Multiplicativity of acyclic digraphs

AbstractA homomorphism of a digraph to another digraph is an edge preserving vertex mapping. A digraph W is said to be multiplicative if the set of digraphs which cannot be homomorphically mapped to W is closed under categorical product. We discuss the necessary conditions for a digraph to be multiplicative. Our main result is that almost all acyclic digraphs which have a Hamiltonian path are nonmultiplicative. We conjecture that almost all digraphs are nonmultiplicative

The Chromatic Number of Product Graphs

This main goal of this paper is to answer the following question: for which graph products can the chromatic number of the product graph be written as a function of the chromatic numbers of the original two products. We approach this\ud question first by examining the Cartesian product of two graphs and showing that the chromatic number of the Cartesian product of two graphs is equal to the maximum of the chromatic numbers of the "factor" graphs. Then the following\ud section introduces the direct product and Hedetniemi's conjecture. This paper also discusses the strong product, lexicographical product, and Cartesian sum and the relationship between the edge sets of these products and the implications of the relationship to computing the chromatic number of these product graphs. We then return to Hedetniemi's conjecture and prove the conjecture\ud for the product of a 4-chromatic graph with an arbitrary finite graph. Finally, using the general product definition, we characterize the products for which the chromatic number of the product graph cannot be written as a function of the\ud chromatic numbers of the "factor" graphs, for all possible products with the\ud exception of the direct product