Question 4. (15 marks) Using the Sequential Coloring Algorithm, determine the number of colors needed to color the following labeled graph. Process the vertices in the order labeled. You need to show sufficient working to demonstrate the use of the algorithm. Comment on the quality of your solution in (a). Calculate the chromatic polynomial of the graph.
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Being provided with the graph below: (i) Determine the chromatic polynomial alongside the chromatic number of the given graph. (ii) Provide a minimal proper coloring of that graph Additional Guideline: This question can be tackled using the known properties of chromatic polynomials. Can you regularly color it with 0, 1, 2 colors? Probably not, so it is p(G, ̀) = ̀(̀ - 1)(̀ - 2)(̀" + à + b) (5 nodes meaning degree 5, leading coefficient 1). Moreover, it must start like ̀⁵ - 8̀⁴ + ⋯ (8 edges), what can you conclude concerning the value of a? Have you tried to color it with 3 colors? Then, you must have found there are 6 ways to do that, and p(G,3)=6 gives you the value of b.
Adi S.
Find the chromatic number of the graph G shown below. Give a coloring of G with χ(G) colors and prove that no proper coloring of G can use fewer colors. (The second copy illustrates how you can add colors to the diagram.)
Sri K.
The chromatic polynomial of graph G is the function PG(k) that gives the number of ways to color the vertices of G using k colors, such that no two adjacent vertices have the same color. (a) If L is a path (linear graph) with vertices, show that Pk(u) = k(k - 1)^(n-1). (b) Let Cn be the cyclic graph with n vertices. Find Pcs(k). (c) Show that for n > 3, Pcn(k) satisfies the recurrence: Pcn(k) = k(k - 1)^(n-1) Pcn-1(k).
Shaiju T.
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