4.1.12. Let n, k be positive integers with n even, k odd, and n > k > 1. Let G be the k- regular simple graph formed by placing n vertices on a circle and making each vertex adjacent to the opposite vertex and to the (k – 1)/2 nearest vertices in each direction. Prove that k(G) = k. (Harary (1962a])

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: 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: Exhibit a perfect matching in the graph below or give a short proof that it has none.…

A: Given,A graph.Aim: To exhibit a perfect matching in the graph given or give a short proof that it…

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: 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 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: (b) Give an example of a disconnected simple graph on n vertices in which every vertex has degree…

A: We need to give an example of a disconnected simple graph on n vertices in which every vertex has…

Q: 3. A simple graph H is the square of a simple graph G if V (H) = V(G) and xy is an edge in H if and…

A: The simple graph H is the square of another simple graph G if V(H)=V(G). xy is an edge in H if and…

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: 5. two. Show that every nontrivial acyclic graph has at least two vertices of degree less than

A: 7. ConclusionWe have proven that in every nontrivial acyclic graph (i.e., a tree or a forest), there…

Q: 5.1.12. (-) Prove or disprove: Every k-chromatic graph G has a proper k-coloring in which some color…

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Q: 9.6 Prove or disprove the following.

A: (a) Every subgraph of a  planar graph is planar.

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: 13(c) Construct a graph G that has all of the following properties: (i) G has exactly 6 vertices,…

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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: 4. Show that K6 is AY equivalent to the Petersen graph

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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: 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 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…