Problem 7. Determine if a loop-free connected undirected graph with eight vertices, where the degrees of the vertices are 1, 1, 1, 2, 3, 4, 5, and 7, exists. If it exists, draw the graph; otherwise, prove that the graph doesn’t exist.

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.

Determine whether the graph is planar or not. To show planarity, give aplanar embedding. To show that a graph is not planar, use Kuratowski’s theorem

Problem 3 1) Consider that you have a graph with 8 vertices numbered A to H, and the following edges: (A, B), (A, C), (B, C), (B, D), (C, D), (C, E), (D, E), (D, F), (E, G), (F, G), (F, H), (G, H) a) Using depth-first search algorithm, what would be the sequence of visited nodes starting at A (show the content of the stack at some steps). b) Same question if the algorithm used in breadth first search. c) What would be the minimum spanning tree rooted at A if a Depth First Search algorithm is used (refer to question a), show few steps in running the algorithm. d) What would be the minimum spanning tree if a Breadth First Search algorithm is used (refer to question b), show few steps of the algorithm.

5. Another Proof Consider an undirected, unweighted, simple graph G. Suppose the longest simple path in G has length 10 and goes from node u to node v. Prove that the degree of node u is at most 10.

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)

Prove that every simple, triangle-free, planar graph is 4-colorable. (Hint: show first that such a graph has a vertex of degree at most 3.)

Please help me solve this problem

Given a Triangle of three vertices A(2,3), B(4,2), C(3,0), Apply the shearing to the Triangle with SH=4

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.