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Graph coloring
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$p_H(x) =$ 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 $i$.


Thm 5:  If $|V(H)|$, $E(H) = \{e_1, e_2, ..., e_m\}$.  $B \subset E(H)$ is a broken cycle if $B = C - e_j$ where $C$ is a cycle in $H$ and $j = max\{i ~|~ e_i \in E(C)\}$.  Then $$p_H(x) = \Sigma_{i=0}^{n-1} (-1)^i a_i x^{n-i}$$ where 
$a_i$ is the number of $i$-subsets of $E(H)$ containing no broken cycles.

 Cor 6:  If $|V(H)|$, $|E(H)| = m$, and girth of $H$ = g, then
$$p_H(x) = \Sigma_{i=0}^{n-1} (-1)^i a_i x^{n-i}$$ where 
$a_i = \left(\matrix{m \cr i}\right)$ for $i \leq g-2$.  If $g$ finite and $H$ has $c_g$ cycles of length $g$, the $a_{g-1} = \left(\matrix{m \cr g-i}\right) - c_g$.


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Edge coloring.
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$\chi'(G) =$ edge-chromatic number.


Thm 7:  $\chi'(G) = \Delta(G)$ or $\Delta(G) + 1$
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Graphs on surfaces.
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Thm:  Every plane graph is 4-colorable.

 $S_p$ = orientable surface with genus $p$ and Euler characteristic, $\chi(S_p) = 2(1-p)$ = 4p-gon with sides identified in a particular way.


 $N_q$ = non-orientable surface with genus $q$ and Euler characteristic, $\chi(N_q) = 2-q$ = 2q-gon with sides identified in a particular way.

$e(G) \leq 3n - 3\chi(M)$ where $n = |V(G)|$ if $G$ can be drawn on the surface $M$.


Thm 9:  Let $k = \chi(G)$.  If $G \subset M$ where $M$ is a closed surface such the $\chi(M) \leq 1$, then $$k \leq h(\chi(M)) = {7 + \sqrt{49 - 24\chi(M)} \over 2}$$ 



Proof:  Let $G$ be a minimal graph of chromatic number $k$.  Then $\chi(G - x) \leq k - 1$ for all $x \in V(G)$.  Since we can color $G - x$ with $k-1$ colors, but we need $k$ colors to color $G$, $deg(x) = k-1$ (else can color $x$ with the color missing in $\Gamma(x)$).  Hence 

$$\delta(G) \geq k-1$$

Suppose $n = |V(G)| \leq h$, then we need at most $h$ colors to color $G$ and thm 9 holds.

\centerline{Suppose $n \geq h + 1$}


Suppose $\delta(G) \leq h - 1$.  Then $k - 1 \leq \delta(G) \leq h - 1$.  Then $k \leq h$ and thm 9 holds.

\centerline{Suppose $\delta(G) \geq h$.}


$h = {7 + \sqrt{49 - 24\chi(M)} \over 2}$ where $\chi(M) \leq 1$.  Hence
$$h  \geq 6$$

$  {nh \over 2} \leq {n \delta(G) \over 2} \leq e(G) \leq 3n - 3 \chi$
 
${nh}  \leq 6n - 6 \chi$

${nh} -6n \leq  -6 \chi$

$(h + 1)(h - 6) \leq n(h -6) \leq  -6 \chi$

$h^2 - 5h - 6  + 6 \chi \leq 0$


$(h - a_1)(h - a_2) \leq 0$.

where $a_i = {5 \pm \sqrt{25 - 4(-6 + 6\chi)} \over 2} = {5 \pm \sqrt{49 - 24\chi} \over 2}$

Hence $h \leq {5 \pm \sqrt{49 - 24\chi} \over 2}$, a contradiction.
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Defn:  $s(M) =$ max$\{i ~|~ G \subset M, \chi(G) = i \}$

Thm 10:  Let $\chi(M) \leq 0$, 
$h = h(\chi(M)) = {7 + \sqrt{49 - 24\chi(M)} \over 2}$,
Suppose $s(M) \geq h$, If $G \subset M$ and
 $G$ = minimal $h$-chromatic graph, then $G = K_h$.

Cor:  $s(M) = h$ iff can draw $K_h$ on $M$

Thm 11:  \hfil \break
$s($torus) = 7, $s($projective plane) = 6, $s($Klein bottle) = 6.

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