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.

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Available from: Venkataraman Yegnanarayanan, Sep 08, 2014
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    • "In[1,14], the relationship between Ψ(G 1 ∨ G 2 ) and Ψ(G 1 ) + Ψ(G 2 ) were discussed. For more results related to this topic, one can refer to[2,10111213. In this paper, the strong pseudoachromatic numbers of C n , P n , K n , W n , L(W n ), F n , L(F n ), and K n1,n2,...,ns are determined. "
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    ABSTRACT: Given a graph G, partition V(G) into k disjoint subsets V-1,..,V-k such that (a) there is at least one edge in G connecting V-i and V-3 (i not equal j), and (b) each C[V-i] (the subgraph of G induced by V-i) contains at least one edge for any i, j is an element of {1,2,..., k}. The maximum number k for which such a partition exists is called the strong pseudoachromatic number of G, and is denoted by psi* (G). In this paper, we determine the strong pseudoachromatic number of some special family of graphs. We also investigate the relationship between the strong pseudoachromatic number of G and its subgraphs obtained by deleting an edge or a vertex of G, and find some bounds for psi* (G). Moreover, we characterize the class of all graphs G that psi*(G v H) =,psi* (G)+ psi*(H) for any G, H is an element of g, where G v H denotes the join of G and H.
    Full-text · Article · Nov 2015 · Utilitas Mathematica
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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]. "
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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.
    Full-text · Article · Apr 2009 · Discrete Mathematics
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    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.
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