The number of degree restricted maps on general surfaces

Abstract

Let D be a finite set of positive integers with maximum bigger than two and m̂ g , n ( m̃ g , n ) be the number on n-edged rooted maps on the orientable (nonorientable) surface of type g whose face degrees (or, dually, vertex degrees) all lie in D. Define m ̂ g(x)= ∑ n⩾0 m ̂ g, nx n , m ̃ g(x)= ∑ n⩾0 m ̃ g, nx n . We show that m̂ g ( x) and m̃ g ( x) are algebraic functions of a certain form. Asymptotic expressions for m̂ g , n and m̃ g , n are also derived for some special sets D.