Suppose a graph G has 22 vertices. What is the minimum number of EDGES that G must contain if Diracs Theorem can be used to guarante a Hamitonian Grat in G?

Q: Let G be a simple graph with 15 vertices and 4 connected components. Prove that G has at least one…

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Q: Let G be a simple plane graph with fewer than 12 faces, in which each vertex has degree at least 3.…

A: Step 1: Consider the provided question, (17) (i)

Q: prove that the maximum girth of a generalized Coxeter graph is 12, no matter what its parameters are

A: Among the generalized Petersen graphs, having n prism G p,1 on the Durer graph G 6.2 the…

Q: 11. Prove: If G is a simple connected graph where the average degree of the vertices is exactly 2,…

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Q: 4. Let P be a plane graph. Show that if each inner face of P has a boundary consisting of an even…

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Q: Prove that every maximal matching in a graph G has at least a'(G)/2 edges.

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Q: 2. Show that any graph with at most 8 vertices is planar.

A: Topic:- planer graph

Q: When two graphs G and H are isomorphic, this means: there exists a bijection between the vertices of…

A: Two graphs G and H are said to be isomorphic, if there exist a bijection f between their vertices…

Q: Give an upper bound on the number e of edges of G in terms of n and g if G is a connected plane…

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Q: (d) Show that the pair of graphs are not isomorphic by showing that there is a property that is…

A: (d)  To prove that the pair of graphs are not isomorphic by showing that there is        a property…

Q: Prove that connecting two nodes u and v in a graph G by a new edge creates a new cycle if and only…

A: Given, Connecting two nodes u and v in a graph G by a new edge creates a new cycle such that if and…

Q: Theorem 1: Suppose G is a connected graph. For every pair of vertices u and v in G, every uv-path…

A: To prove this theorem, first we will recall the definition of partite graph

Q: Determine the largest positive integer k such that χ(H) = χ(G) = k, where H is obtained from a…

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Q: Draw all non-isomorphic graphs of order n with 1<n<4.

A: As per the question we have to draw all the non-isomorphic graphs of order 1, 2, 3, 4 But as there…

Q: Let G = (V, E) be a graph (no loops, no multi-edges, undirected, as always. Let d be the smallest…

A: ANSWER:

Q: An edge is called a bridge if the removal of the edge increases the number of connected components…

A: We draw a graph without loop, find number of bridges by removing edges.

Q: (7) Verify whether it is possible to have a connected Eulerian graph with 3 vertices and 6 edges,…

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Q: Let G be a connected simple graph in which all vertices have degree 4. Prove that it is possible to…

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Q: Prove that a graph with at least three vertices is connected if and only if at least two of the…

A: A graph G is a pair of sets V and E where the elements of the non empty set V are called the…

Q: Prove that a bipartite graph has a unique bipartition iff it is connected.

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Q: If G is a connected planar graph where e = 3v - 6, show that every region is triangular (has three…

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Q: Let G be a connected graph that has an Euler tour. Prove or disprove the following statements. If G…

A: Euler Circuit: A closed walk in a graph G that uses every edge exactly once is called an Euler…

Q: Prove that the two graphs below are isomorphic.

A: Graph G and H are isomorphic if there exists an isomorphism between them.

Q: An outerplanar graph is a graph that admits a planar embedding with all vertices on the boundary of…

A: Given that an outerplanar graph is a graph that admits a planar embedding with all vertices on the…

Q: Draw a directed graph with as few edges as possible that is strongly-connected, has 8 vertices of…

A: Strongly connected directed 8 vertices one has in-degree 3 two have cut degree 3 rest have samller…

Q: 2. Prove that the size of every k-chromatic graph is at least (

A: Consider the provided question, According to you we need to solve only (2). (2) We need to prove…

Q: Prove that cycles are bipartite for an even number of vertexes. But they are not bipartite for the…

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Q: True or false: If a connected graph G can be decomposed into exactly k paths then exactly 2k…

A: To Prove: A connected graph can be decomposed into exactly k paths then exactly 2k vertices of G…

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: Prove that the dual to Eulerian planar graph is bipartite.

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Q: Prove that every simple graph G= (V, E) with |V] > 2 has at least two vertices of equal degree. %3D…

A: We have a simple graph, with V≥2, that is, the number of vertices is greater than or equal to 2.…

Q: 1. Prove that in a finite directed graph if every edge has at least one outgoing edge, then the…

A: 1. It is given that the graph is directed and finite.Therefore it has finite vertex and it is given…

Q: Let G be a pseudograph with 6 vertices. If G is having 3 loops and no parallel or multiple edges,…

A: A pseudograph is a non-simple graph in which both graph loops and multiple edges are permitted.…

Q: A complete m-ary tree of height h has number of vertices mh+1+1 (A m+1 mh+1-1 B h-1 m^ - 1 т-1…

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Q: Let G be a connected graph with at least two nodes. Prove that it has a node such that if this node…

A: Consider a connected graph with at least two vertices. choose vertices u,v so that du,v=maximum if…

Q: Let G be a connected graph of order n = 4 and let k be an integer with 2 ≤ k ≤ n − 2. Prove that if…

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Q: Draw the multigraph G whose adjacency matrix is (0 1 2 0 1 1 1 1 A = 2 1 0 \o 1 0 Show that the…

A: Adjacency matrix We will see this with the example. For the given graph the adjacency matrix is as…

Q: Suppose a graph G contains two distinct paths from a vertex u to a vertex v. Show that G has a…

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Q: Let G be a graph with n vertices and an independent set of size s ≥ 1. What is the maximum possible…

A: Please check the next step for details

Q: 2. Construct a multigraph of order 6 and size 7 in which every vertex is odd.

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Q: For each of the two directed graphs below,

A: Given, Diagram of two directed graphs and . We have to determined that the given two directed…

Q: Give an example of two undirected graphs G1 with nodes {f, s, y, r} and G2 with nodes {t, q, n, p,…

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Q: 5.1.21. Suppose that every edge of a graph G appears in at most one cycle. Prove that every block of…

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Q: Let G be a simple connected plane graph with 6 vertices. (a) (b) (c) (d) What is the largest number…

A: Note - Acc. to Bartleby guidelines, we are supposed to do only first 3 subparts . If you want more…

Q: Prove that if G is a graph of order p and has size greater than ¹) then G is connected. 2

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Q: Show that the Petersen graph is nonplanar. [Hint: show that the graph is edge contractible to Ks).

A: Since you have asked multiple questions, we will solve the first one for you. If you want any…

Q: For any connected graph G of order 54, if x is a vertex of G, then the number of components of G-x…

A: Given: To explain the given statement as follows,

Q: The two directed graphs below are isomorphic. Select one: True False

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