Show that for every finite set $S$ of positive integers, there exists a positive integer $k$ such that the sequence obtain by listing each element of $S$ a total of $k$ time is graphical.
This is one of the example in the book. The book said this can be proven easily by letting $S={a_1,a_2,...,a_n}$ and $k= lcm\{a_1+1,a_2+1,...a_n+1\}$, which make me feel I'm an idiot. Because I still can't see how this gonna help me. I wonder if any one would explain this to me please.
Here is one for the set $S = \{1, 2, 3\}$, with $k = 12 = \operatorname{lcm}\{2, 3, 4\}$ (so there are twelve vertices of order $3$, and just as many of order both $2$ and $1$). Hope that helps you understand what your book means. This is what @genisage meant when they said "lots of little disjoint complete graphs".