  We explore the interplay between algebraic combinatorics and algorithmic
problems in graph theory by defining a polynomial with connections to
correspondence colouring (also known as DP-colouring), a recent generalization
of list-colouring, and the Unique Games Conjecture. Like the chromatic
polynomial of a graph, we are able to evaluate this polynomial at a point,
despite the complexity of computing this polynomial.
  We construct a cover of a graph $X$ by blowing up each vertex to a set of $r$
vertices and joining each pair of sets corresponding to adjacent vertices by a
matching with $r$ edges. To each cover $Y$ of $X$ we associate a polynomial
$\xi(Y,t)$, called the transversal polynomial. The coefficient $t^k$ of
$\xi(Y,t)$ is the number of $k$-edge induced subgraphs of $Y$ whose vertex set
is a transversal of the set system given by the blown-up vertices. We show that
$\xi(Y,t)$ satisfies a contraction-deletion formula, and that if $n=|V_X|$ and
the cover has index $r$, then $\xi(Y,-(r-1)) \equiv 0 \mod r^n$.
