Consider the set of planar graphs where all the internal faces are triangles. If there is an interior point of odd degree the graph cannot be three colored. If every interior point has even degree can it always be three colored? Ideally I'd like a small counterexample.
Welcome to Theoretical Computer Science Stack Exchange
Theoretical Computer Science - Stack Exchange is a question and answer site for professional researchers in theoretical computer science and related fields.
It's built and run by users as a part of the Stack Exchange network of Q&A sites. We welcome you to join us in asking and answering research-level questions in theoretical computer science.
For undergraduate-level questions please visit Computer Science which has a broader scope.
For more information about the scope please check out our FAQ.
We're a little bit different from other sites. Here's how:
Ask questions, get answers, no distractions
This site is all about getting answers. It's not a discussion forum. There's no chit-chat.
Good answers are voted up and rise to the top.
The best answers show up first so that they are always easy to find.
The person who asked can mark one answer as "accepted".
Accepting doesn't mean it's the best answer, it just means that it worked for the person who asked.
Coloring Planar Graphs
Yes, this is a corollary of the Three Color Theorem, see at the bottom here: http://kahuna.merrimack.edu/~thull/combgeom/colornotes.html
This result extends to high dimensions. A triangulation of a d-dimensional sphere so that every vertex has an even degree is (d+1) colorable. See, for example this paper: Jacob E. Goodman and Hironori Onishi, Even triangulations of $S^3$ and the coloring of graphs, Trans. Am. Math. Soc. 246 (1978), 501–510.
Get answers to practical, detailed questions
Focus on questions about an actual problem you have faced. Include details about what you have tried and exactly what you are trying to do.
- Specific research-level questions in theoretical computer science
Not all questions work well in our format. Avoid questions that are primarily opinion-based, or that are likely to generate discussion rather than answers.
Questions that need improvement may be closed until someone fixes them.
Don't ask about...
- Anything not directly related to theoretical computer science
- Questions that are primarily opinion-based
- Questions with too many possible answers or that would require an extremely long answer
- Questions which are not research-level, e.g. exercises in undergraduate textbooks
You earn reputation when people vote on your posts
Your reputation score goes up when others vote up your questions, answers and edits.
As you earn reputation, you'll unlock new privileges like the ability to vote, comment, and even edit other people's posts.
|125||Vote down (costs 1 rep on answers)|
At the highest levels, you'll have access to special moderation tools. You'll be able to work alongside our community moderators to keep the site focused and helpful.
|2000||Edit other people's posts|
|3000||Vote to close, reopen, or migrate questions|
|10000||Access to moderation tools|
Improve posts by editing or commenting
Our goal is to have the best answers to every question, so if you see questions or answers that can be improved, you can edit them.
Use edits to fix mistakes, improve formatting, or clarify the meaning of a post.
Unlock badges for special achievements
Badges are special achievements you earn for participating on the site. They come in three levels: bronze, silver, and gold.
Sign up to get started
Theoretical Computer Science Stack Exchange is part of the Stack Exchange network
Like this site? Stack Exchange is a network of 174 Q&A sites just like it. Check out the full list of sites.