4. This problem is about graphs. In each of the following cases, either draw a graph with the stated property, or prove that no such graph exists. (For the purpose of this problem, a “graph" must have no loops or multiple edges.) (a) A graph on 12 vertices in which every vertex has degree 4. (b) A graph on 45 vertices in which 22 vertices have degree 8 and 23 vertices have degree 7. (c) A graph on 10 vertices in which every vertex has degree 3 and the shortest cycle subgraph has length 5. (d) A graph on 13 vertices in which every vertex has degree at least 7 and there are no cycle subgraphs of length 3.

Graph Complement and Planarity,(a) Let G be a 4-regular plane graph with 10 faces. Determine how many vertices and edges G hasand give a plane drawing of such a graph.  for this problem, you must writecomplete sentences, include all details, show all of your work, and clarify all of your reasoning.

Planarity(b) Let G be a simple graph with exactly 11 vertices. Prove that G or its complement G must benon-planar. Hint: The maximum number of edges in a planar graph with n vertices is 3n − 6. for this problem, you must write complete sentences, including all details, showall of your work, and clarify all of your reasoning

(a) Writeup  Prove Theorem 1 Using Theorem 1, describe an algorithm in words to check bipartiteness and analyze its run time.

4. Prove the following theorem: If a graph has more than two vertices of odd degree, then it does not have an Euler Path.   (Do a proof by contradiction please)

In each part, either draw a graph with the given specifications or explain why no such graph exists.  A graph with 4 vertices that has a Euler circuit but no Hamiltonian circuits.

please explain fully

12) Which of the following problem is an NP-complete problem? Please look at the attached image.

This is a Graph Theory problem.

Q.3 (a) Find the degree sequence of the graphs: K6,5, W, (b) Find All values of n such that the graphs: (i) K. (i) C are bipartite (c) Draw the graph with the given incidence matrix 1 100 1011 1 1 100 00 0 01 1001 0 0011 10 0 00 00 0ol Thope the solution is clear and the line is understandable