How to Control Project Versions on GitHub

First of all, since your specifications are rather unclear, I'm not sure how much of this really applies. But I'll give it a shot...

If your project has one single release cycle, you only need one repository. If your three teams need to release new version independently, it gets a lot messier. But for that I'd need more specific input. At the moment, it sounds to me like you need to stabilize your API, develop the library and manage your back-end development. If that's part of one project, go ahead and use one repository.

As for the branching, I'd recommend to always have one branch that simply represents the latest released and working version. This branch will probably mostly have merge commits. I like the idea of release branches independent of development branches as a release manager could then easily review everything (after feature freeze if you want so), scratch the last few itches, set the version properly, and merge into your production branch. (You could even bind hooks or deployment strategies to that merge in order to automatically push release onto your build system or something).

As I said above, I'd recommend having one release cycle within one repository. If you can have that, only one production branch will be necessary. If the teams really need to release new versions of the same project independently, it feels like there's something wrong with the project management. But that's again without knowing much about the team and project in general.

A style that I like much is the git flow as described by Atlassian. It suggests having one master (production) and one development branch. On development the actual work happens but, that usually splits up into feature branches. What I like is that a development manager (someone with deep technical insight) can manage the development branch. They decide whether to merge new features or not. At some point they will tell a project/release manager that the team's done and that project/release manager can then (together with the development manager) work out the last few glitches on a release branch before it'll be merged into master (and back into development, of course). It's just an interesting separation of concerns that developers are so keen on regarding different aspects of the very same projects. :)

Hope this helps a bit.

I haven't had this situation yet, but that's what I would try:

Try contacting the owner

Maybe they really lost interest, but are willing to transfer the project to somebody else, in particular someone who has already shown considerate commitment.

But perhaps they are just occupied with something else (work, vacations, illness, other projects) and didn't have time to handle your PR, but plan to do so later.

Or maybe they have really stopped work on the project permanently for whatever reason.

Without asking, you won't find it out.

Get in touch with the community

Surely there are other people who have contributed to, or at least used the project. Check who has forked the project (even if they haven't made any change, they might still be interested in seeing this project thriving); check who has reported issues, or commented upon them. Maybe there's also a community outside GitHub, e.g a mailing list, forum, or StackOverflow members.

I'm you eventually really take over the project, you might want their support. And they need to know where the new master repository is.

Continue to make good pull requests

This shows both the owner and the community that you are serious about it, and let's them judge your contributions.