Abstract

For a simple graph G with chromatic number χ( G), the Nordhaus-Gaddum inequalities give upper and lower bounds for χ( G) χ( G c ) and χ( G) + χ( G c ). Based on a characterization by Fink of the extremal graphs G attaining the lower bounds for the product and sum, we characterize the extremal graphs G for which A( G) B( G c ) is minimum, where A and B are each of chromatic number, achromatic number and pseudoachromatic number. Characterizations are also provided for several cases in which A( G) + B( G c ) is minimum.

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