I want to find some example unicyclic graph with degree sequence $(4,3,3,2,2,1,1,1,1,1,1)$.

I think that this graph not exists.

Every 1-degree vertex must be connected to some another $>1$-degree vertex(if graph not connected then solution in the picture)enter image description here

We need to use $\ge 4$ edges in cycle and edge for connected to some another $>1$-degree.

Every $>1$-degree vertex must be connected to another $>1$ too.

$\ge 4$ edges in cycles, $1$ edge for connected cyclic to another and for $>1$ vertices.

But i dont undestand that i need do later.

Thank you.


The degree sequence implies that there are $11$ vertices and $10$ edges. A connected graph with $n$ vertices and $n-1$ edges is a tree and therefore is acyclic. Thus, there is no unicyclic graph with the given degree sequence.


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