(2) Recall the following fact: In any planar graph, there exists a vertex whose degree is s 5 Use...
Q3.a) Show that every planar graph has at least one vertex whose degree is s 5. Use a proof by contradiction b) Using the above fact, give an induction proof that every planar graph can be colored using at most six colors. c) Explain what a tree is. Assuming that every tree is a planar graph, show that in a tree, e v-1. Hint: Use Euler's formula Q3.a) Show that every planar graph has at least one vertex whose degree...
A 2-coloring of an undirected graph with n vertices and m edges is the assignment of one of two colors (say, red or green) to each vertex of the graph, so that no two adjacent nodes have the same color. So, if there is an edge (u,v) in the graph, either node u is red and v is green or vice versa. Give an O(n + m) time algorithm (pseudocode!) to 2-colour a graph or determine that no such coloring...
Recall the definition of the degree of a vertex in a graph. a) Suppose a graph has 7 vertices, each of degree 2 or 3. Is the graph necessarily connected ? b) Now the graph has 7 vertices, each degree 3 or 4. Is it necessarily connected? My professor gave an example in class. He said triangle and a square are graph which are not connected yet each vertex has degree 2. (Paul Zeitz, The Art and Craft of Problem...
Solve all parts please 5. In the following problems, recall that the adjacency matrix (or incidence matrix) for a simple graph with n vertices is an n x n matrix with entries that are all 0 or 1. The entries on the diagonal are all 0, and the entry in the ih row and jth column is 1 if there is an edge between vertex i and vertex j and is 0 if there is not an edge between vertex...
Your teacher is going to give a test where each student is to answer one question. None of the neighboring students should have the same question. How many questions are needed? Graph Coloring Algorithm is used to solve this type of problems. It does not guarantee to use the minimum number of questions, but it guarantees an upper bound on the number of questions. The algorithm never uses more than d+1 questions where d is the maximum degree of vertices...
Notes for lab dc02-Resistors and the Color Code will skip are Part 2 e, g: Part 4; Exercises 2, 4,5,6 and 3. It is important to answer the exercises correctly in each labl you should include the appropriate prefix for the unit in the Numerical Value We will not be Volt using the Volt-Ohm meter (VOM) for this lab, so skip the parts that ask for VOM measurements. The parts we You do need to complete Exercises1 Note that in...