When writing a proof, the goals are to clearly communicate the proof to the reader and to provide enough details so that the reader (and you) can verify its correctness. Pictures are great, if possible (for graph theory, it is very often possible to draw a helpful picture). Pictures should be clearly labeled. One should include, unless it is completely clear, an argument to show that the picture accurately represents the possibilities that can occur given the hypotheses. It is a common mistake to ignore a case because a picture misleadingly shows that this case cannot happen. (Pictures are an extremely important tool for discovering proofs in graph theory and one of the arts of graph theory is learning how to draw pictures following one's intuition, yet still be rigorous. In writeups, pictures should not be relied on as heavily as in proof discovery, but they are still useful for conveying the idea.)

Regarding how much detail to include in proofs, the clarity and level of rigor of the proofs in West are good goals to strive for. Given time limitations, I do not expect that your proofs for homework assignments be as polished as those in West. Most of the graph theory proofs in van Lint and Wilson are too condensed and do not have enough pictures for my taste.

Extreme rigor should mainly be practiced when first learning proofs or when starting out in a new area of math. Details that are expected at the beginning of the course can be omitted later on. A rule of thumb is that if a sentence that gives a proof of an easy fact is more difficult to parse than just to figure out the easy fact, then it's okay to say "it is easy to see that." If you use the phrase "it is easy to see that," it had better be correct and it should be easy for the grader to see.

For example, to prove that a graph is bipartite implies it has no odd cycle, I might say: "To prove the contrapositive, it suffices to show that an odd cycle is not 2-colorable. A path has a unique 2-coloring (up to permuting color classes) and the unique 2-coloring of the longest path contained in an odd cycle does not yield a proper coloring of the cycle." I like the proof in West a little better. Later in the course, for a detail this easy it would be okay to say "It suffices to show that an odd cycle is not 2-colorable. This is easy to verify given that a connected graph has a unique 2-coloring up to permuting color classes." (For this particular fact, later in the course you could just cite it--I mean this to be an example of how much detail to include to prove a fact as easy as this one.)

All this said, correctness will be given more weight than style. Keep in mind though that a well-written proof is easier to check for errors.

Particularly well-written and well-typeset (and correct) solutions will receive extra credit and will be posted online, with your permission.

Some more tips about writeups:

Make your proofs as modular as possible. Find precise statements that help to go from the hypotheses of the problem to the conclusion and label them as facts, lemmas, or claims. This makes it easier for you and the grader to check whether the proof is correct because the statement of the lemma, the proof of the lemma, and the way the lemma is applied in the main proof can each be checked separately for correctness. Defining new terms that allow a proof to be more clearly or compactly written is a good habit to get into.

I have noticed in the writeups so far that it is difficult to find the right amount of detail to include. In some cases, a paragraph is used to explain something where a sentence would suffice. This paragraph can be confusing to read, especially if it is not clear what it is proving. On the other hand, I have seen many incorrect proofs that result from not checking carefully a small detail that seems easy, but turns out to be wrong. If you encounter a detail that seems obvious, but you are not completely sure, then first make sure it is correct by writing out a careful proof and/or trying examples. After this, if you do decide it is obvious, then write a short, simple, and correct sentence for its proof. If the detail turned out to be trickier than expected, then a longer proof is okay. A short precise, correct, and typo-free statement is more valuable than a paragraph with some small mistakes or inaccuracies.

Some comments the grader had:

Please use correct spelling and grammar and proofread your proofs. Watch out for typos that change the meaning of what you write (e.g. T instead of T'). Clarity of your ideas is important. Make sure you answer all parts of a question (e.g. if AND only if).
Don't use symbols without defining them. If you say something like "because $B$ has this property" without EVER defining what $B$ is, that is very hard to parse. Please define symbols BEFORE using them.

Make sure your antecedents are obvious (and correct). Ambiguous pronouns make understanding your proofs harder.

Please don't redefine symbols that are used in the problem statement. If a problem says "Let S be . . ." then avoid using S to mean something different in your proof. It is okay to say "let S be as in the statement of the problem" instead of rewriting all the conditions on S.

If you are using symbols that look very similar, then please write neatly enough (or type) so that it is not ambiguous as to what you wrote.