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Problem 1: (a) Let G = (V, E) be a graph with |V | = n, and let k be an integer, where 1 ≤ k ≤ n. Prove the following theorem: "Suppose the the vertices in V can be ordered v1, v2, ..., vn in such a way that each vertex vi has at most k neighbors among the preceding vertices v1, ..., vi-1. Then G can be colored with at most k + 1 colors." For example, the figure below shows a graph and an ordering of its vertices where each vertex has at most 3 neighbors that precede it in this ordering. So this theorem claims that this graph can be colored with 4 colors.
(b) Prove that the theorem in part (a) above implies the following statement: "If all vertices of a graph G have degree at most D then G can be colored with at most D + 1 colors". (You will cover a different proof of this theorem in the discussion. Here you only need to show how it can be derived from part (a).)
Problem 2: You are given two bipartite graphs G and H below. For each graph determine whether it has a perfect matching. Justify your answer, either by listing the edges that are in the matching or using Hall's Theorem to show that the graph does not have a perfect matching.
Problem 3: Determine which of the following two graphs are planar. Justify your answer. (You need to either show a planar embedding or use Kuratowski's theorem.)
Prove that 2n-1 comparisons are necessary in the worst case to merge two sorted lists containing n elements each.
Prepare a spreadsheet to compute your projected total costs, total revenues, and total profits for giving seminar on cost estimating.
Write an algorithm, using pseudo code, "Consensus algorithm": A group of ten people need to decide which one flavor of ice cream they will all order, out of three options.
Show how the following values would be stored by machines with 32-bit words, using little endian and big endian format. Assume each value starts at address 016. Draw a diagram of memory for each, placing the appropriate values in the correct (and ..
Use a loop with a sentinel to allow a user to enter arbitrary number of values. Compute and display the geometric mean of all values, excluding the sentinel. (Hint: Math.pow(x, 1.0/k) will compute the kth root of x)
Write psuedocode for one of the classic traversal algorithms(preorder, inorder, and postorder) for binary trees. Assuming that your algorithm is recursive, find the number of recursive calls made
Design an algorithm to produce a list of customers from the Glad Rags Clothing Company's customer master file. Each record on the customer master file contains the customer's number.
Discuss the benefits of dynamic over-static arrays. Under what conditions will you choose dynamic arrays?
What is an algorithm? What are the characteristics of a good algorithm and what do you mean by complexity of an algorithm? Explain the meaning of worst case analysis and best case analysis with an example.
question 1write the selection sort algorithm as a template function using the less than operator to compare two
Write a program that inputs the first name, middle initial (without the period), last name, age, salary and sex of new employee. Then displays that person's name with the first name first, middle initial followed by a period, and last name then ag..
Create a shell script to send a customized mail memo to the users listed on the command line through login name, only if they are currently logged on.
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