Does there exist a graph with degree sequence (3,3,3,3,3,3)? I am pretty such graph does not exist since I have tried to draw one without success, but is there a way to prove it?
* /|\ / | \ * * * \ | / \|/ *
Now connect one more vertex to the three on the horizontal midline.
By the way, the Erdős-Gallai theorem gives you a way to determine whether any given degree sequence can be realized in a simple graph.
The two non-isomorphic graphs with these degree sequences can be generated using
geng which comes with nauty. The command is
geng 6 9:9 -d3 -D3
where we have
9:9 means between 9 and 9 edges,
-d3 means minimum degree 3 and
-D3 means maximum degree 3. These can be viewed using
showg, but I prefer to write my own script to draw them: