  We present the first polynomial-time algorithm to exactly compute the number
of labeled chordal graphs on n vertices. Our algorithm solves a more general
problem: given n and omega as input, it computes the number of omega-colorable
labeled chordal graphs on n vertices, using O(n^7) arithmetic operations. A
standard sampling-to-counting reduction then yields a polynomial-time exact
sampler that generates an omega-colorable labeled chordal graph on n vertices
uniformly at random. Our counting algorithm improves upon the previous best
result by Wormald (1985), which computes the number of labeled chordal graphs
on n vertices in time exponential in n.
  An implementation of the polynomial-time counting algorithm gives the number
of labeled chordal graphs on up to 30 vertices in less than three minutes on a
standard desktop computer. Previously, the number of labeled chordal graphs was
only known for graphs on up to 15 vertices.
