Use HAVEL-HAKIMI ALGORITHM
(1) Sort the degree sequence
$$<3,3,3,3,2,2,1,1,1,1>$$
(2) Remove first degree from the above sequence obtained and subtract 1 from that many numbers of degrees ahead of the degree being removed, in this case, subtract from next 3 degrees.
$$<2,2,2,2,2,1,1,1,1>$$
(3) Repeat the previous step again.
$$<1,1,2,2,1,1,1,1>$$
Sort this sequence thereafter, to apply the procedure again.
$$<2,2,1,1,1,1,1,1>$$
(4) Repeat step (2)
$$<1,0,1,1,1,1,1>$$
We can repeat the process again for few more times to obtain a degree sequence consisting of only zeroes. But we can stop right here by saying that the degree sequence obtained is indeed a simple graph as there are even number of 1's in the sequence i.e., even number of vertices with odd degree.
Reference
HTH
