Question

Problem 9: Does the greedy coloring algorithm always use (G) + 1 colors on a graph G? If yes, give a proof of this fact. If no, give an example graph G (say with 4 vertices) where this does not happen [Recall that you need to give an ordering on the vertices as well for which the desired fact happens]. Problem 10: Consider the simple graph G = P4 shown below: 1 2 3 Answer the following questions for this graph. 1. What is the chromatic number x(G)? 2. Find an ordering of the vertices of this graph such that when the greedy coloring algorithm is applied, it does not produce an optimal coloring, i.e., it used more than x(G) colors. 3. Find an ordering of the vertices where the greedy algorithm does produce an optimal coloring, i.e., a coloring with x(G) colors.

          Problem 9: Does the greedy coloring algorithm always use (G) + 1 colors on a graph G? If yes, give a proof of this fact. If no, give an example graph G (say with 4 vertices) where this does not happen [Recall that you need to give an ordering on the vertices as well for which the desired fact happens].
Problem 10: Consider the simple graph G = P4 shown below:
1
2
3
Answer the following questions for this graph.
1. What is the chromatic number x(G)?
2. Find an ordering of the vertices of this graph such that when the greedy coloring algorithm is applied, it does not produce an optimal coloring, i.e., it used more than x(G) colors.
3. Find an ordering of the vertices where the greedy algorithm does produce an optimal coloring, i.e., a coloring with x(G) colors.

problem 9 does the greedy coloring algorithm always use g 1 colors on a graph g if yes give a proof of this fact if no give an example graph g say with 4 vertices where this does not happen 40806

Added by Adam H.

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