  We compute the distance-dependent two-point function of vertex-bicolored
planar maps, i.e., maps whose vertices are colored in black and white so that
no adjacent vertices have the same color. By distance-dependent two-point
function, we mean the generating function of these maps with both a marked
oriented edge and a marked vertex which are at a prescribed distance from each
other. As customary, the maps are enumerated with arbitrary degree-dependent
face weights, but the novelty here is that we also introduce color-dependent
vertex weights. Explicit expressions are given for vertex-bicolored maps with
bounded face degrees in the form of ratios of determinants of fixed size. Our
approach is based on a slice decomposition of maps which relates the
distance-dependent two-point function to the coefficients of the continued
fraction expansions of some distance-independent map generating functions.
Special attention is paid to the case of vertex-bicolored quadrangulations and
hexangulations, whose two-point functions are also obtained in a more direct
way involving equivalences with hard dimer statistics. A few consequences of
our results, as well as some extension to vertex-tricolored maps, are also
discussed.
