  A planar hypermap with a boundary is defined as a planar map with a boundary,
endowed with a proper bicoloring of the inner faces. The boundary is said
alternating if the colors of the incident inner faces alternate along its
contour. In this paper we consider the problem of counting planar hypermaps
with an alternating boundary, according to the perimeter and to the degree
distribution of innerfaces of each color. The problem is translated into a
functional equation with a catalytic variable determining the corresponding
generating function. In the case of constellations - hypermaps whose all inner
faces of a given color have degree $m\geq 2$, and whose all other inner faces
have a degree multiple of $m$ - we completely solve the functional equation,
and show that the generating function is algebraic and admits an explicit
rational parametrization. We finally specialize to the case of Eulerian
triangulations - hypermaps whose all inner faces have degree $3$ - and compute
asymptotics which are needed in another work by the second author, to prove the
convergence of rescaled planar Eulerian triangulations to the Brownian map.
