1. Using induction, prove that any planar graph can be colored in no more than 6 colors where no two vertices connected by an edge share the same color.

Q: a. Prove by Kuratowski's theorem that the two graphs below are not planar E F 2 7 8 b. Show that a…

A: A Kuratowski's theorem state that a finite graph G is planar if it is not possible to subdivide the…

Q: (b) Is it possible to have a 3-regular graph with six vertices? If such a graph is possible, draw an…

A: b) 3 regular graph with 6 vertices is possible. First, connect all 6 vertices to forma cycle Then…

Q: Draw a directed acyclic graph with 6 vertices and 14 edges (IF EXIST), IF Doesn't Exist, Give Your…

A: For a directed acyclic graph, we can have a maximum of n(n-1)/2 number of edges if n is number of…

Q: Show that a, simple planer connected graph with 17 edges and 10 vertices cannot be property coloured…

A: Simple planer Connected graph Having  17 edges and 10 vertices and it cannot property colored with…

Q: Give an example of a graph (with or without weights on the edges) where the betweenness and…

A: Graph: Actually, graph is a pictorial representation or diagram that represents data or values...

Q: Draw a graph with four vertices of degrees 1, 2, 2, and 7 or else explain why no such graph exists.

A: It states that the sum of degrees of the vertices of a graph is equal to twice the number of edges.…

Q: Draw the complete bipartite graph K4, 4. Label its vertices 1, 2, 3, ., n and list the edges in…

A: bipartite graph: In this, vertex are  divided into two disjoint  and independent sets…

Q: 3) The graph k-coloring problem is stated as follows: Given an undirected graph G = (V,E) with N…

A:

Q: 3) The graph k-coloring problem is stated as follows: Given an undirected graph G (V,E) with N…

A: Given an undirected graph G=(V,E) with vertices and M edges and an integer K. Graph k-coloring…

Q: Consider a simple undirected graph with 5 vertices, and let the degrees of these vertices be: 4, 3,…

A: Yes, such graph exists.

Q: Explain the Graph-Coloring problem and draw the state space tree for m = 3 colors and n = 4 vertices…

A: The Answer is

Q: Show the formal notation V(...) and E(...) of the graph that is described by the following adjacency…

A: The answer is

Q: Give an example of each of the following: (4) i) a nonsimple plane graph with its dual a simple…

A: In these questions, we are asked about nonsimple plane graph and its simple dual graph. And we also…

Q: A graph G has 21 edges, 3 vertices of degree 4 and other vertices are of degree 3. Find the number…

A: Introduction :

Q: Show the formal notation V(...) and E(...) of the graph that is described by the following adjacency…

A: ANSWER : In graph theory and computer science, an adjacency matrix is a square matrix used to…

Q: Consider a simple undirected graph with 5 vertices, and let the degrees of these vertices be: 4, 3,…

A: Given: A simple undirected graph with 5 vertices and the degrees of these vertices be 4, 3, 3, 2,…

Q: 2. If G is an undirected graph with 10 vertices, what is the largest number of edges it might have…

A: GIVEN:

Q: If a simple graph G has 8 vertices and 8 edges, how many edges does the complementary graph of G…

A: Given: Number of edges in graph G, |E(G)| = 8 Number of vertices in graph G, n = 8 The number of…

Q: Show the formal notation V(..) and E(..) of the graph that is described by the following adjacency…

A: Given :

Q: Given the following two graphs determine whether they are isomorphic. If so, show the isomorphism…

A: The given graphs are:

Q: a graph with five vertices in which each vertex is of degree 3.

A: It is not possible to draw a graph with five vertices in which each vertex is of degree 3.

Q: Graph Complement and Planarity, (a) Let G be a 4-regular plane graph with 10 faces. Determine how…

A: Answer is given below .

Q: From the graph G given below identify the number of vertices and edges A B C Vertices 3 and Edges 4…

A: In a diagram of a graph, a vertex is usually represented by a circle with a label, and an edge is…

Q: Say that a graph G has a path of length three if there exist distinct vertices u, v, w, t with edges…

A: Given that, Graph G contains a path of length 3 and the vertices are u, v, w, t with edges (u,v),…

Q: (a) Given the following graph G(V, E), find the shortest path from A to other vertices in the graph.…

A: here is your answer

Q: Let G be the graph below, with 11 vertices and 20 edges. Clearly explain why x(G) = 4.

