This is the form of a solemn vow in that time. Abner is asking God to judge him severely if he does not keep his vow. Your language may have a way of expressing vows. AT: "I am asking God to punish me if I do not"
# the house of Saul
Here "house" refers to Saul's family and supporters who survived Saul's death. AT: "the family and supporters of Saul"