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Báo cáo toán học: "On the number of genus one labeled circle trees"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: On the number of genus one labeled circle trees. | On the number of genus one labeled circle trees Karola Meszaros Massachusetts Institute of Technology karola@math.mit.edu Submitted Sep 25 2005 Accepted Sep 17 2007 Published Oct 5 2007 Mathematics Subject Classihcation 05C30 05C75 05C10 05A99 Abstract A genus one labeled circle tree is a tree with its vertices on a circle such that together they can be embedded in a surface of genus one but not of genus zero. We dehne an e-reduction process whereby a special type of subtree called an e-graph is collapsed to an edge. We show that genus is invariant under e-reduction. Our main result is a classihcation of genus one labeled circle trees through e-reduction. Using this we prove a modihed version of a conjecture of David Hough namely that the number of genus one labeled circle trees on n vertices is divisible by n or n 2. Moreover we explicitly characterize when each of these possibilities occur. 1 Introduction Graphical enumeration arises in a variety of contexts in combinatorics 2 and naturally so in the realm of combinatorial objects with interesting topological properties 5 . We provide a new classification of genus one circle trees and address a question raised by Hough 3 about their number. Our study is motivated by numerous results in the study of partitions and trees of a certain genus as well as results about the genuses of maps and hypermaps 1 6 7 8 . The following two definitions are discussed in 3 in great detail we shall use the definition of a labeled circle tree throughout the paper whereas we shall mostly use an alternate less technical definition for the genus of a circle tree. Definition 1. A labeled circle tree l-c-tree on n points is a tree with its n vertices labeled 1 through n on a circle in a counterclockwise direction and its edges drawn as straight lines within the circle. THE ELECTRONIC JOURNAL OF COMBINATORICS 14 2007 R68 1 Definition 2. The genus g of a l-c-tree T on n points is defined to be g a 1 2 n-1-z a z a 1 ơ where a is the matching