tailieunhanh - Báo cáo toán học: "Compositions of Graphs Revisited"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Compositions of Graphs Revisited. | Compositions of Graphs Revisited Aminul Huq Department of Mathematics Brandeis University MS 050 Waltham MA 02453 aminul@ URL aminul Submitted Apr 1 2007 Accepted Jul 15 2007 Published Jul 19 2007 AMS classification 05A05 05C30 05A15 05A18. Abstract The idea of graph compositions which was introduced by A. Knopfmacher and M. E. Mays generalizes both ordinary compositions of positive integers and partitions of finite sets. In their original paper they developed formulas generating functions and recurrence relations for composition counting functions for several families of graphs. Here we show that some of the results involving compositions of bipartite graphs can be derived more easily using exponential generating functions. Keywords compositions bipartite graph Stirling number. 1 Introduction A composition of a graph G is a partition of the vertex set of G into vertex sets of connected induced subgraphs of G. Knopfmacher and Mays 2 found an explicit formula for C Km n the number of compositions of the complete bipartite graph Km n in the form m 1 C Km n 52 amzn i 1 n 1 where ai j is an array defined via the recurrences am 0 0 for any nonnegative integer m a0 1 1 a0 n 0 for any n 1 and otherwise am n m 1 1 E m 1 .A i i 0 7 7 am 1 i n 1 am 1 i n 1 We will derive this result using exponential generating functions and also show that we can express the coefficients am i explicitly in terms of the Stirling numbers of the second kind. We first need to describe some basic properties of exponential generating functions in two variables. We will use Stanley s notation 4 throughout this paper. THE ELECTRONIC JOURNAL OF COMBINATORICS 14 2007 N15 1 2 Exponential generating function in two variables Proposition 1. Given functions fig N X N K where K is a field of characteristic 0 we define a new function h N X N K by h X Y X f S T g U V where X and Y are finite sets and the sum is over all S T U and V such that X S U and Y T V . X and Y are .