If a judge doesn't swear in the litigants and proceeds with questioning, does it constitute a mistrial?
Does it constitute a mistrial? No. An oath is taken in SOME judicial proceedings or in relation to SOME matters connected with judicial proceedings.
What is the name of your state, D M Gibson?
Here are links to the Federal Rules of Evidence Rule 603, to Alabama's Rule of Evidence Rule 603 (to pick one state out of 50), and to the different types of oaths and for what and when they are used:
Federal Rules of Evidence, Rule 603, Oath or Affirmation: http://www.law.cornell.edu/rules/fre/rule_603
Alabama Rules of Evidence, Rule 603, Oath or Affirmation: http://judicial.alabama.gov/library/rules/ev603.pdf
The 'Lectric Law Library, Legal Definition of Oath: http://www.lectlaw.com/def2/o027.htm
Are you hoping for a mistrial in a case you are involved in, D M Gibson?