Clarification of definition of class $DisNP$
A disjoint NP pair consists of two NP languages $A,B$ which are disjoint. For example, $A$ could consist of all graphs which are $k(n)$-colorable (where $n$ is the number of vertices and $k$ is an arbitrary function), and $B$ could consist of all graphs having a clique of size $k(n)+1$.

Disjoint pairs of NP languages are used in proof complexity. For example, the disjoint pair discussed above (for $k(n) = n^c$) is used to prove exponential lower bounds on the size of proofs in the cutting planes proof system. See for example these notes. (For the state-of-the-art regarding cutting planes, see the recent paper Random CNFs are Hard for Cutting Planes.)