7. Prove that if G is a 2-connected, {claw, Pa}-free graph of order at least 13, then G is either a cycle or chorded pancyclic.

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: 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: 4. Prove: "Show that the number of edges in a simple planar graph of order n is at most 3n – 6".…

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

A: Topic:- planer graph

Q: Let G be a simple graph on 9 vertices, and assume we know that the sum of all degrees in G is at…

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Q: Prove that having n vertices, where n is a positive integer, is an invariant for graph isomorphism.

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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: 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: 7. Answer the following questions for the two graphs below. 13 I8 Figur 2: G2 Figur 1: G1 (a) State…

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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: 18. Let G be a graph with n vertices and exactly n-1 edges . Prove that G has either a vertex of…

A: Now, we need to prove that if G is a graph with n vertices and (n-1) edges then G has either a…

Q: Prove that a simple 2-connected graph G with at least four vertices is 3-connected if and only if…

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: Euler cycle.

A: AnswerSuppose we have some simple graph G, with odd n>, 1 nodes, and that this graphhas an Euler…

Q: Let G be a connected graph with n(G) ≥ 2. Show that G at least 2 vertices which are not cut-vertices…

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Q: Let G =(V,E) be a finite graph on n ≥ 3

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Q: Let Vn be the set of connected graphs having n edges, vertex set [n], and exactly one cycle. Form a…

A: Given Let Vn be the set of connected graphs having n edges, vertex set [n], and exactly one cycle.…

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: 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: Show that there is a graph G with exactly 5 nodes where both G and its complement, have chromatic…

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Q: 12. Determine whether the given pairs of graphs is isomorphic. Exhibit an isomorphism (i.e., find a…

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Q: Show that if a simple graph G has n vertices and at least n edges, then G must contain a cycle.

A: We have to Show that if a simple graph G has n vertices and at least n edges, then G must contain a…

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: Let G be a connected graph. Prove: (a) If G contains a cycle C which contains an edge e, then G – e…

A: Part (a):   A connected graph is a graph in which there is a path joining each pair of vertices, in…

Q: Prove that the graphs in the figure below are isomorphic: (2 6. a

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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: 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: 4.1.5. (-) Let G be a connected graph with at least three vertices. Form G' from G by adding an edge…

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Q: Show that any connected simple planar graph on n vertices must have at most n/2 vertices of degree…

A: Consider the equation

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: Prove that if a graph on n vertices is connected, and has n-1 edges, it must be acyclic.

A: A graph G is said to be a connected graph, if there is a path between any two vertices in that…

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

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Q: Prove the following. 1. Prove that the number of odd vertices of a graph is even.

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Q: 4 Let G be a digraph with 2 or more vertices which is strongly connected and in which every node has…

A: Given that G be a digraph with 2 or more vertices which is strongly connected and in which every…

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: Show that it is not possible to creak graph the" degree with g vertices such that every a of vertex…

A: Given that  The no of vertices = 9 degree of each vertices =3 To show such graph is not possible.

Q: 5. Let G be a connected graph without loops and of order at least 11. Prove that G or its complement…

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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: 5.1.7. (-) Construct a graph G that is neither a clique nor an odd cycle but has a vertex ordering…

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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 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: Draw a single graph G which satisfies all of the following properties. - All vertices of G are of…

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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: Show all possible connected sub graphs of the following graph. How many of them are there refer to…

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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: 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…