.

$\chi(G) = 2$ iff $G$ bipartite iff $G$ does not contain an odd cycle.


$\chi(G) \geq 3$ iff $G$ contains an odd cycle.


Let $\alpha(G) = $ independence number of $G$ = maximal size of an independent set of vertices.

$\chi(G) \geq max\{ \omega(G), |G|/\alpha(G) \}$.

Greedy algorithm:  
\u
Order the vertices, $x_1, ..., x_n$.  Let $f(x_1) = 1$.  If $f(x_i)$ defined
for $i < j$, let $f(x_j) = min\{ c \in {\cal N} ~|~ $ if $x \in \{x_1, ..., x_j-1\} \cap \Gamma(x_j)$, then $f(x) \not= c \}$

\vfill
\eject

Greedy algorithm implies $\chi(G) \leq \Delta(G) + 1$ (where $\Delta(G)$ = maximum degree of vertices in $G$.

Suppose $\chi(G) = k + 1$ and $\chi(H) \leq k$ for all induced subgraphs $H$ of $G$.  If $deg(x_0) = \delta(G)$, then $\chi(G - x_0) \leq k$. Hence $\delta(G) \geq k$.


Thm 1:  let $k = max \{\delta(H) ~|~ H$ induced subgraph of $G \}$, then $\chi(G) \leq k + 1$.



Thm 2:  If $H_0$ is an induced subgraph of $G$ and suppose for all $H$ such that $V(H) \not= V(H_0)$ and $H_0 \subset H \subset G$, there exists $x \in V(H) - V(H_0)$ such that $d_H(x) = k$, then $\chi(G) \leq max\{k+1, \chi(H_0)\}$

Suppose there exists $K_n \subset G$ such that $G - K_n$ is a disjoint union of $H_i$.  Then $\chi(G) = max\{\chi(H_i \cup K_n) \}$

$\chi(K_n) = n = \Delta(G) + 1$

$\chi(C_{2n + 1}) = 3 = \Delta(G) + 1$


Suppose $G$ is connected.  If $G$ is not $\Delta$-regular, then \break
$max \{\delta(H) ~|~ H$ induced subgraph of $G \} \leq \Delta(G) - 1$


Thm 3:  Let $G$ be connected.  Suppose $G$ is not a complete graph nor an odd cycles.  Then $\chi(G) \leq \Delta$.

\eject

$p_H(x) =$ number of colorings of a graph $H$ using colors $\{1, ..., x\}$


If $\chi(H) = k$, then $p_H(x) = 0$ for all $x < k$.

If $ab \in G'$, then $p_{G'-ab}(x) = p_{G'}(x) + p_{G'/ab}(x)$

$\chi(G' - ab) = min\{\chi(G'), \chi(G'/ab) \}$



Thm 4:  If $|V(H)| = n \geq 1$, $|E(H)| = m$, and $H$ has $k$ components. , then $$p_H(x) = \Sigma_{i=0}^{n-k} (-1)^i a_i x^{n-i}$$ where $a_0 = 1$, $a_1 = m$, $a_i \in {\cal N} = \{1, 2, 3, ... \}$ for all $i$.

\end