Extremal graphs in some coloring problems.

Department of Mathematics, Annamalai University, Annamalainagar 608 002, India
Discrete Mathematics (Impact Factor: 0.58). 01/1998; 186:15-24. 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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    Southeast Asian Bulletin of Mathematics 05/2011; 35:431-438.
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    ABSTRACT: A "clique minor" in a graph G can be thought of as a set of connected subgraphs in G that are pairwise disjoint and pairwise adjacent. The "Hadwiger number" h(G) is the maximum cardinality of a clique minor in G. This paper studies clique minors in the Cartesian product G*H. Our main result is a rough structural characterisation theorem for Cartesian products with bounded Hadwiger number. It implies that if the product of two sufficiently large graphs has bounded Hadwiger number then it is one of the following graphs: - a planar grid with a vortex of bounded width in the outerface, - a cylindrical grid with a vortex of bounded width in each of the two `big' faces, or - a toroidal grid. Motivation for studying the Hadwiger number of a graph includes Hadwiger's Conjecture, which states that the chromatic number chi(G) <= h(G). It is open whether Hadwiger's Conjecture holds for every Cartesian product. We prove that if |V(H)|-1 >= chi(G) >= chi(H) then Hadwiger's Conjecture holds for G*H. On the other hand, we prove that Hadwiger's Conjecture holds for all Cartesian products if and only if it holds for all G * K_2. We then show that h(G * K_2) is tied to the treewidth of G. We also develop connections with pseudoachromatic colourings and connected dominating sets that imply near-tight bounds on the Hadwiger number of grid graphs (Cartesian products of paths) and Hamming graphs (Cartesian products of cliques).
    New York Journal of Mathematics 11/2007; · 0.44 Impact Factor
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    ABSTRACT: For a graph G on n vertices with chromatic number χ(G), the Nordhaus–Gaddum inequalities state that , and . 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.
    Discrete Mathematics. 01/2009;

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