I have found that the Art Gallery Problem engages middle- and high-school students, and quickly leads to the unknown, which itself can be eye-opening to students. (On the latter point, students tend to think of mathematics as settled, so it is nice for them to reach unsolved problems they can comprehend, which abound at the interface between geometry and graph theory.)
Proving the traditional art gallery theorem (that $\lfloor n/3 \rfloor$ guards suffice and are sometimes necessary to cover an $n$-wall gallery) introduces triangulations and the chromatic number of a graph. There are many sources, including the recent book (if I may self-promote) Discrete and Computational Geometry.
Addendum. May I also recommend "How to Guard an Art Gallery and Other Discrete Mathematical Adventures", by T.S. Michael, whom I had the pleasure of teaching two decades before his book was
published.
Trying to cover the material in a course is usually bad. You don't have much time, and even if the students learn the material, they might just be bored when they see the material later in a normal class.
I think you have an opportunity to let the students do some mathematics, and to see some of its beauty and challenge. Don't make the mistake of trying to get them to work on open problems, but do give them activities related to the following parts of mathematics:
- Guess what is true.
- Prove what you believe is true.
- Communicate.
It was a big surprise to me when I learned in the PROMYS program that these are what mathematicians do as opposed to the types of things I had seen in mathematics classes, which concentrated on learning techniques and applying them. I do not recommend trying to follow PROMYS because you have much less time.
For example, you can start with Pascal's triangle. Have them look for patterns. Suggest looking at the even-odd pattern, or the sums of every element in a row, every other element or every third. Estimate the size of $2n \choose n$, and ask how you might compute ${30 \choose 15} \approx 2^{27}$ with $32$-bit numbers so that you can't compute $15! \approx 2^{40}$ directly. (Recursion? ${30 \choose i}/{30 \choose i-1}$? Prime factorization?) Ask for generalizations to multinomial coefficients. Show some connections to other areas of mathematics, e.g., ask how many faces a hypercube has of each dimension, or mention the Central Limit Theorem. You can show how you might prove some of these patterns with induction, or bijective arguments, or by evaluating $(x+y)^n$ at particular values of $x$ and $y$. Have them read an article covering related material in something like the College Journal of Mathematics or Quantum so that they see some good exposition, and that mathematics is still active. Let them flesh out and present parts of the article. Then have them write something about what they have learned.
An advantage over some competitions which emphasize distinguishing the most exceptional individuals is that there is something for everyone in Pascal's Triangle (or in the exploration of similar rich subjects of mathematical study). If your students have a lot of talent and energy, they can work on complicated material. If they are not as advanced, then they can still make real progress on simpler patterns.
An historical overview in parallel might complement the hands-on study.
Best Answer
If I may forgiven for self-promotion, you might examine How To Fold It: The Mathematics of Linkages, Origami, and Polyhedra (Cambridge University Press, 2011). All of its topics are accessible to high-school students, but all fall outside the high-school curriculum. See also
howtofoldit.orgfor some (not yet well-organized) supplementary material.