Demonstrating b-coloring of generalized Jahangir graphs for representing complex manufacturing process

Abstract

A graph’s b-coloring admits proper coloring and has the extra characteristic of having a dominating node in each color-class in the graph. [Formula: see text] the b-chromatic number, is the largest integer k for which G can be colored with k colors using the b-coloring method. G is said to be b-continuous if b-coloring exists for [Formula: see text] meeting the inequality [Formula: see text] The b-spectrum [Formula: see text] of a graph G is the set of all integers k for which a b-coloring of G exists using k colors. b-Chromatic number, b-continuity and b-spectrum of generalized Jahangir graphs and that of line graph of generalized Jahangir graphs are determined in this work and the concept of b-coloring of the generalized Jahangir graph has also been extended to represent complex manufacturing processes to enhance visualization. Investment casting is a highly complex manufacturing process widely accepted for manufacturing high-valued metallic components. The concept of b-coloring has been employed to represent investment casting. This has created a great platform to combine the approach of graph theory with a complex manufacturing process, which can be explored to perform various tasks associated with scheduling and optimization in future work.

Keywords

• b-Coloring
• b-continuity
• b-spectrum
• generalized Jahangir graph
• line graph
• investment casting