1. Some special topic ideas

2. 6 2 3 13 5 11 10 8 9 7 12 4 14 15 1
Magic Squares
34 is magic
Magic Square—Where all the rows, columns, and main diagonals add up to the same “magic” number.
Semimagic Square – A magic square that fails to be magic only due to one or both of the main diagonals not being equivalent to the magic constant. Panmagic Square – A magic square where the main diagonals are equivalent to the magic constant, as well as the wrapped-diagonals. These wrapped diagonals will be discussed in the first algorithm. A Panmagic Square is also referred to as a Diabolical Square or a Pandiagonal Square. Bimagic Square – A magic square where each individual component is replaced by its square, and the result is still magic. Trebly Magic Square – A magic square whose cube replaces the original component, and the result is still magic. Multiplication Magic Square – A square that is magic under multiplication, rather than addition.

3. The Chromatic Polynomial of a Graph
counts the ways to properly vertex color a graph
=# proper vertex colorings of G in n colors
Recursively
Let e be an edge of G . Then, + = G
G\e
G-e

4. Knight’s Tour Problem Description
The Knight's Tour is a classic chess problem which was studied (and probably solved) over 1000 years ago.  The famous mathematician, Euler, published the first rigorous mathematical analysis of the problem in 1759.  
Here's the problem:  From an arbitrary starting position, move a  Knight chess  piece around a chess board visiting all other squares on the board exactly once.  

5. Possible Presentation Topics
·       The game of life:
Cellular Autonoma in general
Any original work
·        An application of some aspect of graph theory to a real world problem
·        Any theorem in the book which isn’t proved in the text (usually references are given)
·        Any topic in the book past section 7.2
·        Scheduling theory
·        Steiner triple systems
·        Kuratowski’s theorem (proving that K3,3 and K5 are the excluded minors for planar graphs)
·        Combinatorial knot invariants
·        Latin and room squares
·        Tournaments
·        Traveling salesman problem

6. More possible topics · Coloring problems
·        Analysis of Instant Insanity Game or NIM (chapter 10)
·        Combinatorial games (see
·        Graph embeddings
·        Hypergraphs
·        Domination
·        Random Graphs
·        Graph or geometric thickness
·        Graph drawing:
·        Grey code and its applications
·        Origami mathematics (See Tom Hull’s homepage:

7. Open Problems
Color code—this is sort of the opposite of the grey code problem. Here, assume you have n objects (for any n) and you want to color them using RGB so that the colors are as distinct as possible. I have two industry project that need a solution to this ASAP. It may already be done, but we haven’t been able to find it. If you have some CS background, and want a problem that someone will actually use, think about trying this.

8. The Cycle Double Cover Conjecture
The cycle double cover conjecture states that every bridgeless graph has a family of cycles such that each edge appears in exactly two of the cycles. Recall that a bridge is an edge whose removal disconnects the graph, i.e. a bridge is the same thing as an isthmus (only easier to say). For example, if
then the following family of cycles gives a double cover of G, since each edge of G appears in exactly one red cycle and exactly one blue cycle.
G = ,

9. Fibonacci Graphs G7 Decycled G1 G2 G3 G4 G5 G6 G7 2 1 3 4 5 6 72 1 3 4 5 6 7 G6 G7
G7 Decycled 5 3 4 2 1 6 7

10. Tower of Hanoi
The game:
The math:

11. Neighborhoods in digraphs
Let D be a simple (no loops or multiple edges) digraph. Prove that there is always some vertex v so that , where is the number of vertices at distance 1 along a directed path from v, and is the number of vertices at distance 2 along a directed path from v. A B C D

12. Some more open questions
Any open questions suggested by your special topic (this has the advantage that you have already studied the backgound of the problem)
Bill Martin’s problem (originally posed by Kaneko?): Let D be a simple directed graph. Prove there is some vertex v such that the number of vertices of distance 2 from v is greater or equal to the number of vertices of 1.
Can you always color the edges of a graph so that there are no monochromatic 4 cycles? If not, can you categorize the graphs for which it is and isn’t possible?
Problems in Jensen and Toft’s graph coloring problems book (in my office).
Erdos on graphs—his legacy of unsolved problems (book in my office).
Dan Archdeacon’s website at is an excellent source of open problems. 

13. A few more resources…
Chung, F. R. K. Open problems of Paul Erdös in graph theory. J. Graph Theory 25 (1997), no. 1, (at UVM library)
Colbourn, Charles J. Some open problems on reliability polynomials. (English. English summary) Proceedings of the Twenty-fourth Southeastern International Conference on Combinatorics, Graph Theory, and Computing (Boca Raton, FL, 1993). Congr. Numer. 93 (1993), (at UVM library, or I may have a rough draft of this paper in my files)
Graph labeling—see