(46) Prove that if G is a simple graph on n vertices, and for any two vertices X and Y of G, it is true that dz +d 2 n, then G has a Hamiltonian it is true that d + d 2 n, then G has a
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(47) Prove that the statement of the previous exercise is not true assume that-de+42 n-1. (46) ...
Write down true (T) or false (F) for each statement. Statements are shown below If a...
Write down true (T) or false (F) for each statement. Statements are shown below If a graph with n vertices is connected, then it must have at least n − 1 edges. If a graph with n vertices has at least n − 1 edges, then it must be connected. If a simple undirected graph with n vertices has at least n edges, then it must contain a cycle. If a graph with n vertices contain a cycle, then it...
Graph 2 Prove the following statements using one example for each (consider n > 5). (a)...
Graph 2 Prove the following statements using one example for each (consider n > 5). (a) A graph G is bipartite if and only if it has no odd cycles. (b) The number of edges in a bipartite graph with n vertices is at most (n2 /2). (c) Given any two vertices u and v of a graph G, every u–v walk contains a u–v path. (d) A simple graph with n vertices and k components can have at most...
SE 9.1.CV-5 DE Question Help Fill in the blank so that the resulting statement is true...
SE 9.1.CV-5 DE Question Help Fill in the blank so that the resulting statement is true (x + 3) The graph of 64 (y - 4) + 40 , = 1 has its center at 49 Fill in the blanks so that the resulting statement is true. Consider the standard form of the equation ² y² + The graph passes through the ordered pairs The value of as is 5, so the y-intercepts are which are the vertices
solve with steps 1. (20 points) True or false. Justify. Every planar graph is 4-colorable /2 The number of edges in a simple graph G is bounded by n(n 1) where n is the number of vertices. The number...
solve with steps
1. (20 points) True or false. Justify. Every planar graph is 4-colorable /2 The number of edges in a simple graph G is bounded by n(n 1) where n is the number of vertices. The number of edges of a simple connected graph G is at least n-1 where n is the number of vertices. Two graphs are isomorphic if they have the same number of vertices and 1) the same mumber of edges
1. (20 points)...
Problem 1. Consider a graph G with n vertices: • Ifq is the size of the...
Problem 1. Consider a graph G with n vertices: • Ifq is the size of the largest independent set in graph G, show that ax(G) n. • Use the previous result to prove that x(G)(n - d) > n and thuse x(G) > n/(n - d), where d is the minimal degree of the vertex in G. • Prove that x(G) + X(G) Sn +1 (Hint: Induction). • x(G)x(G) 2 n. • Use previous inequality to show that x(G) +...
Find the logical mistakes in these proofs, and explain why the mistakes you've identified cause problems in their a...
Find the logical mistakes in these proofs, and explain why the mistakes you've identified cause problems in their arguments. (b)Claim: Suppose that G is a graph on n 3 vertices in which the degree of every vertex is exactly 2. Then G is a cycle Proof. We proceed by induction on n, the number of vertices in G. Our base case is simple: for n - 3, the only graph with 3 vertices in which all vertices have degree 2...
Find the logical mistakes in these proofs, and explain why the mistakes you've identified cause problems in their a...
Find the logical mistakes in these proofs, and explain why the mistakes you've identified cause problems in their arguments. (b)Claim: Suppose that G is a graph on n 3 vertices in which the degree of every vertex is exactly 2. Then G is a cycle Proof. We proceed by induction on n, the number of vertices in G. Our base case is simple: for n - 3, the only graph with 3 vertices in which all vertices have degree 2...
Question 1# (a) Let G be a connected graph and C a non-trivial circuit in G. Prove directly that if an edge e fa, b is...
Question 1# (a) Let G be a connected graph and C a non-trivial circuit in G. Prove directly that if an edge e fa, b is removed from C then the subgraph S C G that remains is still connected. "Directly' means using only the definitions of the concepts involved, in this case connected' and 'circuit'. Hint: If z and y are vertices of G connected by path that includes e, is there an alternative path connecting x to y...
Prove the Binomial Theorem, that is Exercises 173 (vi) x+y y for all n e N...
