PLEASE HELP ME WITH NUMBER 50 Find the (a) crossing number, and (b) the thicknes

Question

PLEASE HELP ME WITH NUMBER 50

Find the (a) crossing number, and (b) the thickness for each of the following graphs. (i) K_5 (ii) K_6 (iii) K_7 (iv) K_3, 4 (v) K_4, 4 (vi) K_4 Let A, B, C, D, G, and H be the graphs (a) Calculate the chromatic numbers for A, B, C, D, G, and H. For each, give an example of a vertex coloring of the corresponding graph using exactly x colors. (b) Which of A, B, C, D, G, and H have the property that removing a single vertex will reduce the chromatic number? (c) Classify all graphs with chromatic number (i) 1, and (ii) 2. (d) What are the chromatic numbers of (i) K_n, (ii) K_m, n, (iii) C_n (iv) W_n, (v) Q_n, (vi) The Petersen graph.

Explanation / Answer

50 a)

A = 3

B = 4

C = 4

D = 5

G = 3

H = 6

b)

A and C has the property where removing 1 edge will reduce the chromatic number

c) chromatic number 1 graphs are single nodes without any edges.

chromatic number 2 graphs are trees.

d) Kn = n

Km,n = n

Cn = 3

Wn = n+1

Qn = 2

Petersen's graph = 4

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