Muhammad Kashif

Chromatic Polynomials and Chromaticity of Some Linear H-hypergraphs Muhammad Kashif

For a century, one of the most famous problems in mathematics was to prove the four-color theorem. In 1912, George Birkhoff proposed a way to tackling the four-color conjecture by introduce a function P(M, t), defined for all positive integer t, to be the number of proper t-colorings of a map M. This function P(M, t) in fact a polynomial in t is called chromatic polynomial of M. If one could prove that P(M, 4)>0 for all maps M, then this would give a positive answer to the four-color problem. In this book, we have proved the following results: (1) Recursive form of the chromatic polynomials of hypertree, Centipede hypergraph, elementary cycle, Sunlet hypergraph, Pan hypergraph, Duth Windmill hypergraph, Multibridge hypergraph, Generalized Hyper-Fan, Hyper-Fan, Generalized Hyper-Ladder and Hyper-Ladder and also prove that these hypergraphs are not chromatically uniquein the class of sperenian hypergraphs. (2) Tree form and Null graph representation of the chromatic polynomials of elementary cycle, uni-cyclic hypergraph and sunflower hypergrpah. (3) Generalization of a result proved by Read for graphs to hypergraphs and prove that these kinds of hypergraphs are not chromatically unique.