Question

A 2-coloring of an undirected graph with n vertices and m edges is the assignment of one of two colors (say, red or green) to each vertex of the graph, so that no two adjacent nodes have the same color. So, if there is an edge (u,v) in the graph, either node u is red and v is green or vice versa. Give an O(n + m) time algorithm (pseudocode!) to 2-colour a graph or determine that no such coloring exists. (Hint: Use BFS)

 The following shows examples of graphs that are and are not 2-colourable: 

Answer 1

Algorithm to find out whether a given graph is Birpartite or not (2-colorable or not )using Breadth First Search (BFS).
step 1. Assign RED color to the source vertex (putting into set U).
step 2. Color all the neighbors with GREEN color (putting into set V).
step 3. Color all neighbor's neighbor with RED color (putting into set U).
step 4. This way, assign color to all vertices such that it satisfies all the constraints of m way coloring problem where m = 2.
step 5. While assigning colors, if we find a neighbor which is colored with same color as current vertex,
then the graph cannot be colored with 2 vertices (or graph is not Bipartite)

As,we are using BFS, Time complexity is O(m+n) where m is number of vertices and n is number of edges.