Article

# Extremal graphs in some coloring problems

Department of Mathematics, Annamalai University, Annamalainagar 608 002, India
(Impact Factor: 0.56). 05/1998; 186(1-3):15-24. DOI: 10.1016/S0012-365X(97)00216-1
Source: DBLP

ABSTRACT

For a simple graph G with chromatic number χ(G), the Nordhaus-Gaddum inequalities give upper and lower bounds for χ(G)χ(Gc) and χ(G) + χ(Gc). 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(Gc) 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(Gc) is minimum.

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Available from: Venkataraman Yegnanarayanan, Sep 08, 2014
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• "Nordhaus–Gaddum inequalities have been established for numerous other graph parameters, such as the independence and edge-independence number [3] [8], list-colouring number [7] [10], diameter, girth, circumference, and edge-covering number [25], connectivity and edge-connectivity number [6], achromatic and pseudoachromatic number [1] [26], and arboricity [18] [23]. "
##### Article: Nordhaus–Gaddum inequalities for the fractional and circular chromatic numbers
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ABSTRACT: For a graph G on n vertices with chromatic number X(G), the Nordhaus-Gaddum inequalities state that [2 root n] <= chi(G) + chi((G) over bar) <= n + 1, and n <= chi(G) . chi((G) over bar) <= left perpendicular(n + 1/2)(2)right perpendicular. Much analysis has been done to derive similar inequalities for other graph parameters, all of which are integer-valued. We determine here the optimal Nordhaus-Gaddum inequalities for the circular chromatic number and the fractional chromatic number, the first examples of Nordhaus-Gaddum inequalities where the graph parameters are rational-valued. (c) 2008 Elsevier B.V. All rights reserved.
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Southeast Asian Bulletin of Mathematics 02/2000; 24(1):129-136. DOI:10.1007/s10012-000-0129-z
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##### Article: Graph colourings and partitions
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