Extremal graphs in some coloring problems

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


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.

Download full-text


Available from: Venkataraman Yegnanarayanan, Sep 08, 2014
  • Source
    • "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]. "
    [Show abstract] [Hide abstract]
    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.
    Discrete Mathematics 04/2009; 309(8-309):2223-2232. DOI:10.1016/j.disc.2008.04.052 · 0.56 Impact Factor
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: The pseudoachromatic number of a graph G is the maximum size of a vertex partition of G (where the sets of the partition may or may not be independent) such that, between any two distinct parts, there is at least one edge of G. This parameter is determined for graphs such as cycles, paths, wheels, certain complete multipartite graphs, and for other classes of graphs. Some open problems are raised.
    Southeast Asian Bulletin of Mathematics 02/2000; 24(1):129-136. DOI:10.1007/s10012-000-0129-z
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: In this paper we have investigated mainly the three colouring parameters of a graph G, viz., the chromatic number, the achromatic number and the pseudoachromatic number. The importance of their study in connection with the computational complexity, partitions, algebra, projective plane geometry and analysis were briefly surveyed. Some new results were found along these directions. We have rede0ned the concept of perfect graphs in terms of these parameters and obtained a few results. Some open problems are raised.
    Theoretical Computer Science 07/2001; 263(1-2):59-74. DOI:10.1016/S0304-3975(00)00231-0 · 0.66 Impact Factor
Show more