[FEATURE] Addition of Graph Coloring Problem in Graphs directory

[Charuhas10]

Detailed description

Proposal

Addition of Graph Coloring Problem algorithm under in Graphs directory

Overview

The graph coloring problem asks to assign colors to the vertices of a graph in such a way that no two adjacent vertices share the same color. The objective is often to color the graph with as few colors as possible.

Issue details

The greedy coloring algorithm is a straightforward approach to solve this problem. Here's a basic outline of the greedy algorithm:

1. Ordering the Vertices: Although not strictly necessary, the algorithm can start by ordering the vertices in some fashion. Different orderings may produce different results. A common ordering is by the degree of the vertices, but the simplest is just the order in which the vertices are given.
2. Coloring: Start with the first vertex and assign it the first color. Then move to the next vertex.
   For each subsequent vertex, look at its neighbors and determine what colors have already been assigned. Assign the smallest possible color that hasn't been used by its neighbors.

More Details

https://en.wikipedia.org/wiki/Graph_coloring

Context

This change is crucial as it addresses a common problem in graph theory solved using greedy approach, enhancing the repository's comprehensiveness

Possible implementation

Pseudo Code:

Algorithm GreedyGraphColoring(G):
    Input: A graph G with V vertices
    Output: A color assignment for each vertex

    Initialize an array color[] of size V and set all to -1 (indicating uncolored)
    Initialize an array available[] of size V and set all to False (indicating all colors are initially available)
    
    Assign color[0] = 0  // Assign the first color to the first vertex
    
    FOR vertex u from 1 to V-1 DO
        FOR each vertex i from 0 to V-1 DO
            IF there's an edge between u and i AND color[i] is not -1 THEN
                SET available[color[i]] = True
            END IF
        END FOR

        // Find the first available color
        clr = 0
        WHILE clr < V AND available[clr] is True DO
            INCREMENT clr
        END WHILE

        Assign color[u] = clr

        Reset available[] to False for the next iteration
    END FOR

    RETURN color

END Algorithm

Additional information

No response

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