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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    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 01/2001; 263:59-74. · 0.49 Impact Factor
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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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