Graph Colouring is a chief element in graph theory with tremendous applicability in computer science like data mining, clustering, networking, image segmentation etc. And a variety of implementations in aircraft scheduling, register allocation, sudoku, mobile networking, etc. Various algorithms were contrived for vertex colouring. This paper defines the HB colour matrix method and its kinds. There are three types of such matrices, Vertex HB colour matrix (VHBCM), Edge HB Colour matrix (EHBCM), and Region HB colour matrix (RHBCM). Also, the HB colour matrix algorithm is developed using a special assignment method, which gives the chromatic number of the given graph. Further, the algorithm is used to develop the python program, giving time complexity O(n) and space complexity O(n2). Also, the output of the python program for some standard graphs is calculated. The Similar algorithm can be developed for edge and face colouring of the graphs. Colouring of the graph further can be extended to perfect colouring.
The Perfect colouring (jp) of a graph is an assignment of colours to all elements (vertices, edges, and regions) of the graph such that no two adjoint elements receive the same colour. In this paper, we determined the tight bounds of perfect colouring as/“(G) </P(G) </“(G) + 4, where/“(G) is total colouring of the graph G. Depending on these bounds, the perfect colouring is divided into five different kinds, and the results of these for some standard graphs are presented in the paper.
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