tailieunhanh - Báo cáo toán học: "Towards the Distribution of the Size of a Largest Planar Matching and Largest Planar Subgraph in Random Bipartite Graphs"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Towards the Distribution of the Size of a Largest Planar Matching and Largest Planar Subgraph in Random Bipartite Graphs. | Towards the Distribution of the Size of a Largest Planar Matching and Largest Planar Subgraph in Random Bipartite Graphs MARCOS KIWI MARTIN LOEbL Depto. Ing. Matematica and Dept. of Applied Mathematics and Ctr. Modelamiento Matematico UMI2807 CNRS Institute of Theoretical Computer Science ITI University of Chile Charles University Correo 3 Santiago 170-3 Chile Malostranske nam. 25 118 00 Praha 1 e-mail mkiwi@ Czech Republic e-mail loebl@ Submitted May 31 2007 Accepted Oct 16 2008 Published Oct 20 2008 Mathematics Subject Classification 05A15 Abstract We address the following question When a randomly chosen regular bipartite multigraph is drawn in the plane in the standard way what is the distribution of its maximum size planar matching set of non-crossing disjoint edges and maximum size planar subgraph set of non-crossing edges which may share endpoints The problem is a generalization of the Longest Increasing Sequence LIS problem also called Ulam s problem . We present combinatorial identities which relate the number of r-regular bipartite multigraphs with maximum planar matching maximum planar subgraph of at most d edges to a signed sum of restricted lattice walks in Zd and to the number of pairs of standard Young tableaux of the same shape and with a descend-type property. Our results are derived via generalizations of two combinatorial proofs through which Gessel s identity can be obtained an identity that is crucial in the derivation of a bivariate generating function associated to the distribution of the length of LISs and key to the analytic attack on Ulam s problem . Finally we generalize Gessel s identity. This enables us to count for small values of d and r the number of r-regular bipartite multi-graphs on n nodes per color class with maximum planar matchings of size d. Our work can also be viewed as a first step in the study of pattern avoidance in ordered bipartite multi-graphs. Keywords Gessel s identity longest increasing .