Question: =+23. Consider a random graph with n nodes. Between every pair of nodes, independently introduce an edge with probability p. The graph is said to

=+23. Consider a random graph with n nodes. Between every pair of nodes, independently introduce an edge with probability p. The graph is said to be k colorable if it is possible to assign each of its nodes one of k colors so that no pair of adjacent nodes share the same color.

The chromatic number X of the graph is the minimum value of k.

Demonstrate that Pr[|X − E(X)| ≥ λ] ≤ 2e−λ2/(2n). (Hint: Consider the martingale Xi = E(X | Y1,...,Yi), where Yi is the random set of edges connecting node i to nodes 1,...,i − 1.)

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