If a graph G is k-colorable and k<n, for any integer n then it is ___________

If a graph G is k-colorable and k<n, for any integer n then it is ___________ ans. n2 nodes (k3+n3+1) nodes n-colorable (k+n)-colorable

(a) Structural part of a proof that proceeds directly: 1. Assume G is a finite graph that is planar and has no loops. 2. Show that G can be colored with at most 4 colors. 3. Use the fact that any planar graph can be colored with at most 4 colors. 4. Demonstrate that G satisfies the conditions for being colored with at most 4 colors. 5. Conclude that the chromatic number of G is at most 4. (b) Structural part of a proof that proceeds via the contrapositive: 1. Assume G is a finite graph with a chromatic number greater than 4. 2. Show that G is either not planar or has loops. 3. Use the fact that any graph with a chromatic number greater than 4 must contain a subgraph that is not planar or has loops. 4. Demonstrate that G contains such a subgraph. 5. Conclude that if the chromatic number of G is greater than 4, then G is either not planar or has loops. ####

What does the chromatic number of a graph represent?Select one:a. The minimum number of colors needed to color the graph so that no two adjacent vertices have the same color.b. The number of vertices in the graph.c. The number of colors used to color each edge of the graph.d. The total number of possible colorings for a given graph.

Kevin is a computer science enthusiast exploring the intricacies of graph theory. He is actively developing a program that, given an adjacency matrix, calculates the chromatic number of an undirected graph. The chromatic number is the minimum count of colors needed to color the graph vertices, ensuring neighboring vertices have distinct colors.Your role is to support Kevin in implementing this algorithm, facilitating his understanding of chromatic numbers and their application in practical problem-solving.Input format :The first line contains an integer v, representing the number of vertices in the graph.The next v lines contain the adjacency matrix of the graph, where each line contains v space-separated integers (0 or 1).Output format :The output prints "Chromatic Number of the graph is: " followed by an integer representing the chromatic number of the graph.Refer to the sample output for the formatting specifications.Code constraints :1 ≤ v ≤ 50Sample test cases :Input 1 :40 1 1 11 0 1 01 1 0 11 0 1 0Output 1 :Chromatic Number of the graph is: 3Input 2 :50 1 0 1 01 0 1 0 10 1 0 1 01 0 1 0 10 1 0 1 0Output 2 :Chromatic Number of the graph is: 2Input 3 :50 1 1 0 01 0 1 1 01 1 0 1 10 1 1 0 10 0 1 1 0Output 3 :Chromatic Number of the graph is: 3write an accurate code for this problem in c following given input and output formats

Prove that simple graph with n vertices and k components can have at most (n-k) (n-k+l)/2 edges.