How to auto merge pull request on github?

GitHub recently shipped this auto-merge feature in beta. To use this, you can enable it in the repo settings. Just keep in mind you will need to add branch protection rules as well.

See the documentation for more info.

https://docs.github.com/en/free-pro-team@latest/github/collaborating-with-issues-and-pull-requests/automatically-merging-a-pull-request

You can use Mergify to do this.

It allows to configure rules and define criteria for your pull request to be automatically merged. In your case, setting something like "Travis check is OK and one reviewer approved the PR" would allow the PR to be automatically merged.

(Disclosure: I'm part of the Mergify team.)

You can most probably add an after_success action to your .travis.yml that would merge the PR using GitHub API. I do not know of any ready to use script for this, but there is no reason for it to be hard. Special care needed for authentication ...