  The primary objectives of this paper are to present the construction of
bivariate fractal interpolation functions over triangular interpolating domain
using the concept of vertex coloring and to propose a double integration
formula for the constructed interpolation functions. Unlike the conventional
constructions, each vertex in the partition of the triangular region has been
assigned a color such that the chromatic number of the partition is 3. A new
method for the partitioning of the triangle is proposed with a result
concerning the chromatic number of its graph. Following the construction, a
formula determining the vertical scaling factor is provided. With the newly
defined vertical scaling factor, it is clearly observed that the value of the
double integral coincides with the integral value calculated using fractal
theory. Further, a relation connecting the fractal interpolation function with
the equation of the plane passing through the vertices of the triangle is
established. Convergence of the proposed method to the actual integral value is
proven with sufficient lemmas and theorems. Sufficient examples are also
provided to illustrate the method of construction and to verify the formula of
double integration.
