How many simple Eulerian graphs (up to isomorphism) on 5 vertices are there? I have considered Euler's Theorem which states that for the simple connected graph: G is Eulerian if and only if the degree of each vertex is even.
Therefore I came up with (4,4,4,4,4) and (2,2,2,2,2). However, how would I check if there are more?
Note: I need to use degree sequences.
 A: There are 34 simple graphs with 5 vertices, 21 of which are connected (see link). There are four connected graphs on 5 vertices whose vertices all have even degree.


*

*The $5$-cycle with degree sequence $(2,2,2,2,2)$: 

*The complete graph on $5$ vertices with degree sequence $(4,4,4,4,4)$: 

*The butterfly/hourglass graph with degree sequence $(2,2,2,2,4)$: 

*The following graph with degree sequence $(2,2,2,4,4)$: 
A word of warning: In general, it's not good enough to just specify the degree sequence as non-isomorphic graphs can have the same degree sequences. 
Edit. If you want to start with a given degree sequence and determine whether there is a simple graph with that degree sequence, then the Erdos-Gallai theorem is of interest. It states the following.
A sequence of non-negative integers $d_1\geq \cdots \geq d_n$ can be represented as the degree sequence of a simple graph if and only if $d_1+d_2+\cdots + d_n$ is even, and
$$\sum_{i=1}^k d_i \leq k(k-1) + \sum_{i=k+1}^n \min(d_i,k)$$
holds for every $k$ with $1\leq k \leq n$.
