RIOPLATENSE 2013 Problem 3

A division of a group of people into various groups is called $k$ -regular if the number of groups is less or equal to $k$ and two people that know each other are in different groups. Let $A$ , $B$ , and $C$ groups of people such that there are is no person in $A$ and no person in $B$ that know each other. Suppose that the group $A \cup C$ has an $a$ -regular division and the group $B \cup C$ has a $b$ -regular division. For each $a$ and $b$ , determine the least possible value of $k$ for which it is guaranteed that the group $A \cup B \cup C$ has a $k$ -regular division.