Let denote the set of all normalized analytic univalent functions in the open unit disc D. Let f(z), F(z) and φ(z) be analytic in |z| < r. We say that f(z) is majorized by F(z) in we say that f(z) is subordinate to F(z) in where .

Let be the set of all locally univalent (f’(z) ≠ 0) analytic functions in D with order ≦α which are of the form f(z) = z +… . The family is known as the universal linear invariant family of order α [6]. A concise summary of and introduction to properties of linear invariant families which relate to the following material is contained in [1]. The present paper contains the proofs of some of the results announced in [1]

In this paper we derive a functional equation whose solution would give the sum of the chromatic polynomial P(M, λ) over certain classes of rooted planar maps M called “ triangulations” and “near-triangulations”. For an integral colour-number λ this sum is the number of λ-coloured rooted maps of the kind considered, but the sum can also be discussed for non-integral λ.