Determine the largest positive integer k such that χ(H) = χ(G) = k, where H is obtained from a nonempty graph G by subdividing each edge of G exactly once.

Q: 5. Let G = (V, E) be a finite, connected graph. Show that if no vertex of G has degree less than 2,…

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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: 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: 5. Prove that every graph of order 6 with chromatic number 3 has at most 12 eges.

A: Solution :- As per guidelines.i solve 5th number.

Q: Let G be a graph of order n and let k be an integer with 1≤k≤n−1.Prove that if every vertice of G…

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

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Q: Show that the following two graphs are isomorphic 2. 3 10 1. 9. •5 a. f g,

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

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Q: Prove that every connected graph with m vertices has at least (m-1) edges, for every m>1

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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: 7. Suppose a graph G = (V, E) has n vertices and m edges (i.e., |V| = n and |E| = m). Further…

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Q: 7. Prove : Every graph with radius r and girth with 2r +l is regular

A: To Determine :-  Every graph with radius r and girth with 2r +1 is regular.

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 finite graph on n ≥ 3

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Q: Prove that every graph G with n vertices and chromatic number k = x(G) has at most · (n2 – ) edges.…

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Q: 1. Find the 2 transformations that occurred to map quadrilateral KLMN to quadrilateral K"L"M"N" in…

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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: Q10. Using Corollary 3.1.24 of the book, to prove that a graph G is bipartite if and only if a(H) =…

A: Given: α(H)=β'(H) To find: Isolated vertices

Q: Check that each of the following maps is a group convergence and find its core: E) h : Z18 → Z3…

A: Given h :Z18→Z3 where h([x]18) =[2x]3

Q: Find the invers e Fenction of nd graph (メノ)

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Q: 4.1.14. (!) Let G be a connected graph in which for every edge e, there are cycles C1 and C2…

A: The given question is related with graph theory. Given that G is a connected graph in which for…

Q: 15. Let G be a graph on n vertices (with no loops or multiple edges). Prove that if all vertices in…

A: 1. A graph G is an ordered pair (V(G), E(G)) consisting of a non-empty set V(G) of vertices and a…

Q: Let G be a graph on n vertices (with no loops or multiple edges). Prove that if all vertices in G…

A: 1. A graph G is an ordered pair (V(G), E(G)) consisting of a non-empty set V(G) of vertices and a…

Q: Prove that the two graphs below are isomorphic.

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

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

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Q: Draw the graph of the following map: xn+1= {3xn if x1/3  Show also that the periodic orbits are…

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Q: Suppose G is a connected graph with n > 1 vertices and n − 1 edges. Prove that G has a vertex of…

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Q: Suppose that a subgroup H of S5 contains a 5-cycle and a 2-cycle.Show that H = S5.

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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: 16. Let G be a graph with n vertices , t of which have degree K and the others have degree K+1…

A: Consider the given information. By the theorem of graph theory, the sum of the degree of a graph is…

Q: d. e

A: A graph is said to be strongly connected if we can find a path from each vertex to every other…

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: Let G be a 2-connected graph. Prove that if G has no even cycles, then G is isomorphic to an odd…

A: k-Connected graphs: A graph is said to be k-connected if it has more than k vertices and remains…

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: 4.1.17. Prove that deleting an edge cut of size 3 in the Petersen graph isolates a vertex.

A: Edge Connectivity: λ(G) or κ′(G) is the minimum size of edge set F s.t. G\F has more than one…

Q: Which of the following graphs contain a subdivision of K3,3 as a subgraph

A: The solution is given below in detail in which we found no graph contains a subdivision of K3, 3 1)

Q: Let G=(V, E) be a graph such that • V| ≥ 3, G has exactly 2 leaves, and in G all the non-leaf…

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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: =. Find the smallest 3-regular graph G such that (G) 1.

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Q: Let G be a graph with v vertices and e edges. Let M be the maximum degree of the vertices of G, and…

A: We are given a G graph G with v number of vertices and e number of edges in the graph. we are also…

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: 5. Let G be a graph with n vertices, t of which have degree k and the others have degree k+1. Prove…

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Q: 8. If we remove an edge e from a graph G and G- e is still connected, then show that e lies along…

A: Given: Remove an edge e from a graph G and G-e is connected.

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,