Design a polynomial-time algorithm for the graph 2-coloring problem: deter-mine whether vertices of a given graph can be colored in no more than two colors so that no two adjacent vertices are colored the same color.

4-Clique Problem The clique problem is to find cliques in a graph. A clique is a set of vertices that are all adjacent - connected - to each other. A 4-clique is a set of 4 vertices that are all connected to each other. So in this example of the 4-Clique Problem, we have a 7-vertex graph. A brute-force algorithm has searched every possible combination of 4 vertices and found a set that forms a clique: https://en.wikipedia.org/wiki/Clique_problem   You should read the Wikipedia page for the Clique Problem (and then read wider if need be) if you need to understand more about it. Note that the Clique Problem is NP-Complete and therefore when the graph size is large a deterministic search is impractical. That makes it an ideal candidate for an evolutionary search. For this assignment you must suppose that you have been tasked to implement the 4-clique problem as an evolutionary algorithm for any graph with any number of vertices (an n-vertex graph). The algorithm succeeds if it finds a…

Note: Solve the following question and please don't repost previous answers and don't give a computer generated answer.   Problem: Draw a simple, connected, directed, weighted graph with 9 vertices and 17 edges, each with unique edge weights. Identify one vertex as a “start” vertex and illustrate a running of Dijkstra’s algorithm on this graph.

An undirected graph G = (V,E) is said to be k-colorable if all the vertices of G can be colored using k different colors such that no two adjacent vertices have the same color. Design an algorithm that runs in time O(n + e) to color a graph with two colors or determine that the graph is not 2-colorable.

an undirected graph is called complete if every shares an edge with every other vertex.

Write an algorithm using depth-first search that determines if a graph is bipartite. What is the complexity of your algorithm?

Why does the "Bellman-Ford Algorithm" get used instead of the "Dijkstra Algorithm" to find a graph's negative weight cycle?