4. If in a certain month the number of Mondays is greater than the number of Tuesdays, then the year in which this occurs is ( ).
(A) 2010
(B) 2012
(C) 2014
(D) 2016
4. D.

From the problem, we know that the number of days for Wednesday, Thursday, Friday, Saturday, and Sunday in the month are not more than the number of days for Tuesday. A month can have 30 days, 31 days, 28 days, or 29 days.
We will discuss the four scenarios below.
(1) A month has 30 days. If the 29th of this month is a Monday, then the 30th is a Tuesday, in which case, the number of Mondays and Tuesdays in the month is the same; If the 30th of this month is a Monday, then the 29th is a Sunday, in which case, the number of Sundays and Mondays in the month is more than the number of Tuesdays, which contradicts the problem.
(2) A month has 31 days. Similarly, this leads to a contradiction.
(3) A month has 28 days. Then the number of Mondays and Tuesdays is the same. This contradicts the problem.
(4) A month has 29 days. If the 1st of this month is a Monday, then the 29th is also a Monday, with 5 Mondays and 4 Tuesdays. Therefore, the scenario that fits the problem can only occur in a month with 29 days.
Thus, the year that fits the problem can only be in February of a leap year.
Since the leap year is divisible by 4, and 2010 and 2014 are common years, we exclude options A and C.

Note that, in February, "only the number of Mondays is more than the number of Tuesdays," in which case, February 1st is a Monday, and the January 1st of that year is a Friday. It is easy to calculate that January 1, 2012, is a Sunday, which is a contradiction, so we exclude option B.

In fact, since May 12, 2013, is a Sunday, we can deduce that January 1, 2016, is a Friday.