Question

Recall that a colouring of the vertices of an undirected graph is called proper if it does not assign the same colour to any pair of neighbouring vertices, and 3COL is the problem of deciding whether a graph is 3-colourable. Let X be the following problem. • Input: An undirected graph G. • Question: Does G have a proper 3-colouring that assigns red to more than 99% of the vertices? Prove that 3COL ≤p X (that is, 3COL is polynomially reducible to X) by giving an efficient algorithm that solves 3COL using a black-box algorithm for X

Answer 1

A Venn diagram for the given experiment can be created with two overlapping circles representing events C (green, blue, purple) and P (red, yellow, blue), and their intersection, which contains the outcome blue (C AND P). The union of these sets (C OR P) includes green, blue, purple, red, and yellow.

Step-by-step explanation:

To represent the experiment outcomes and the events C and P using a Venn diagram, we would draw two overlapping circles within a larger rectangle that represents the universal set of all outcomes. This universal set includes black, white, red, orange, yellow, green, blue, and purple.The first circle represents event C, which includes the outcomes green, blue, and purple. The second circle represents event P, which includes the outcomes red, yellow, and blue. The overlap between circles C and P includes only the outcome blue, which is C AND P. The union of circles C and P, which is C OR P, includes all the elements that are in either circle, so it will include green, blue, purple, red, and yellow.Each circle is drawn so that it overlaps with the other only if they share outcomes. The outcome blue, being shared, is placed in the intersecting region of the two circles. The rest of the outcomes are placed in their respective circles or outside the circles if they are not part of events C or P.