MATH/CSCI Graph Theory Help:) Thank you for helping me I appreciate it, will definitely upvote the correct solution! Please read the question carefully and answer it fully! If you handwrite the solution, please be as legible as possible:) take care and have.a great day!!

(a) (20 points) Let P be the Petersen graph (see Figure 2 ). Find (P) and prove that your answer is correct. Show all of your work and clearly explain, using words, your reasoning. Your solution should be coherent and precise. Do not leave out any details whatsoever. Write complete sentences. Lack of clarity and precision will result in point deductions. Figure 2: The Petersen Graph Problem 2, continued b) (15 points) Define 2(G) to be the minimum number of colors needed to color the vertices of simple graph G so that vertices that are distance exactly two apart receive different colors (but adjacent vertices are allowed to have the same color or not). This is a variation on standard proper vertex coloring. Let W6 be the 6-wheel graph shown below. Find 2(W6) and prove that your answer is correct. Clearly explain your work and don't leave out any details. Your solution should be coherent and precise. Do not leave out any details whatsoever. Write complete sentences. Lack of clarity and precision will result in point deductions. Figure 3: 6-wheel or W6 3. (10 points) TRUE or FALSE (circle one). There exists a connected planar graph G such that in a plane embedding of G all faces have size 6 . Do not justify your