I assume that you are asking about an arrest in which the police do not also have a search warrant. If this is the case, the police have the right to search the arrestee and they have the right to search in ways that will protect them from possible harm. However, as seen in the Chimel v. California case, the police may not search a whole house. In that way, the arrestee has the expectation of privacy in the rest of his or her house.
When a person is arrested at home, the police may search the area in his or her immediate control. They may also search to the extent that is necessary to ensure that they will not be attacked by anyone else who may be hiding in the house. However, they may not simply act as if they have a search warrant and search the whole house.