According to the problem in the picture, how do I Prove that 3-Color ≤p 4-Color? We say a graph G = (V, E) has a k-coloring for some positive integer k if we can assign k differentcolors to vertices of G such that for every edge (v, w) E E, the color of v is different to the colorw. More formally, G = (V, E) has a k-coloring if there is a function f : V → {1, 2, ..., k} suchthat for every (v, w) E E, ƒ(v) # 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

It is defined as a common data structure that consists of a finite set of nodes (or vertices) and a…

say a R-colorin assign colors to vertices of G such that for every edge (v, w) E E, the color of v is different to the c w. More formally, G = (V, E) has a k-coloring if there is a function f : V → {1,2, ., k} suc that for every (v, w) E E, f(v) # 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

say a R-colorin assign colors to vertices of G such that for every edge (v, w) E E, the color of v is different to the c w. More formally, G = (V, E) has a k-coloring if there is a function f : V → {1,2, ., k} suc that for every (v, w) E E, f(v) # 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

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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A directed graph G= (V,E) consists of a set of vertices V, and a set of edges E such that each element e in E is an ordered pair (u,v), denoting an edge directed from u to v. In a directed graph, a directed cycle of length three is a triple of vertices (x,y,z) such that each of (x,y) (y,z) and (z,x) is an edge in E. Write a Mapreduce algorithm whose input is a directed graph presented as a list of edges (on a file in HDFS), and whose output is the list of all directed cycles of length three in G. Write the pseudocode for the mappers/reducers methods. Also, assuming that there are M mappers, R reducers, m edges and n vertices -- analyze the (upper-bound of the) communication cost(s).
Let G be a directed graph where each edge is colored either red, white, or blue. A walk in G is called a patriotic walk if its sequence of edge colors is red, white, blue, red, white, blue, and soon. Formally ,a walk v0 →v1 →...vk is a patriotic walk if for all 0≤i<k, the edge vi →vi+1 is red if i mod3=0, white if i mod3=1,and blue if i mod3=2. Given a graph G, you wish to find all vertices in G that can be reached from a given vertex v by a patriotic walk. Show that this can be accomplished by efficiently constructing a new graph G′ from G, such that the answer is determined by a single call to DFS in G′. Do not forget to analyze your algorithm.
Consider a set of real number pairs that represent a collection of V intervals on the real line. An interval graph with one vertex for each interval is defined by such a collection, with edges connecting vertices in the event that the associated intervals cross (have any points in common). Create a programme that creates V random intervals, each of length d, in the unit interval and then creates the interval graph that goes with it.Use a BST, as a tip.
An undirected graph G = (V, E) is called “k-colorable” if there exists a way to color the nodes withk colors such that no pair of adjacent nodes are assigned the same color. I.e. G is k-colorable iff there existsa k-coloring χ : V → {1, . . . , k}, such that for all (u, v) ∈ E, χ(u) ̸= χ(v) (the function χ is called a properk-coloring). The “k-colorable problem” is the problem of determining whether an input graph G = (V, E) isk-colorable. Prove that the 3-colorable problem ≤P the 4-colorable problem.
Consider an undirected graph G = (V;E). An independent set is a subset I V such that for any vertices i; j 2 I, there is no edge between i and j in E. A set i is a maximal independent set if no additional vertices of V can be added to I without violating its independence. Note, however, that a maximal independent sent is not necessarily the largest independent set in G. Let (G) denote the size of the largest maximal independent set in G. 1) What is (G) if G is a complete graph on n vertices? What if G is a cycle on n vertices?
Let G = (V, E) be a graph with vertex-set V = {1, 2, 3, 4, 5} and edge-set E = {(1, 2), (3, 2), (4, 3), (1, 4), (2, 4), (1, 3)}. (a) Draw the graph. Find (b) maximal degree, i.e. ∆(G), (c) minimal degree, i.e. δ(G), (d) the size of biggest clique, i.e. ω(G),(e) the size of biggest independent set, i.e. α(G), ter(f) the minimal number of colours needed to color the graph, i.e. χ(G).
Discrete mathematics. Let G = (V, E) be a simple graph4 with n = |V| vertices, and let A be its adjacency matrix of dimension n × n. We want to count the L-cycles : such a cycle, denoted by C = u0u1 · · · uL with uL = u0 contains L distinct vertices u0, . . . , uL-1 et L edges E(C) = {uiui+1 | 0 ≤ i ≤ L − 1} ⊆ E. Two cycles are distinct if the edge sets are different : C = C' if and only if E(C) = E(C'). We define the matrices D, T, Q, the powers of A by matrix multiplication : D = A · A = A2, T = A · D = A3, Q = A · T = A4. Consider the values on the diagonals. Prove that the number of 3-cycles N3 in the whole graph is N3 = 1/6 ∑ u∈V Tu,u
Discrete mathematics. Let G = (V, E) be a simple graph4 with n = |V| vertices, and let A be its adjacency matrix of dimension n × n. We want to count the L-cycles : such a cycle, denoted by C = u0u1 · · · uL with uL = u0 contains L distinct vertices u0, . . . , uL-1 et L edges E(C) = {uiui+1 | 0 ≤ i ≤ L − 1} ⊆ E. Two cycles are distinct if the edge sets are different : C = C' if and only if E(C) = E(C'). We define the matrices D, T, Q, the powers of A by matrix multiplication : D = A · A = A2, T = A · D = A3, Q = A · T = A4. Consider the values on the diagonals. Prove that for any vertex u ∈ V with degree d(u), d(u) = Du,u.
Let G = (V, E) be a directed graph, and let wv be the weight of vertex v for every v ∈ V . We say that a directed edgee = (u, v) is d-covered by a multi-set (a set that can contain elements more than one time) of vertices S if either u isin S at least once, or v is in S at least twice. The weight of a multi-set of vertices S is the sum of the weights of thevertices (where vertices that appear more than once, appear in the sum more than once).1. Write an IP that finds the multi-set S that d-cover all edges, and minimizes the weight.2. Write an LP that relaxes the IP.3. Describe a rounding scheme that guarantees a 2-approximation to the best multi-set
(a) Show that a graph G with at least three vertices is 2-connected if and only if any vertex and any edge of G lie on a common cycle of G: (b) Show that a graph G with at least three vertices is 2-connected if and only if any two edges of G lie on a common cycle
Throughout, a graph is given as input as an adjacency list. That is, G is a dictionary where the keysare vertices, and for a vertex v,G[v] = [u such that there is an edge going from v to u].In the case that G is undirected, for every edge u − v, v is in G[u] and u is in G[v]. 3. Write the full pseudocode for the following problem.Input: A directed graph G, and an ordering on the vertices given in a list A.Output: Is A a topological order? In other words, is there an i, j such that i < j and there is an edge fromA[j] to A[i]?
For the following simple graphs G=(V,E) (described by their vertex and edge sets) decide whether they are bipartite or not. If G is bipartite, then give its bipartition, and if it is not explain why. 1) V={a,b,c,d,e,f,g}, E={ag,af,ae,bf,be,bc,dc,dg,df,de} 2) V={a,b,c,d,e,f,g,h}, E={ac,ad,ah,bc,bh,bd,ec,ef,eg,hf,hg}