Python: Graph Colouring A simple method to find a colouring of a graph, with vertices {0,1,…,?−1}, is the following greedy algorithm: For ? in 0, 1, 2, ..., n - 1: Find the smallest colour (positive integer) ? which is not a colour of any neighbour of ?. Assign ? as the colour of ?. i) Implement the method above as a function greedy_colouring which takes a networkx Graph G (with vertices {0,1,…,?−1}) and returns a list C of the colours of G, where C[i] is the colour of vertex ?. ii) As a test, the Hoffman-Singleton graph (nx.hoffman_singleton_graph()) should require 6 colours using this method. If you have a graph G, you can draw it with coloured vertices using the code below: ten_colours = ['#CC6677', '#332288', '#DDCC77', '#117733', '#88CCEE', '#882255', '#44AA99', '#999933', '#AA4499', '#DDDDDD'] nx.draw_networkx(G, node_color=[ten_colours[i - 1] for i in greedy_colouring(G)], with_labels=False) iii) One way of refining the greedy algorithm is to sort the vertices in descending order of degre

Python: Graph Colouring A simple method to find a colouring of a graph, with vertices {0,1,…,?−1}, is the following greedy algorithm: For ? in 0, 1, 2, ..., n - 1: Find the smallest colour (positive integer) ? which is not a colour of any neighbour of ?. Assign ? as the colour of ?. i) Implement the method above as a function greedy_colouring which takes a networkx Graph G (with vertices {0,1,…,?−1}) and returns a list C of the colours of G, where C[i] is the colour of vertex ?. ii) As a test, the Hoffman-Singleton graph (nx.hoffman_singleton_graph()) should require 6 colours using this method. If you have a graph G, you can draw it with coloured vertices using the code below: ten_colours = ['#CC6677', '#332288', '#DDCC77', '#117733', '#88CCEE', '#882255', '#44AA99', '#999933', '#AA4499', '#DDDDDD'] nx.draw_networkx(G, node_color=[ten_colours[i - 1] for i in greedy_colouring(G)], with_labels=False) iii) One way of refining the greedy algorithm is to sort the vertices in descending order of degree before colouring them. This is a heuristic which may reduce the number of colours required; the idea is that high-degree vertices will be the most difficult ones to colour if you leave them until the end. (a) Update your function greedy_colouring above so that it has an additional optional keyword argument use_heuristic with default value False, which determines whether it uses the heuristic (edit the function above, rather than copying it, arguments should be defined like so: def greedy_colouring(G, use_heuristic = False), extra argument must be optional; your function should work if you only provide G, do not have to sort vertices with the same degree in any particular order.) (b) Generate 1000 random Erdős–Rényi graphs with 100100 vertices and edge-probability 0.20.2. For each one, calculate the difference in the number of colours used with and without the heuristic. Make a good histogram of these differences, and comment briefly on it. (if it takes more than 30 seconds, improve the efficiency of the code.)