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CS 173, Fall 2014: Skills list for seventh examlet

- Graph coloring
- Know that a (vertex) coloring of a graph is labelling of each
vertex with a color, such that adjacent vertices always have different
colors.
- Know that the chromatic number of a graph is the minimum number
of colors required to color it.
- Know the shorthand notation χ(G).
- Know that graph coloring is useful for (among other things) register allocation
in compilers for programming languages like C and Java.
- Know that if D is the maximum degree of any vertex in a graph G,
then G can be colored with D+1 colors.
- Know that the "greedy" coloring algorithm often
produces pretty good colorings fast, but is not guaranteed to produce
the best solution.

- Two-way bounding
- Understand the difference between an exact result, an upper bound, and a
lower bound.
- Prove an equality by establishing upper and lower bounds. E.g.
prove that the chromatic number of a graph G is n.
- Prove a set equality by proving inclusion in both directions.

- Summations
- Know how to read summation notation, i.e. convert between it
and a list of terms with ellipsis (...). Same for product notation.
- Extract the first or last term in a sum or product.
- Rewrite a sum or product so that its index variable starts
at a different number.
- Know the closed forms for two key summations:
sum of the first n integers and sum
of (1/2)
^{k} (for k from 1 to n).

- (Easy) Induction
- Given a claim, identify/state the key parts of an inductive proof:
the induction variable, the claim
P(n), base case, inductive step, inductive hypothesis, conclusion of the inductive step.
- Be able to state a (strong) inductive hypothesis. (We always use strong hypotheses
in this class, but some of you may have used weak hypotheses in other classes.)
- Use induction to prove a formula (equality or inequality)
is correct for all integers starting at some base case.
- Use induction to prove that a non-formula claim (e.g. a divisibility relation)
holds for all integers starting at some base case.
- Any proofs for this examlet will have very simple structure.
In particular, the truth of the claim at n=k depends only on the claim being true at n=k-1,
with only one base case required.