The day that is directly before June 1 is May 31; however if you are referring to the day of week as of this year's upcoming May 31, then the day is Thursday.
June 1, 1975 was a Sunday
June 1 1990 was a Friday.
1 June 1991 was a Saturday.
June 1 1987 was a Monday.
1 June 1796 was a Wednesday.
Fathers' Day is the third Sunday of June, which makes it the Sunday that falls after June 14 and before June 22.
June 1 1973 was on a Friday.
1 June 1951 was a Friday.
June 1 1975 was on a Sunday.
June 1, 1945 was on a Friday.
June 1 2005 was a Wednesday.
June 1, 1953 was on a Monday.