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

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. φ(G), 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 ∀k, meeting the inequality χ(G)≤k≤φ(G). The b-spectrum Sb(G) 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.

图的b着色（b-coloring）需满足正常着色（proper coloring）的基本要求，且额外具备一项核心特性：图的每一个颜色类（color-class）中均存在至少一个支配点（dominating node）。φ(G)即b色数（b-chromatic number），是可通过b着色方法为图G分配k种颜色的最大整数k。若对于所有满足χ(G)≤k≤φ(G)的整数k，均存在对应的b着色方案，则称图G为b连续的（b-continuous）。图G的b谱（b-spectrum）Sb(G)，是所有可使图G实现k色b着色的整数k构成的集合。
本文完成了广义雅兴格尔图（generalized Jahangir graphs）及其线图（line graph）的b色数、b连续性与b谱的求解工作；同时将广义雅兴格尔图的b着色概念拓展应用于复杂制造流程的可视化表征，以提升流程直观性。熔模铸造（investment casting）是一类被广泛应用于高价值金属构件制备的复杂制造工艺，本文借助b着色概念对熔模铸造流程进行建模表征，由此搭建起将图论方法与复杂制造流程相结合的研究平台，未来可依托该平台开展调度与优化等相关领域的多项研究工作。