## Welcome to Theoretical Computer Science Stack Exchange

Theoretical Computer Science Stack Exchange is a question and answer site for theoretical computer scientists and researchers in related fields. It's built and run by you as part of the Stack Exchange network of Q&A sites. With your help, we're working together to build a library of detailed answers to every question about theoretical computer science.

We're a little bit different from other sites. Here's how:

This site is all about getting answers. It's not a discussion forum. There's no chit-chat.

Just questions...

up vote

Good answers are voted up and rise to the top.

The best answers show up first so that they are always easy to find.

accept

Accepting doesn't mean it's the best answer, it just means that it worked for the person who asked.

# Coloring Planar Graphs

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.

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.

## Tags make it easy to find interesting questions

All questions are tagged with their subject areas. Each can have up to 5 tags, since a question might be related to several subjects.

Click any tag to see a list of questions with that tag, or go to the tag list to browse for topics that interest you.

# Coloring Planar Graphs

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.

## You earn reputation when people vote on your posts

+5 question voted up
+2 edit approved

As you earn reputation, you'll unlock new privileges like the ability to vote, comment, and even edit other people's posts.

ReputationPrivilege
15 Vote up
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 Vote to close, reopen, or migrate questions Access to moderation tools
see all privileges

## 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.

• 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

## 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.

You can always comment on your own questions and answers. Once you earn 50 reputation, you can comment on anybody's post.

Remember: we're all here to learn, so be friendly and helpful!

Yes, this is a corollary of the Three Color Theorem, see at the bottom here: http://kahuna.merrimack.edu/~thull/combgeom/colornotes.html

Also, the corollary is not mentioned in Hull's notes, only the 3-color theorem itself. But from a 3-connected graph G with triangular internal faces and even internal vertices one can form a maximal planar graph 2G with even vertices simply by stitching two copies of G on the outer face. If G is not 3-connected, one can 3-color its 3-connected components independently. - David Eppstein Dec 31 '10 at 6:12

## 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.