A domino is a rectangle divided into two squares with each square numbered one of 0,1,2,3,4,5,6. Two squares on a single domino can have the same number. Show that distinct dominoes can be arranged in a circle so that touching dominoes have adjacent squares with identical numbers.

This is how I interpreted this question. But the solution says something else and I don't know how they got it.

This is my diagram

Now we can turn this into a graph, G, by letting the dominoes be the edges and each of the numbers be vertices.

Clearly, G is a connected graph and each vertex's degree is 2. Thus G has a Euler's cycle, therefore the dominoes can be arranged in a circle so that touching dominoes have adjacent squares with identical numbers.

Now the solutions says this: (My questions are in brackets)

"We model the situation as a graph G with seven vertices labeled 0,1,2,3,4,5,6. The edges represent the dominoes. There is one edge between each distinct pair of vertices (Why do you connect every single vertex to each other?) and there is one loop at each vertex (Why do you have a loop at each vertex?). Notice that G is connected. Now the dominoes can be arranged in a circle so that touching dominoes have adjacent squares with identical numbers iff G contains an Euler cycle (Why does G have to contain an Euler cycle so that adjacent squares will have identical numbers?). Since the degree of each vertex is 8, then G has an Euler cycle. Therefore, the dominoes can be arranged in a circle so that touching dominoes have adjacent squares with identical numbers."

Many thanks.