In Algorithms Given a graph G = ( V , E ) , a K – coloring is a function c : V – > $ 1 , 2 , … , K ; such that c ( U ) * c ( V ) forevery edge ( U , V ) { E . In other words the number 1 , 2 . .. . K represent the k colors and adjacentvertices must have different colors . The decision problem K – COLOR asks if a graph can becolored with at most K colors .a . The 2 – COLOR decision problem is in P. Describe an efficient algorithm to determine if agraph has a 2 – coloring . What is the running time of your algorithm ?"b . It is known that the 3 – COLOR decision problem is NP- complete by using a reduction fromSAT . Use the fact that 3 – COLOR is NP- complete to prove that 4 – COLOR is NP- complete .

# Algorithms on different colors

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