Show that the following graphs are isomorphic

Question

I have the following problem to resolve:

If $X=\{1,2,3,4,5\}$ and $V$ is the set of all the subsets of 2 elements of $X$. If $A$ is the set of pairs of elements of $V$ that are disjoint (as subsets of $X$). Show that the graph $G=(V,A)$ is isomorphic to the graph shown below:

From this, we know that:

$$X=\{1,2,3,4,5\}$$

And from what I understand:

$$ V=\{\{1,2\},\{1,3\},\{1,4\},\{1,5\},\{2,1\}\{...\}\} $$

But this seems incorrect as since $X$ appears to be representing the nodes of the graph, that set of subsets above would be $A$ and here is where I start getting confused.

Any suggestions on better understanding the concept of the problem and at the same time the steps in resolving it?

Answer 1

This is the Petersen graph. The set of vertices are the two element subsets of $\{1,2,3,4,5\}$ and two vertices are adjacent iff they are disjoint as sets. For example there is an edge between $\{1,2\}$ and $\{3,4\}$ (since $\{1,2\}\cap \{3,4\}=\varnothing$ but not an edge between $\{1,2\}$ and $\{2,3\}$ (since $\{1,2\}\cap \{2,3\}\neq\varnothing$). This graph has $\binom{5}{2}=10$ vertices.

Answer 2

You just need to label the vertices and observe that the edges are "disjoint pairs". It is of course the Petersen graph.