First, let's discuss the structure of your paper.  Generally, your paper
should have a title, abstract, introduction, main body, conclusion,
acknowledgements, and a bibliography.  So what should each of these parts
look like?  When you begin writing a paper, you must keep in mind that
there are hundreds, and perhaps thousands, of papers published every year
on graph theory and combinatorics.  A professional mathematician only has
time to consistently skim a handful of journals every month, and will read
just a few of these articles in detail.  So if you want to generate
interest in your results, you need to provide the reader with compelling
reasons to read your paper beyond the title and the abstract.

What does this mean?  First, your title should be short, concise, and
descriptive.  Titles like "On a conjecture of Erdos" and "A new graph
labeling problem" won't (and shouldn't) excite the reader; much better to
title a paper "Incidence-coloring cubic graphs" or "Well-covered graphs
and the roller coaster conjecture" (after all, could you really resist
finding out what the intriguingly-named roller coaster conjecture is all
about?).  Your abstract should expand on the title, giving a one or two
sentence description of the problem/topic, and then stating your main
results.

Then comes the introduction.  In a very real sense, your introduction
should be your abstract expanded from one paragraph to three or four.  
One preferred approach is the following:  in the first paragraph, give a
history of the problem.  What is the motivation?  Who has thought about
this topic?  In the second (and possibly third) paragraph, describe the
problem, defining relevant terms, and talk about existing results.  In the
final paragraph, outline your paper, stating what (type of) results you
will prove in each section.  In the introduction, you should try to avoid
giving proofs - this will rarely serve to interest the reader.  Also, many
people prefer to state theorems and definitions "in-line" (not set off
from the rest of the text), possibly reserving more formal statements for
the main body of the paper; the idea is to keep the introduction
accessible and readable, keeping the main material in the body of the
paper.

The body of your paper should simply be divided into sections in some
natural way.  If possible, it is nice to have some text/commentary in
between theorems; a paper that goes Theorem, Proof, Theorem, Proof, ...
with no notes in between will not flow well, and it will be harder for the
reader to maintain interest and focus.  Finally, in the conclusion, you
could recap the results you have proved.  Then, ideally, you should
suggest open problems, conjectures, or further directions for research.  
This is highly advisable; after all, if the topic is a dead end, why
should people care about it?  (In particular, Joe would never select a
paper for the program if it didn't have any open problems.)

Please keep in mind that while you have surely seen research papers that
have violated these guidelines (no introduction and/or conclusion, no
commentary, etc), that does not make it right.  Those authors simply did
not care to carefully present their research to their audience - you
should strive to do a better job.

Let's move on to your exposition.  You should realize that exposition is
an art that many (indeed, most) mathematicians are not particularly good
at.  But with some diligent practice, anyone can improve their
presentation.  The most important aspect of good exposition is clarity.  
We have all read papers or textbooks in which the author crams as much
information as possible onto one page and makes no efforts to enlighten
the reader as to what is really going on.  This is NOT desirable.  You
should try to make your mathematics transparent to the reader.  It is true
that concision is encouraged, but only up to the point that it begins to
obscure your writing.  Do not become too attached to your first
formulation of a proof; always strive to improve or revise it, perhaps
making it shorter, but more importantly, clarifying it.

If the proof of a theorem seems complicated or long, try breaking into
multiple lemmas or propositions.  These may be technical lemmas, or they
may be propositions that are of interest independent of the main theorem.  
While you are writing, constantly be thinking carefully about notation,
variable names, order of proof, extremal cases, possible generalizations
or simplifications, and your wording.  And as always, you are encouraged
to provide as many specific examples as possible.  Remember that while you
have spent (at least) a summer thinking about the problem, your reader may
be seeing it for the first time, so that things that may appear obvious to
you are in fact quite difficult statements that require proof for the
reader.

So to sum up, when writing a research paper, you should have two
objectives.  First, you must get your audience interested in your paper.  
Then, when they actually read the paper, you must explain your research so
that they understand (and can appreciate what you have done).  This is an
ever-evolving process; continually strive for clarity and interest.  
Writing mathematical papers well is a difficult skill to learn, but pays
off in the long run.

---Geir