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

Q: Prove or disprove: There exist two connected graphs G and H both of order at least 3 and neither of…

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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 the (3,4)-complete bipartite graph. (a) List all non-isomorphic subgraphs with 5 vertices…

A: Part(a) Solution:Let be the (3,4) complete bipartite graph K3,4      These are the 11…

Q: Prove that for every coloring (n colors) of a complete graph on 2n+1 vertices there's a…

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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: 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: Prove on the number of vertices that every graph G with n edges has a bipartite subgraph with at…

A: We will first assume that  P(n) : every graph G with n edges has a bipartite subgraph with atleast…

Q: 4.1.3. (-) Let G be an n-vertex simple graph other than Kn. Prove that if G is not k-connected, then…

A: Given: Let G be an n-vertex simple graph other than Kn. Prove that if G is not k-connected, then G…

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: 4. a. Prove or disprove that the given graphs are isomorphic.

A: Given graphs are Isomorphic Graph: Two graphs are called isomorphic if the number of vertices and…

Q: Suppose G is a simple graph with n > 1 vertices such that every vertex has degree at least (n –…

A: Let G be the simple graph with the vertices n≥1 and every vertex has the degree of at least n-12.…

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: 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: Prove that if  v0 and v1  are distinct vertices of a graph  G  = (V,E) and a path exists in G from…

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Q: 4. Prove: "The complement of the graph G is an Eulerian graph whenever G is also an Eulerian graph…

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Q: Prove that if  v0 and v1  are distinct vertices of a graph  G  = (V,E) and a path exists in G from…

A: Let v0 and v1 be distinct vertices of a graph G=V,E. Assume that there exists path in G from v0 to…

Q: simple path in G from v0 to v1

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Q: b) Is the following graph isomorphic to Q4? If yes, give an isomorphism from Q4 to this graph.…

A: 2b ) Given graph is Here number of vertices in the given graph is 12 and number of vertices in Q4…

Q: G is a graph on n 21 vertices such that both G and its complement have an Euler trail. (Recall that…

A: To do: Draw a simple graph G such that "G is a graph with n≥1 vertices such that both G and its…

Q: Prove that any graph has at least two vertices with the same degree. A complete bipartite graph on (…

A: There are multiple questions here.   Question (1) Prove that any graph has at least two vertices…

Q: Let k be a fixed positive integer. Every k-regular bipartite graph has a perfect matching. Prove or…

A: The given statement is "Every k-regular bipartite graph has a maximum matching". Since this…

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

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Q: Prove that any graph has at least two vertices with the same degree. A complete bipartite graph on (…

A: To prove: Any graph has at least two vertices with the same degree. A complete bipartite graph has…

Q: Let G be a simple graph. Show that the relation R on theset of vertices of G such that uRv if and…

A: Given: A simple graph is G. To prove: That the set of vertices of G is such that uRv if and only if…

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

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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: 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: (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: Discrete math. Graph theory

A: The given statement is not true. Counter example is shown below.

Q: 4. Prove: a) In a graph, if two vertices are connected by a walk then they are connected by a path.…

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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: Suppose G is a simple graph with n >1 vertices such that every vertex has degree at least (n – 1)/2.…

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Q: Show that every simple graph on n vertices subgraph of K.

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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: 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: 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: Prove that if  v0 and v1  are distinct vertices of a graph  G  = (V,E) and a path exists in G from…

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

A: Given that G is a 2-connected graph. If G is a cycle then there is nothing to prove. Suppose G is…

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…