  The identification of phase transition points, beta_c, with the percolation
thresholds of suitably defined clusters of spins has proved immensely fruitful
in many areas of statistical mechanics. Some time ago Kertesz suggested that
such percolation thresholds for models defined in field might also have
measurable physical consequences for regions of the phase diagram below beta_c,
giving rise to a ``Kertesz line'' running between beta_c and the bond
percolation threshold, beta_p, in the M, beta plane.
  Although no thermodynamic singularities were associated with this line it
could still be divined by looking for a change in the behaviour of high-field
series for quantities such as the free energy or magnetisation. Adler and
Stauffer did precisely this with some pre-existing series for the regular
square lattice and simple cubic lattice Ising models and did, indeed, find
evidence for such a change in high-field series around beta_p. Since there is a
general dearth of high-field series there has been no other work along these
lines.
  In this paper we use the solution of the Ising model in field on planar
random graphs by Boulatov and Kazakov to carry out a similar exercise for the
Ising model on random graphs (i.e. coupled to 2D quantum gravity). We generate
a high-field series for the Ising model on $\Phi^4$ random graphs and examine
its behaviour for evidence of a Kertesz line.
