Realizing the Chromatic Numbers and Orders of Spinal Quadrangulations of Surfaces

A method is suggested for the construction of quadrangulations of the closed orientable surface with given genus \( g \) and either (1) with a given chromatic number or (2) with a given order allowed by the genus \( g \). In particular, N. Hartshfield and G. Ringel’s results [J. Comb. Theory, Ser. B 46 (1989), 84-95] are generalized by way of generating minimal quadrangulations of infinitely many other genera.