7n where x(G)is the cromatic number of G -2n Using extremal graph theory , prove thot for any G graph x(G 7n where x(G)is the cromatic number of G -2n Using extremal graph theory , prove thot fo...
6. Prove that for any graph G of order n an x(G) Sn + 1-a(G) α(G) 6. Prove that for any graph G of order n an x(G) Sn + 1-a(G) α(G)
(10) Prove ONLY ONE of the following statements using the principle of mathematical induction 7n n(n+3) (11) Give a recurrence definition of the following sequence: an 2n +1, n 1,2,3,..
Prove or Disprove: For any natural number n, 7 divides (gn – 2n).
8. Prove that max { x ( 2 -*): x is a real number} < 1 using the MAX/MIN method. 9 Sunoco thot Sand Taro qubooto ful
Let G be a graph and let µ(G) be the Mycielskian of the graph. Prove the following: ω(µ(G)) = ω(G), where ω(G) is the clique number of the graph, G.
Problem 5. Prove the following result for any number a and discrete random variable X. 티(X-a 21 = Var(X) + (E(X)-a)2 You must start your proof by using the definition of the expected value of a function of a discrete random variable, i.e. where g(x)- (x-a)
Question 18. Prove that for any graph G, β(G) ≤ 2α' (G)
Bounds on the number of edges in a graph. (a) Let G be an undirected graph with n vertices. Let Δ(G) be the maximum degree of any vertex in G, δ(G) be the minimum degree of any vertex in G, and m be the number of edges in G. Prove that δ(G)n2≤m≤Δ(G)n2
Question 8: For any integer n 20 and any real number x with 0<<1, define the function (Using the ratio test from calculus, it can be shown that this infinite series converges for any fixed integer n.) Determine a closed form expression for Fo(x). (You may use any result that was proven in class.) Let n 21 be an integer and let r be a real number with 0<< 1. Prove that 'n-1(2), n where 1 denotes the derivative of...
(a) Let G be a graph with order n and size m. Prove that if (n-1) (n-2) m 2 +2 2 then G is Hamiltonian. (b) Let G be a plane graph with n vertices, m edges and f faces. Using Euler's formula, prove that nmf k(G)+ 1 where k(G) is the mumber of connected components of G. (a) Let G be a graph with order n and size m. Prove that if (n-1) (n-2) m 2 +2 2 then...