For each of the following degree sequences, either draw a graph with this degree sequence, or prove that no such graph exists. $(i) (1,2,2,3,4,5) $ $(ii) (1,2,2,3,4,4) $ $(iii) (1,2,2,3,5,5) $
$(i)$ - By handshaking lemma this isn't possible as a graph must have an even number of vertices with odd degree but it has $3$ vertices with odd degree and $3$ is odd and not even.
$(ii)$ See the image attached below. Is there an easier way of doing this though because it took me $5$ minutes to figure out. I think I overcomplicated it.
$(iii)$ Not possible. Let $V = ${${A,B,C,D,E,F}$}$ $ Imagine that $A$ and $B$ have degree $5$. This means that vertex $A$ must be adjacent to vertices $B,C,D,E,F$ and vertex $B$ must be adjacent to vertices $A,C,D,E,F$. This means the vertices $C,D,E,F$ are adjacent to $A$ and $B$, so they all have minimum degree of $2$. This means no vertex of the vertex set $V$ can have degree $1$, so the graph can't be drawn.
Can someone check my justification for $(iii)$ please ?