Prove the Binomial Theorem, that is Exercises 173 (vi) x+y y for all n e N C) Recall that for all 0rS L is divisible by 8 when n is an odd natural number vii))Show that 2 (vin) Prove Leibniz's Theorem for repeated differentiation of a product: If ande are functions of x, then prove that d (uv) d + +Mat0 for all n e N, where u, and d'a d/v and dy da respectively denote (You will need to...
Exercise 1.6.4: Prove the following by induction: (a) “k - n(n+1)(2n +1) k= 1 (b) If...
Exercise 1.6.4: Prove the following by induction: (a) “k - n(n+1)(2n +1) k= 1 (b) If n > 1, then 13-n is divisible by 3. (c) For n 3, we have n +4 <2". (d) For any positive integer n, one of n, n+2, and 11+ 4 must be divisible by 3. (e) For all n e N, we have 3" > 2n +1. ()/Prove that, for any x > -1 and any n e N, we have (1+x)" 21+1x.
SE 9.1.CV-5 DE Question Help Fill in the blank so that the resulting statement is true (x + 3) The graph of 64 (y - 4) + 40 , = 1 has its center at 49 Fill in the blanks so that the resulting statement is true. Consider the standard form of the equation ² y² + The graph passes through the ordered pairs The value of as is 5, so the y-intercepts are which are the vertices
solve with steps
1. (20 points) True or false. Justify. Every planar graph is 4-colorable /2 The number of edges in a simple graph G is bounded by n(n 1) where n is the number of vertices. The number of edges of a simple connected graph G is at least n-1 where n is the number of vertices. Two graphs are isomorphic if they have the same number of vertices and 1) the same mumber of edges
1. (20 points)...
Problem 1. Consider a graph G with n vertices: • Ifq is the size of the largest independent set in graph G, show that ax(G) n. • Use the previous result to prove that x(G)(n - d) > n and thuse x(G) > n/(n - d), where d is the minimal degree of the vertex in G. • Prove that x(G) + X(G) Sn +1 (Hint: Induction). • x(G)x(G) 2 n. • Use previous inequality to show that x(G) +...
Find the logical mistakes in these proofs, and explain why the mistakes you've identified cause problems in their arguments. (b)Claim: Suppose that G is a graph on n 3 vertices in which the degree of every vertex is exactly 2. Then G is a cycle Proof. We proceed by induction on n, the number of vertices in G. Our base case is simple: for n - 3, the only graph with 3 vertices in which all vertices have degree 2...
Find the logical mistakes in these proofs, and explain why the mistakes you've identified cause problems in their arguments. (b)Claim: Suppose that G is a graph on n 3 vertices in which the degree of every vertex is exactly 2. Then G is a cycle Proof. We proceed by induction on n, the number of vertices in G. Our base case is simple: for n - 3, the only graph with 3 vertices in which all vertices have degree 2...
Question 1# (a) Let G be a connected graph and C a non-trivial circuit in G. Prove directly that if an edge e fa, b is removed from C then the subgraph S C G that remains is still connected. "Directly' means using only the definitions of the concepts involved, in this case connected' and 'circuit'. Hint: If z and y are vertices of G connected by path that includes e, is there an alternative path connecting x to y...
Prove the Binomial Theorem, that is Exercises 173 (vi) x+y y for all n e N C) Recall that for all 0rS L is divisible by 8 when n is an odd natural number vii))Show that 2 (vin) Prove Leibniz's Theorem for repeated differentiation of a product: If ande are functions of x, then prove that d (uv) d + +Mat0 for all n e N, where u, and d'a d/v and dy da respectively denote (You will need to...
Exercise 1.6.4: Prove the following by induction: (a) “k - n(n+1)(2n +1) k= 1 (b) If n > 1, then 13-n is divisible by 3. (c) For n 3, we have n +4 <2". (d) For any positive integer n, one of n, n+2, and 11+ 4 must be divisible by 3. (e) For all n e N, we have 3" > 2n +1. ()/Prove that, for any x > -1 and any n e N, we have (1+x)" 21+1x.
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