Drawing Graph on 5 vertices. Draw a graph on 5 vertices that contains no clique of size three (that is, no triangle) and no anti-clique of size three (that is, three vertices none of which is connected to any other).


Here are a few graphs which I made. The problem I'm having is I don't quite understand what the question is asking me for. Like user96614 suggested, a ring could work too. So am I suppose to just come up with some graph with 5 vertices where no 3 vertices are together and no 3 vertices are alone? 
 A: The problem is probably asking you to find a graph such that for an three points , those three points don't form the triangle $C_3$ or its complement, the edgeless graph on 3 vertices. So the cycle $C_5$ works.
You can think of cliques as a collection of people where everyone is friends with each other and of anti-cliques as a collection of people where no two people in this collection are friends. 
A: A ring (i.e. the cycle graph $C_5$) works and in fact, it is the only graph which will work. 
Suppose there is an isolated vertex. Then for there to be no anti-triangle, each of the remaining $4$ vertices must be joined to each other. But this necessarily induces a triangle. Therefore the graph we require is connected.
Now suppose the graph is cycle free. Then it is a tree. If the tree has three leaves then we are done since those three leaves form an anti-triangle. If the tree has two leaves then it is the path graph $P_5$. Taking the middle vertex and the two leaves also leaves us an anti-triangle.
Therefore the graph must have a cycle. This cycle cannot be $C_3$ since our graph must be triangle-free. Therefore there is either a $C_4$ or a $C_5$ in the graph. In the first case, let four of the vertices form a $4$-cycle. It is straightforward to check that there is no way to connect the remaining vertex to the cycle without introducing a triangle or an anti-triangle.
Finally, this leaves us with graphs which contain $C_5$. The cycle itself will work, but adding any diagonal to the cycle will introduce a triangle. Therefore $C_5$ is the only $5$ vertex graph without a triangle or an anti-triangle.
