Question

Let G be a bipartite graph with n nodes and k connected compo­ nents. You (mutually) independently color each of the nodes of G red or black with equal probabilities. What is the probability that your coloring is a valid 2-coloring of G? (Hint: the answer does not depend on the number of edges.)

Answer 1

Given that the graph G has to be independently coloured with equal probabilities of red and black which is only possible with even number of vertices.For example let us assume a rectangle which consists of four vertices and we can colour two of its vertices in red and the remianing two with black.So from the above example it is evident that the vertices should be even.We also know that every acyclic graph can be bipartite but a cyclic graph to be biparite its all cycles should be even.A bipartite graph is something which consists of two sets in which each set is coloured with the same colour.So, for a bipartite to be two coloured the probability is always greatre than 1/2.