Let G be a critical 2-connected graph; this means that G is 2-connected but no graph G e for e E E(G) is 2-connected. (a) Prove that at least one vertex of G has degree 2. (b) For each n, find an example of a critical 2-connected graph with a vertex of degree at least n.

Q: Let H be a graph If H has at least one vertex of degree greater than zero then there are two of even…

A: Given a graph representing H. Statement: If H has at least one vertex of degree greater than zero…

Q: Let Gbe a simple graph with at least 11l vertices, and let G be its complement. (i) Prove that Gand…

A: Given: G is a simple graph with at least 11 vertices and Ḡ be its complement.

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: Show for every even integer n ≥ 2 that there exists a graph G of order n such that G has n/2 pairs…

A: According to the question, In graph theory, two vertices and u are similar if there is an…

Q: The following pairs of simple graphs are non-isomorphic. For each pair, give a precise reason why it…

A:

Q: A graph is symmetric with respect to the ________ if, whenever (x, y) is on the graph, (−x, y) is…

A: We have to fill in the blanks: A graph is symmetric with respect to the --- if, whenever x, y is on…

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 graph of order n ≥ 3. Prove that G is 2-connected if and only if for any vertex v and…

A: Proof:Sufficiency:Suppose that G is 2-connected and for any vertex v and edge e of G, v and e lie on…

Q: (a) Prove that if G is a graph of minimum degree 3, then G contains a path of length 4. (b) Prove…

A: G is a graph of minimum degree 3.G is a graph of minimum degree 5.

Q: (3) If possible, define graphs with the following properties. If not, explain why. (a) A graph with…

A: (a) The graph is defined.(b) The graph is not defined.(c) The graph is defined.Explanation:Step 1:We…

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

A:

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

A: This is the problem of graph theory.

Q: Call a graph H regular if there exists k ≥ 1 such that dµ(v) = k for all v € V(H). (e) Does there…

A:

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

A: