Choosing one number from each set so that the difference between maximum and minimum is minimized
It's not really a graph problem, I think. Problems in which you try to choose some objects to minimize the difference between the min and max chosen object are called "minimum range" problems — e.g. look up "minimum range cut", which is a graph problem — but this one looks like you're trying to find the minimum range basis of a partition matroid with one partition set for each of your groups.

In general minimum range matroid basis problems can be solved in $O(n\log n)$ time, plus $O(n)$ steps of a subroutine that finds a basis of a set with corank one: sort the elements from smallest to largest and then process the elements one by one. While you process the elements maintain an independent set $I$; when you process an element $e$, add $e$ to $I$ and, if that addition causes $I$ to become dependent, kick out the minimum weight element in the unique circuit of $I$. The minimum range basis is one of the sets $I$ that you found in this process: the one that has full rank and has as small a range between min and max as possible.

For your partition matroid special case it's easy to find the circuit in $I$ when it becomes dependent: a circuit happens when $e$ belongs to the same group as an element $f$ that you previously added, and $f$ is the element you should kick out of $I$. So the whole algorithm takes $O(n\log n)$ time, or possibly faster depending on how much time it takes to do the sorting step.