I need to construct a graph that has the following properties: the vertices of G are the edges of a complete graph $K_5$ on 5 vertices. The vertices of G are adjacent if and only if the corresponding edges of $K_5$ have an endpoint in common. I then need to determine the chromatic number of this graph.
Can someone help me with at least the construction of the graph?
I'm not sure what you're looking for, where the construction is concerned; it is fairly explicit. But, let me try to rephrase a little bit, and perhaps that will help.
We start off with $K_5$; to make things precise, let's say that $K_5$ has vertex set $\{1,2,3,4,5\}$, and the edges of $K_5$ are all pairs $\{i,j\}$ where $i,j\in \{1,2,3,4,5\}$ and $i\neq j$.
So, the VERTEX set of your new graph should be $$ V=\{\{1,2\},\{1,3\},\{1,4\},\{1,5\},\{2,3\},\{2,4\},\{2,5\},\{3,4\},\{3,5\},\{4,5\}\}; $$ the edge set consists of all pairs $e,f\in V$ such that $e\neq f$ and $e\cap f\neq\varnothing$. So, for instance, there is an edge connecting $\{1,2\}$ and $\{2,3\}$, but not connecting $\{1,2\}$ and $\{3,4\}$.
Does that help?