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

Q: Suppose that an edge e ::= {v1, v2 } is on a cycle in a graph G where v1, v2 are vertices of G. If e…

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Q: A simple grарh that contains exactly опe edge between each раir of distinct vertices is called

A: A simple graph that contains exactly one edge between each pair of distinct vertices is called

Q: Find the edge chromatic number of Kₙ when n is a positive integer.

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Q: Consider the problem Q defined below. Prove whether or not problem Q is in NP An undirected graph…

A: Given an undirected graph G(V, E) and a positive integer k.

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: Consider the following directed graph. E

A: a)From the given graph The vertex…

Q: Use Ore's theorem to prove that any n-vertex graph G for which |E(G)|> (",') +2 is Hamiltonian.

A: answer is in next step

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: Prove that every Cayley graph is vertex-transitive.

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Q: Prove that every graph G on {1, ... n} which is acyclic and has n – 2 edges has exactly 2 connected…

A: A graph with v vertices and e edges has at least v-e connected components. By induction on e, If e=0…

Q: Draw the graph for the funclion Y= SRal and then find (Pp) & (Re)

A: See the attachment

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

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Q: simple path in G from v0 to v1

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Q: show that if Gòe connected grapht that is mot complete, then contains two vertices u and v such that…

A: This problem is related with graph theory. Here, we have to show that if G is a connected graph that…

Q: Show that if u and v are the only odd-degree vertices in G, then there is a uv path G.

A: We prove this by induction on the number of vertices in G.   Case 1:   If the graph consists of…

Q: Prove or disprove that if G has m edges and n vertices, then the degeneracy of G is at least m

A: Given  be a graph with m edges and n vertices. Required to prove that the degeneracy of G is at…

Q: Let G be a bipartite graph with bipartition (X, Y). For SCX, let E, be the set of edges in G…

A: Given below the detailed solution

Q: Prove or disprove the following. Let G = (V, E) be a simple undirected graph without loops, where…

A: Definition Degree of vertex: The degree of a vertex is the number of edges connecting it.…

Q: Prove that every graph G on {1,... n} which is acyclic and has n 2 edges has exactly 2 connected…

A: Given: To prove the given statement as follows:

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: Prove that if u is a vertex of odd degree in a graph, then there exists a path from u to another…

A: We have to prove :       If u is a vertex of odd degree in a graph, then there exists a path from…

Q: Let G = (V, E) be a finite bipartite graph with bipartition (A, B) where DI

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Q: If a graph G has a subgraph H with 1. ĮV(H)| = |V(G)| 2. H is connected 3. Je(H)| = |V(H)| %3D 4. Vu…

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Q: Prove or disprove (by giving a counterexample) that a graph G with n 2 4 vertices and ("5') +3 edges…

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Q: 3. A labeled Ks graph with labeled edges appears below. Which edge permutation is induced by the…

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Q: Suppose that G is a k-regular graph of n vertices, and m edges. The vertex connectivity of G is…

A: Sol

Q: Let G = (V, E) be a k-connected graph, let æ E V, and let B C V\{x} such that |B| > k. Show that…

A: Fan: Let v∈V(G) and let A∈V(G)\{v}. A v,A-fan of size k is a collection of k paths P1,P2,...,Pk so…

Q: Prove or disprove the following. Let G = (V, E) be a simple undirected graph without loops, where…

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Q: If G is a Hamiltonian graph, then G has no cut-vertex. True or false? Justify

A: In order to find the solution of the given question, we will use the concept of Hamiltonian graph…

Q: Prove that if G is a simple graph with n > 3 vertices and (",') +2 edges, then G is Hamiltonian. b)

A: In the given question we have to prove that a simple graph with vertices more than or equal to three…

Q: 4) Draws the G graph that has the following adjacencymatrix: [o 1 1 1 1 0 1 0 a) 1 10 0 [o 3 0 27 3…

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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: prove If G is a plane graph with n vertices, m edges and r regions, then n - m + r = 1 + k(G).

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Q: [1] Write down the vertex-connectivity K(G) and the edge-connectivity 2(G) for each of the following…

A: Vertex connectivity: The minimum number of vertices required to disconnect the graph G is called…

Q: Show that every simple graph on n vertices is isomorphic to a subgraph of Kn.

A: We need to prove that every simple graph on n vertices is isomorphic to a subgraph of Kn (complete…

Q: Prove that every graph G on {1, ...n} which is acyclic and has n – 2 edges has exactly 2 connected…

A: A graph is said to be acyclic if it has no cycles and acyclic if it contains no cycle. For a graph…

Q: Let G be a simple connected graph with exactly 8 vertices, one of them has degree 7, and the other 7…

A: Consider G be a simple connected graph with exactly 8 vertices, one of them has degree 7, and…

Q: For the given graph G, which of the following statements is false? A B) (F a) G is a simple directed…

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Q: Which of the following is the adjacency ma- trix of the given graph? 1 2 4 3 [1 2 2 1 2 0 1 2 11 1 0…

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Q: Consider a graph G where the vertices are points (a, b) with integer coordinates, and {(ai, bi),…

A: Given: Let us consider graph G, in which the vertices are integer-coordinated points (a,b), and (ai,…

Q: If a graph G has a subgraph H with 1. [V(H)| = |V(G)| 2. H is connected 3. Je(H)| = |V(H)| %3D 4. Vu…

A: The necessary condition for a graph to be Hamiltonian circuit is that the graph must be connected,…

Q: Let G be a n vertices graph ( n 2 3 ), such that d(u) + d(v) 2 n, for every pair of non adjacent…

A: Let us start proving step by step

Q: 7. Give an (i) a non-planar graph that is not homeomorphic to K5or K3,3 example of

A: To find: A graph which is non planar and not homomorphic to K5 or K3,3. If the edges of the graph…

Q: Are the following directed graphs strongly connected or not? Explain why in detail? 3 4 A

A: A directed graph is said to be strongly connected if there is a path between any pair of vertices in…

Q: (3) Determine whether or not each of the following graphs is planar. If planar, show a planar…

A: As per our company guidelines, we can solve one question at a time. If you want other question to be…

Q: Prove that if a plane graph G has n vertices, e edges, f faces and k components, then n−e+ f = k +1.

A: Euler’s Theorem: According to Euler's Theorem, V-E+F=2 Where, V is the number of vertices. E is the…

Q: prove that the degree of a regular complete tripartite graph kr,s,t with n vertices is given by 2n/3

A: Since the graph is tripartite with vertex division of r, s, t, we have three clusters of vertices…

Q: Let G be a simple graph with nonadjacent vertices v and w, and let G+e denote the simple graph…

A: Given that, v and w are non-adjacent vertices in G. (1): G+e the simple graph formed by adding the…