Give an example of a simple connected graph G for which G is planar, has no subgraph isomorphic to K4, and for which χ(G) = 4. You must prove with full and complete details, and a carefully reasoned argument that χ(G) = 4.

Q: Prove that if G is a graph with no isolated vertices, then α'(G) +β'(G) = n(G)

A: Here α'(G) is largest size of matching in G. β'(G) is the smallest size of an edge cover. We have to…

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

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Q: A simple graph G on n vertices with m edges is overfull if m> [n/2]A. Show that if G is overfull,…

A: Given that: A simple graph G on n vertices with m edges is overfull if m>n2△. The objective is to…

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

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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: Let G be a connected graph that has exactly 4 vertices of odd degree: V1, V2, V3 and V4. Show that…

A: Let G be a connected graph which has 4 vertices of odd degree. The objective is to show that there…

Q: (d) Suppose that a graph G is regular of degree r, where r is odd. (i) Prove that G has an even…

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

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Q: Prove that if the graph G is discontinuous, then the complementary graph G is continuous.

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Q: Let G be a simple graph with n vertices, which is regular of degree d. By considering the number of…

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Q: 9. Verify that Euler's formula does not hold for a graph that is not connected: count the vertices,…

A: The given question is related with graph theory. We have to verify that Euler's formula does not…

Q: 3) Draw three non-isomorphic graphs of order 5 and size 5.

A: Three non isomorphic graphs of order 5 and size 5 are:

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: 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: Prove: For any graph r with six vertices ror r complementary contains C3

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Q: 1. For the pair of graphs shown, • if the graphs are not isomorphic, explain why not; • if the…

A: if you find my answer helpful please put thumbs up. thank you. the required solution is given below…

Q: Prove that the two graphs below are isomorphic.

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

Q: (b) Show that G may not necessarily have three cycle subgraphs by giving an example of a graph G,…

A: Consider the equation

Q: Find a subgraph H of the graph G given below that has 4 vertices and 3 edges.

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Q: True/False: If the graphs G1 and G2 are isomorphic, and G, is connected, then G2 must be connected…

A: If G1 and G2 are isomorphic , and G1 is connected

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

A: This is a problem of Graph Theory.

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: Show that the petersen graph is nonplanar.( Hint :show that the graph is contradictible to k5.

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Q: For any connected graph G of order 55, if x is a vertex of G, then the number of components of G-x…

A: A connected graph is graph that is connected in the sense of a topological space, i.e., there is a…

Q: 3.1.5. (-) Prove that a (G) > AG)+1 n(G) for every graph G.

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Q: 2. Determine if the following two graphs are isomorphic. а) b)

A: (a) let us suppose  graph 1 graph 2 we have to find if graph is isomorphic or not

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: 6. (In this problem, y denotes the domination umber of a graph.) Let G be a graph such that (G) > 3.…

A: Let G be a graph. Such that γ(G)≥3. Prove that diam(G)≤2.

Q: 6. Find the strongly connected components of the following graph. 4 7. Use paths either to show that…

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Q: 4.1.10. (1) Find (with proof) the smallest 3-regular simple graph having connectivity 1.

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Q: Let H be a connected planar graph with at least 3 vertices. Prove that f is greater than or equal to…

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Q: The number of non-isomorphic simple graphs on 8 vertices, all of which have degree 2, is Select one:…

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Q: Prove: “The complement of the graph G is an Eulerian graph whenever G is also  an Eulerian graph of…

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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: Prove that in any simple graph G with n vertices and m edges, 2m ≤ n^2 − n.

A: We need to prove that in any simple graph G with n vertices and m edges, 2m≤n2−n

Q: Provide an easy-to-understand explanation for the direct proof problem below: For every graph that…

A: Given, for every graph that is undirected (no self-loops or multi-edges) with n nodes.

Q: O For odd values of n>1, Kn is Eulerian. O For odd values of na 1, Kn has an Eulerian trail. ) K2…

A: The above statement is true as “A graph is Eulerian if and only if the degree of each vertex is…

Q: Give an example of a non-planar graph that is not contractible to K5or K3,3.

A: Solution :-

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: 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: 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: 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…