I am currently working on an assignment for my discrete mathematics lecture. Here we are currently looking at triangulations, that is, graphs which have the following attributes: G is a graph with minimum degree of 3. G is a triangulation if every area in the graph is surrounded by exactly 3 edges.
Now, let vi be the number of vertices of degree 3. Show that the following is true:
3 * v3 + 2 * v4 + v5 = v7 + 2 * v8 + 3 * v9 + ... + (X - 6) vX + 12
Whereas X is the maximum degree in G.
Now, I was wondering whether this notation is at all exact. I understand what is implied by the "..." in there and see how the whole thing is supposed to work, but is it clear enough that you could, for example, say that when X = 4, you don't even include the (X - 6)vX part? Because if you do, this is easily shown to be wrong. If you don't, it probably is true. But is that something that is to be considered given through this notation? Or can I just give an example where it isn't true? I am simply not sure whether this is in any ways precise, when alternatively it would be easily written as a sum instead.
Thank you very much for reading!