Back to math

I've taken a hiatus from posting here for a brief period. That's ok though since no one reads this anyway. However I haven't just been laying on my couch and watching television. I've been working a on problem presented by a friend of mine. He has a (currently nine part) description of the problem that you can read here:

http://toomai.wordpress.com/

called "Partitions of a Graph." I've been spending my free time trying to write a formal description of the problem and present some of the things we've proved so far. I just this morning found a class of graphs that allows me to say this:

Theorem: For every positive even integer n = 2m, there exists a Q1 graph G of order n such that \delta(G) = m-1.
Proof: The graph K_{(m+1),(m-1)} is Q1 and prohibits the partition (2, 2, ..., 2).

That's as much detail as I'm going to give here, because as you can tell above, I much prefer doing my formal writing in LaTeX. I will however try and figure a way to upload a pdf for those of you that are interested, (all zero of you).

The great thing about the theorem though, is that it allows me to say this.

Corollary: For any given positive integer n, there exist Q1 graphs with minimum degree greater than or equal to n.

and

Corollary: For any given positive integer n, there exist Q1 graphs with connectivity of degree greater than or equal to n.

These two facts seem rather counterintuitive, even though the example presented is fairly simple. It just adds to my interest in the problem.