tailieunhanh - Báo cáo toán học: "Regular spanning subgraphs of bipartite graphs of high minimum degree"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Regular spanning subgraphs of bipartite graphs of high minimum degree. | Regular spanning subgraphs of bipartite graphs of high minimum degree Bela Csaba y Submitted Aug 1 2007 Accepted Oct 5 2007 Published Oct 16 2007 Mathematics Subject Classihcation 05C70 Abstract Let G be a simple balanced bipartite graph on 2n vertices ỗ ỗ G n and p0 S 2S 1. If ỗ 1 2 then G has a LponJ-regular spanning subgraph. The statement is nearly tight. 1 Introduction In this paper we will consider regular spanning subgraphs of simple graphs. We mostly use standard graph theory notation V G and E G will denote the vertex and the edge set of a graph G respectively. The degree of x 2 V G is denoted by degG x we may omit the subscript Ỗ G is the minimum degree of G. We call a bipartite graph G A B with color classes A and B balanced if IAI I B . For X Y c V G we denote the number of edges of G having one endpoint in X and the other endpoint in Y by e X Y . If T c V G then G t denotes the subgraph we get after deleting every vertex of V T and the edges incident to them. Finally Kr s is the complete bipartite graph on color classes of size r and s for two positive integers r and s. If f V H Z is a function then an f -factor is a subgraph H of the graph H such that degH x f x for every x 2 V H . Notice that when f r for some r 2 Z then H0 is an r-regular subgraph of H. There are several results concerning f -factors of graphs. Perhaps the most notable among them is the theorem of Tutte 7 . Finding f-factors is in general not an easy task even for the case f is a constant and the graph is regular see eg. 1 . In this paper we look for f -factors in not necessarily regular bipartite graphs with large minimum degree for f r. Analysis and Stochastics Research Group of the Hungarian Academy of Sciences University of Szeged Hungary and Department of Mathematics Western Kentucky University Bowling Green KY USA yPartially supported by OTKA T049398 and by Hungarian State Eotvos Fellowship. THE ELECTRONIC JOURNAL OF COMBINATORICS 14 2007 N21 1 Theorem 1 Let G A B be a .