Use the Reduction Algorithm and find the chromatic polynomial P_k(G) of the graph shown below. In how many ways can we color this graph using four different colors?
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In this case, both vertices A and B have degree 3. Show more…
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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).
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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.
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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.)
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