6. Consider the complete graph Kn for n ≥ 3. Color r of the vertices in K, red and the remaining n − r(= g) vertices green. For any two vertices v, w in Kn, color the edge {v, w} (1) red if v, w are both red, (2) green if v, w are both green, or (3) blue if v, w have different colors. Assume that r≥ g. 1 a) Show that for r = 6 and g = 3 (and n = 9) the total number of red and green edges in K9 equals the number of blue edges in K9. b) Show that the total number of red and green edges in K, equals the number of blue edges in K, if and only if n = r+g, where 9, r are consecutive triangular numbers. [The triangular numbers are defined recursively by t₁ = 1,tn+1 tn = n(n+1)/2. Hence t₁ = 1, t2 = 3, t3 = 6,...] = = t + (n+1), n > 1; so

Consider eight points on the Cartesian two-dimensional x-y plane. a g C For each pair of vertices u and v, the weight of edge uv is the Euclidean (Pythagorean) distance between those two points. For example, dist(a, h) : V4? + 1? = /17 and dist(a, b) = v2? + 0² = 2. Because many pairs of points have identical distances (e.g. dist(h, c) V5), the above diagram has more than one minimum-weight spanning tree. dist(h, b) = dist(h, f) Determine the total number of minimum-weight spanning trees that exist in the above diagram. Clearly justify your answer.

Let DIST (u, v) denote the distance between vertex u and v. It is well known that distances in graphs satisfy the triangle inequality. That is, for any three vertices u, v, w, DIST (u, v) ≤ DIST (u, w) + DIST (w, v). Let D∗ denote the distance between the two farthest nodes in G. Show that for any vertex s    D∗  ≤  2 max DIST (s, v).

4. Let n ∈ Z+ with n ≥ 4, and let the vertex set V ′ for the complete graph Kn−1 be{v1, v2, v3, . . . , vn−1}. Now construct the loop-free undirected graph Gn = (V, E) from Kn−1 asfollows: V = V ′ ∪ {v}, and E consists of all the edges in Kn−1 except for the edge {v1, v2}, whichis replaced by the pair of edges {v1, v} and {v, v2}.a) Determine deg(x) + deg(y) for all nonadjacent vertices x and y in V .b) Does Gn have a Hamilton cycle?c) How large is the edge set E?d) Do the results in parts (b) and (c) contradict Corollary 11.6? PS:Please do not use ChatGPT and type the correct answer!

Given N cities represented as vertices V₁, V2, un on an undirected graph (i.e., each edge can be traversed in both directions). The graph is fully-connected where the edge eij connecting any two vertices vį and vj is the straight-line distance between these two cities. We want to search for the shortest path from v₁ (the source) to VN (the destination). ... Assume that all edges have different values, and €₁,7 has the largest value among the edges. That is, the source and destination have the largest straight-line distance. Compare the lists of explored vertices when we run the uniform-cost search and the A* search for this problem. Hint: The straight-line distance is the shortest path between any two cities. If you do not know how to start, try to run the algorithms by hand on some small cases first; but remember to make sure your graphs satisfy the conditions in the question.

Consider a weighted, directed graph G with n vertices and m edges that have integer weights. A graph walk is a sequence of not-necessarily-distinct vertices v1, v2, ... , Vk such that each pair of consecutive vertices Vi, Vi+1 are connected by an edge. This is similar to a path, except a walk can have repeated vertices and edges. The length of a walk in a weighted graph is the sum of the weights of the edges in the walk. Let s, t be given vertices in the graph, and L be a positive integer. We are interested counting the number of walks from s to t of length exactly L. Assume all the edge weights are positive. Describe an algorithm that computes the number of graph walks from s to t of length exactly L in O((n+ m)L) time. Prove the correctness and analyze the running time. (Hint: Dynamic Programming solution)

We are given an undirected connected graph G = (V, E) and vertices s and t.Initially, there is a robot at position s and we want to move this robot to position t by moving it along theedges of the graph; at any time step, we can move the robot to one of the neighboring vertices and the robotwill reach that vertex in the next time step.However, we have a problem: at every time step, a subset of vertices of this graph undergo maintenance andif the robot is on one of these vertices at this time step, it will be destroyed (!). Luckily, we are given theschedule of the maintenance for the next T time steps in an array M [1 : T ], where each M [i] is a linked-listof the vertices that undergo maintenance at time step i.Design an algorithm that finds a route for the robot to go from s to t in at most T seconds so that at notime i, the robot is on one of the maintained vertices, or output that this is not possible. The runtime ofyour algorithm should ideally be O((n + m) ·T ) but you will…

How do I do this?  We say a graph G = (V, E) has a k-coloring for some positive integer k if we can assign k different colors to vertices of G such that for every edge (v, w) ∈ E, the color of v is different to the color w. More formally, G = (V, E) has a k-coloring if there is a function f : V → {1, 2, . . . , k} such that for every (v, w) ∈ E, f(v) 6= f(w).3-Color problem is defined as follows: Given a graph G = (V, E), does it have a 3-coloring?4-Color problem is defined as follows: Given a graph G = (V, E), does it have a 4-coloring?Prove that 3-Color ≤P 4-Color.(hint: add vertex to 3-Color problem instance.)

4. Let G (V, E) be a directed graph. Suppose we have performed a DFS traversal of G, and for each vertex v, we know its pre and post numbers. Show the following: (a) If for a pair of vertices u, v € V, pre(u) < pre(v) < post(v) < post(u), then there is a directed path from u to v in G. (b) If for a pair of vertices u, v € V, pre(u) < post(u) < pre(v) < post(v), then there is no directed path from u to v in G.

(3) Question 3: Given an undirected graph G and an edge uv in G. Design an algorithm that runs in O(|E|+|V)) time that decides if there is a cycle that contains UV. (4) Question 4: Problem 3 in page 245 of (Erickson]. Hint: Use topological sort. O S PC D00 吕口 F3 000 F4 F5 F7 F8 F9 F10 F2 # $ & 一 3 4 5 6 7 8 9. W E R T Y U FIL S D F H J K L V В N M and command B.