A: The chromatic number of a graph is the minimum number of colour needed to colour all the vertices of…

Q: Either draw the undirected graph with the given specifications or explain why no such graph exists.…

A: It is defined as a nonlinear data structure, compared to arrays, linked lists, stacks and queues…

Q: 11. With the above information provided, draw a graph for the data provided. Indicate the  weights…

A: The first question will be solved. Please upload another question again. 11) 1) Weighted directed…

Q: The Petersen graph is most commonly drawn as a pentagon with a pentagram inside, with five spokes.…

A: Chromatic number is 3

Q: (a) Given the following graph G(V, E), find the shortest path from A to every other vertices in the…

A: here  is your answer

Q: In each of the following questions V=[n]={1,2,...,n}. What is the number of (simple undirected)…

A: Given: In each of the following questions V=[n]={1,2,...,n}. What is the number of (simple…

Q: Give, with justification, a formula for the minimum number of edges that must be added to a general…

A: In order for an Euler Circuit to exist in the graph, every node must have an even degree because…

Q: 1. Consider the directed acyclic graph (DAG) D shown below. (a) Write down the adjacency matrix A…

A: Adjacency Matrix Representation: If a directed graph G consists of n vertices then the adjacency…

Q: 1- Compare the required space for graph representation in terms of edges and vertices using a) list…

A: As per our guidelines, we are supposed to answer only one question. Kindly report other questions as…

Q: 1- Compare the required space for graph representation in terms of edges and vertices using a) list…

A: We are authorized to answer one question at a time since you have not mentioned which question you…

Q: The following conditions must be met for a graph to include an Euler circuit: (1) it must be linked,…

A: Path of Euler An Euler route is a path that only passes over each graph's edge once. An Euler…

Q: a) How many edges does a K50 graph have? Answer: b) What is the degree of each vertex of a K50…

A: GIVEN:

Q: a) How many edges does a K50 graph have? Answer: b) What is the degree of each vertex of a K50…

A: A graph is consists of a number of edges and vertices. it is represented as G = (V, E). A bipartite…

Q: Let G be a connected planar graph with 32 vertices, 50 edges and degree of each region is k. Find…

A: the value of region as k is calculated in step 2.

Q: Graph Coloring, continued (b) Find the chromatic number of the cubical graph (shown below). Be sure…

A: Chromatic Number: Chromatic Number is the minimum number of colors required to color any graph such…

Q: Consider the statement “Every connected graph has one or more vertices of degree 2." Each of the…

A: The number of edges connected to the vertex. Any connected graph has minimum 1 edge. If the graph…

Q: ) Let G be a simple plane graph with 11 vertices. (i) Prove that G has a vertex of degree at most 4.…

A: SUMMARY: -Hence, We discussed all the points.

Q: |(a) Draw an undirected graph G1 represented by the following adjacency matrix: 0 1 1 0 1 1 0 0 0 1…

A:

Q: 4) For the given graph below, prove that this graph does not contain any negative cyc 3 6. -2 -2 4…

A: About negative cycle: ========================================================================

Q: All of the following statements are false. Provide a counterexample for each one. iii. If it can be…

A: The true statement for the both statements are given below. iii. If it can be shown that there is…

Q: Choose a graph G with with vertices v0, v1, .., v9 and find a minimum spanning tree for G: 1. Use…

A: Choose a graph G with vertices v0,v1,.......,v9 and find a minimum spanning tree for G: 1. Use…

Q: Show that there are eleven nonisomorphic simple graphs on four vertices.

A: The answer is

Q: Let DIST (u, v) denote the distance between vertex u and v. It is well known that distances in…

A: Given that, DIST(u, v) is the distance between the vertex u and v. If we consider any three vertices…

Q: he requirements for a graph to contain an Euler circuit are that (1) the graph must be connected,…

A: Euler path An Euler path is a path that moves through every edge of a graph only once. An Euler…

Q: Let G be a graph such that every vertex has degree 4 and the number of edges is 12. How many…

A: Actually, graph consisting of edges and nodes.