A domino is a $2\times1$ rectangle. On each half of the domino is a number denoted by dots. In the figure, we show the ten dominoes whose pairs of numbers correspond to pairs chosen from $\{1,2,3,4,5\}$ (we do not include dominoes where the numbers are the same). Notice that we have arranged the ten dominoes in a ring so that where two dominoes meet, they show the same number.
Of interest to us is the following question: for what values of $n \geq 2$ is it possible to form a domino ring that uses all of the $C(n,2)$ dominoes that correspond to the pairs from $\{1,2,\ldots,n\}$. Let us solve the question by viewing it as a graph problem.
a. Recall that $K_n$ is the complete graph on $n$ vertices. Please draw $K_5$, naming your vertices $1,2,3,4,5$.
b. Write down an Eulerian circuit of $K_5$. Translate this Eulerian circuit into a ring of dominoes using pieces shown in the example. Conversely, starting with the example above, describe the corresponding Eulerian circuit of $K_5$.
c. Briefly argue that there is a bijection between the Eulerian circuits of $K_5$ and the domino rings that can be formed from the $C(5,2)$ dominoes above.
d. Regarding part (c), for what values of $n$ is it possible to form a domino ring using all $C(n,2)$ dominoes that correspond to the pairs from ${1,2,\ldots,n}$? Why?