Show that when the graph is a tree 2 color has a problem

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Reference no: EM131122648

Consider the k-color problem, which is to assign one out of k colors to each node of a graph so that for every arc (i, j), nodes i and j have different colors.

(a) Suppose we want to choose the colors of countries in a world map so that no two adjacent countries have the same color. Show that if the number of available colors is k, the problem can be formulated as a k-color problem.

(b) Show that the k-color problem has a solution if and only if the number of nodes can be partitioned in k or less disjoint subsets such that there is no arc connecting a pair of nodes from the same subset.

(d) Show that if each node has at most k - 1 neighbors, the k-color problem has a solution.

Reference no: EM131122648

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