Consider the following labelled graph: (a) State an upper-bound and a lower-bound on the vertex-chromatic number of the graph, and give reasons for your (b) Find an optimal colouring of the graph. (c) Is the graph an interval graph? Provide a reason for your answer. answers (d) Beginning with the matching M - [(a, b)), extend M into a maximum matching by repeatedly finding augmenting paths. At each step you should increase the number of edges in the matching by one. Recall, an augmenting path is an alternating path that joins two "exposed" vertices (e) Find a minimum node-cover for the given graph. How does the size of the node-cover correspond to the matching you found in the previous question? Why? (f) State a lower-bound on the edge-chromatic number of the graph, and give a reason for your answer. How do we know that this lower-bound is actually equal to the edge-chromatic number in this example? (g) Find an edge-colouring of cardinality four in the graph by repeatedly finding and removing matchings involving all vertices of maximum degree. All steps must be explicitly shown