If a graph with six vertices is embedded in S², what is the maximum number of edges it can have. Prove your answer and illustrate the maximal case.

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Q: Is hypercube graph Q4 planar? If not can you provide any bounds on it crossing number?

A: No, hypercube graph Q4 is not a planner graph.

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Q: 3. а) Define the following terms as used in graph theory: i) regular graph ii) simple circuit

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Q: Descrete math. Graph theory

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Q: Show that the maximum number of edges in a graph with n-vertices is *C2 (d)

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Q: Suppose G is a simple graph (no loops; no parallel edges) with n = 14 vertices and m = 7 edges. What…

A: GIVEN  vertices n=14 edges m=7 EXPLANATION as we know  every graph has atleast n-m component…

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Q: Show that every graph with n vertices and k edges, n> k has n – k components.

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Q: (i) Use Euler's formula to prove that, if G is a connected planar graph of girth 5 with n vertices…

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Q: How many different labelled graphs are there on the vertex set [n]? Justify your claim.

A: Given Vertex set n Let vertex V=1, 2, 3, ........., n Each graph G with a vertex V is determined by…

Q: How many connected components does the following graphs have?

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Q: Use the Handshaking Lemma to show that a graph G always has an even number of vertices with odd…

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Q: What would be the most appropriate qualification of the following graph? It is a directed…

A: 1. Given graph is We observe that there are multiple arcs with same source and destination in the…

Q: What is the maximum number of edges you can have in a graph with 7 vertices? Show all steps of your…

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Q: Suppose a planar graph G is not connected and has exactly 2 components. What is the value of n – m+…

A: To obtain the result, use Euler's formula for planar graphs. Let G be a connected planar graph with…

Q: What is the most edges a graph can have if it has p vertices and has a bridge?

A: Given: A simple graph with p-vertices.

Q: Consider the following directed graph. a. Give the indegree of each vertex. b. Give the outdegree of…

A: The vertices are {a , b , c , d , e} (a) The indegree of the vertex is the number of edges coming to…

Q: For many applications of matchings, it makes sense to use bipartite graphs. You might wonder,…

A: As per our guidelines we can solve only first three parts. If you want other subpart to be answered…

Q: Draw an undirected graph that has exactly 11 edges and at least 5 vertices, in which two of the…

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Q: If G1 is a simple graph with 20 edges and G has 25 edges, how many vertices does G1 have?

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Q: Show that if G is a simple graph with n vertices (where n is a positive integer) and each vertex has…

A: Hello. Since your question has multiple parts, we will solve first question for you. If you want…

Q: The complete tripartite graph K,, consists of three sets of vertices (of sizes r, s and t), with an…

A: Given that The complete tripartite graph Kr,s,t The graph of K2,2,2 in the following graph Hence,…

Q: Use the pigeonhole principle to prove that a graph has at least two vertices with the same degree

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Q: 3. (MS) Select all that are true. a. Every graph has a vertex cover of size 0. b. Every graph has a…

A: Vertex cover V' of an undirected graph G=V,E is defined as a subset of V such that  if uv∈E implies,…

Q: Find, with a proof, the number of distinct graphs with vertex set (n).

A: In the given question we have find the number of the distict graph with n verex.it's answer is "The…

Q: 2) What is the maximum number of edges possible for any bipartite graph between A and B?

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Q: Out(4) G3. List the set of vertices for the graph above:

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Q: 1. Say two methods for testing the non planarity of graphs.

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Q: Do the following: 1. Find a graph with 12 edges having six vertices of degree three and the…

A: We need to find a graph with 12 edges, such that 6 vertices have degree three and remaining all the…

Q: If G is a graph, then the sum of its vertex degrees is nonnegative and even.

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Q: 2. Suppose G is a planar graph having 67 edges. If each face of G has length at least 6, then what…

A: From the given information. The number of edges (e) = 67 The number of vertices at each face (v) = 6

Q: If a graph has 5 vertices then 'By the Handshaking Theorem', each vertex can have degree 9. True…

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Q: If G is a simple graph with at least two vertices. prove that G must contain two or more vertices of…

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Q: Recall that the handshaking lemma says that the total degree of a loopless graph G is twice the…

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Q: For both graphs drawn below, verify Euler's formula: (# of nodes) - (# of edges) + (# of loops) = 1.

A: Use the given information.

Q: If G has at least one vertex of degree greater than zero then there are two of even degree as long…

A: This is the problem of graph theory.

Q: The maximum number of edges in a bipartite graph on 15 vertices is ____.

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Q: If a graph with six vertices is embedded in S?, what is the maximum number of edges it can have.…

A: Consider the provided question, A graph G is embedable on S2 if it is embedable on a plane. Since,…

Q: It is obvious that for a bipartite graph to contain a perfect matching, it is necessary that |A| =…

A: Given That: It is obvious that for a bipartite graph to contain a perfect matching, and also it is…

Q: 1. (a) What is the maximum number of edges that a bipartite planar graph with 10 vertices can have?…

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Q: 4. Is the graph below bipartite? If so, list the vertices in the one part and the vertices in the…

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Q: If G is a (not necessarily simple) graph with n vertices where each vertex has degree greater than…

A: Definitions: Simple graph: A simple graph is an undirected graph with no parallel edges and…

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Q: 2) Find, with a proof, the number of distinct graphs with vertex set [n].

A: 2) The number of distinct graphs of vertex set for given n should be found. Let us take n=3.…

Q: Suppose G is a simple undirected graph with n vertices. If G is self-complementary, prove that…

A: ↦A Simple undirected graph G is meant by a graph which have no multiple loop. there exist at most…

Q: Let G be a simple undirected graph with 16 edges. If all vertices have degree at most four, which of…

A: Let G be a simple graph with n vertices and e edges. Use the following formula, to determine the…