Description

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.