How to argue that my list of graphs with four edges (without loops or vertices of degree $0$) is complete? I am trying to draw all graphs with four edges without loops or vertices of degree $0$. I have drawn many graphs with four edges. Can anyone give me idea how to argue that this list is complete?
 A: Here's an idea. Graphs can be encoded using numbers, and using that encoding you can simply enumerate all of the finitely many possibilities.
Denote your four edges as $E_1$, $E_2$, $E_3$, $E_4$. Each endpoint is a vertex. Also, the two endpoints of $E_i$ are different vertices (because there are no loops); let me denote those two endpoints as $\partial_- E_i$, $\partial_+ E_i$. Also, each vertex is an endpoint (because there are no vertices of degree $0$), and so there are at most 8 vertices.
Denote the vertices as $V_1, V_2, V_3, V_4, V_5, V_6, V_7, V_8$ (there might be fewer than $8$ vertices, in which case some of those notations will be unused).
Now make a table. The table has four rows labelled $1,2,3,4$, one row per edge. The table has two columns $-$, $+$. For each $i=1,2,3,4$, we have $\partial_- E_i = V_j$ and $\partial_+ E_i = V_k$ for some $j \ne k$; in row $i$ of the table we enter $j$ in column $-$ and $k$ in column $+$.
By enumerating all possible tables, you obtain an enumeration of all possible graphs. This is not the most efficient method of enumeration, as I'm sure you can see, because there will be lots of duplicates. So there is room for improvement. Nonetheless perhaps you can also see that it does give you a way to check whether your list is complete.
