TITLE: Domino Trains Questions
QUESTION [6 upvotes]: A train is an end-to-end arrangement of dominoes such that the adjoining halves of neighboring dominoes have the same number of dots.
A "double-$n$'' domino set has one of each possible domino using integers from $0$ to $n$, where order is not a distinguishing feature (so there aren't separate $0$-$2$ and $2$-$0$ dominoes, for example). 
Let $f(n)$ be the smallest number of trains that can be formed from the dominoes in a double-$n$ set, such that each domino is used in exactly one train. What are the values of $f(12)$ and $f(15)$?

How can I approach this problem with graph theory?

REPLY [4 votes]: We need to look at Eulerian paths, and use the well-known result about them using odd and even vertices. This makes $f(12)=1$ and $f(15)=8$.
The trains are fairly basic in the odd case (an even number of vertices) - we 'disconnect' $\dfrac{k+1}{2}$ single edge disjoint graphs, the remaining graph is Eulerian and can have a path joined to it to make a train, the rest form individual trains.
A perhaps more interesting answer can be arrived at by defining a train as a rotationally symmetric path for the complete graph, for example for $n=8$ we have:

If we rotate the image four times we arrive at:

which gives us the four minimum trains as required.