BJP will likely win in Karnataka

Who has the right to form the government in Karnataka after the election results?

The procedure follows the parliaments after the Westminster model and provides that the governor, after advice, invites the largest parliamentary group (here the BJP) to try to form an administration.

Since the BJP does not have a majority, it would either have to reach an agreement with another party or rely on the support of a sufficient number of rebels from other parties. If the largest party cannot form an administration after a certain period of time (often two weeks), the governor (also on recommendation) either invites the next larger group (in this case INC) to form an administration or to convene new elections.

I haven't followed this very carefully, but I understand that the BJP is unlikely to get a majority, but a coalition of INC and JDS could be formed. The parties would have to decide on the details of the coalition. It could be a formal coalition led by the INC (as the main party in the coalition), but some ministries give it to the JDS, or a loose agreement where the JDS supports the INC in critical votes, but does so Independently reserves the right to vote in other matters.

These issues can take several weeks to resolve, so a new administration is unlikely to form quickly. The public service remains in place and can ensure that the basic function of government is not overly disrupted.

To update

On May 17th, the BJP were asked to try to educate and govern. BS Yeddyurappa became chairman of the board, but was unable to demonstrate a majority in the assembly. After 6 days he was removed and on May 23rd a coalition led by HD Kumaraswamy took over from the JD (S).