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G H H26 Let r be the number of regions in a plane diagram of a disconnected planar graph G Suppos...
Do in Computing Mathematics or Discrete Mathematics 3. (8 pts) A graph is called planar if...
Do in Computing Mathematics or Discrete
Mathematics
3. (8 pts) A graph is called planar if it can be drawn in the plane without any edges crossing. The Euler's formula states that v - etr = 2, where v, e, and r are the numbers of vertices, edges, and regions in a planar graph, respectively. For the following problems, let G be a planar simple graph with 8 vertices. (a) Find the maximum number of edges in G. (b) Find...
5. Let G is a simple planar graph containing no triangles. (i) Using Euler's formula, show...
5. Let G is a simple planar graph containing no triangles. (i) Using Euler's formula, show that G contains a vertex of degree at most 3. (ii) Use induction to deduce that G is 4-colorable-(v).
5. Let G is a simple planar graph containing no triangles. (i) Using Euler's formula, show that G contains a vertex of degree at most 3. (ii) Use induction to deduce that G is 4-colorable-(v).
(a) Let G be a graph with order n and size m. Prove that if (n-1)...
(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...
1. A planar graph is said quadrag enary if all the faces of the graph are quadrilaterals (all fa...
1. A planar graph is said quadrag enary if all the faces of the graph are quadrilaterals (all faces are formed by 4 arcs). Let n be the number of vertices of the graph. Find a formula that gives the number of arcs of a graph quadrag enary according to n. Also find a formula that gives the number of faces of a quadrag enary graph according to n. You must clearly write your two formulas. You have to prove...
3. Let y: K + Aut(H) be a homomorphism. Write (k) = Ok. Let G be...
3. Let y: K + Aut(H) be a homomorphism. Write (k) = Ok. Let G be a group. A function d: K + H is called a derivation if dikk') = d(k) (d(k')). Show that d: K + H is a derivation if and only if V: K + H y K given by v(k) = (d(k), k) is a homomorphism. 4. Suppose that a: G + K is a surjective homomorphism and that 0: K + G is a...
Let G = (V. E) be an undirected, connected graph with weight function w : E → R
Problem 4 Let G = (V. E) be an undirected, connected graph with weight function w : E → R. Furthermore, suppose that E 2 |V and that all edge weights are distinct. Prove that the MST of G is unique (that is, that there is only one minimum spanning tree of G).
Let G = (V, E) be a finite graph. We will use a few definitions for...
Let G = (V, E) be a finite graph. We will use a few definitions for the statement of this problem. The Tutte polynomial is defined as the polynomial in 2 variables, 2 and y, given by: Definition 1 Tg(x,y) = (x - 1)*(A)-k(E)(y - 1)*(A)+|A1-1V1 ACE where for A CE, k(A) is the number of connected components of the graph (V, A). For this problem we will need the following definition: Definition 2 (Acyclic Graph) A graph is called...
(18) Let f and g be functions from R to R that have derivatives of al orders. Let h(k) denote the...
(18) Let f and g be functions from R to R that have derivatives of al orders. Let h(k) denote the kth derivative of any function. Prove using the product rule for derivatives, the fact that and induction that k +1 k=0 (19) The Fibonacci numbers are defined recursively by Fn+2 = Fn+1 Prove that the number of subsets of { 1, 2, 3, . . . , n} containing no two successive integers is E, (20) Prove that 7n...
1 Let R be a region bounded between two curves on the r, y-plane. Suppose that you are asked to find the volume of the...
1 Let R be a region bounded between two curves on the r, y-plane. Suppose that you are asked to find the volume of the solid obtained by revolving the region R about the r-axis If you slice the region R into thin horizontal slices, i.e., parallel to the r-axis, in setting up the Riemann sum, then which method will come into play? A. Disc method B. Washer method C. Either disc or a washer method depending on the shape...
a graph theory homework questions parts c,d,e,f 6. Let G be the fllowing graph: 1) Fig, 7.7.1 (n) Does G have a perfect matching? (b) Find four maximum matchings in G. (c) Is there any maximum mat...
a graph theory homework questions
parts c,d,e,f
6. Let G be the fllowing graph: 1) Fig, 7.7.1 (n) Does G have a perfect matching? (b) Find four maximum matchings in G. (c) Is there any maximum matching in G that contains the edge cl? (d) Find four maximal matchings (for definition, see Problem 7.6.20) that are not maximum. (e) Find in G (1) a maximum independent set, (ii) a minimum v-cover, and iii) n minimum c-cover. (f) Find the values...
Do in Computing Mathematics or Discrete
Mathematics
3. (8 pts) A graph is called planar if it can be drawn in the plane without any edges crossing. The Euler's formula states that v - etr = 2, where v, e, and r are the numbers of vertices, edges, and regions in a planar graph, respectively. For the following problems, let G be a planar simple graph with 8 vertices. (a) Find the maximum number of edges in G. (b) Find...
5. Let G is a simple planar graph containing no triangles. (i) Using Euler's formula, show that G contains a vertex of degree at most 3. (ii) Use induction to deduce that G is 4-colorable-(v).
5. Let G is a simple planar graph containing no triangles. (i) Using Euler's formula, show that G contains a vertex of degree at most 3. (ii) Use induction to deduce that G is 4-colorable-(v).
(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...
3. Let y: K + Aut(H) be a homomorphism. Write (k) = Ok. Let G be a group. A function d: K + H is called a derivation if dikk') = d(k) (d(k')). Show that d: K + H is a derivation if and only if V: K + H y K given by v(k) = (d(k), k) is a homomorphism. 4. Suppose that a: G + K is a surjective homomorphism and that 0: K + G is a...
Let G = (V, E) be a finite graph. We will use a few definitions for the statement of this problem. The Tutte polynomial is defined as the polynomial in 2 variables, 2 and y, given by: Definition 1 Tg(x,y) = (x - 1)*(A)-k(E)(y - 1)*(A)+|A1-1V1 ACE where for A CE, k(A) is the number of connected components of the graph (V, A). For this problem we will need the following definition: Definition 2 (Acyclic Graph) A graph is called...
(18) Let f and g be functions from R to R that have derivatives of al orders. Let h(k) denote the kth derivative of any function. Prove using the product rule for derivatives, the fact that and induction that k +1 k=0 (19) The Fibonacci numbers are defined recursively by Fn+2 = Fn+1 Prove that the number of subsets of { 1, 2, 3, . . . , n} containing no two successive integers is E, (20) Prove that 7n...
1 Let R be a region bounded between two curves on the r, y-plane. Suppose that you are asked to find the volume of the solid obtained by revolving the region R about the r-axis If you slice the region R into thin horizontal slices, i.e., parallel to the r-axis, in setting up the Riemann sum, then which method will come into play? A. Disc method B. Washer method C. Either disc or a washer method depending on the shape...
a graph theory homework questions
parts c,d,e,f
6. Let G be the fllowing graph: 1) Fig, 7.7.1 (n) Does G have a perfect matching? (b) Find four maximum matchings in G. (c) Is there any maximum matching in G that contains the edge cl? (d) Find four maximal matchings (for definition, see Problem 7.6.20) that are not maximum. (e) Find in G (1) a maximum independent set, (ii) a minimum v-cover, and iii) n minimum c-cover. (f) Find the values...
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