tailieunhanh - Báo cáo toán học: "A note on the number of edges guaranteeing a C4 in Eulerian bipartite digraph"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: A note on the number of edges guaranteeing a C4 in Eulerian bipartite digraphs. | A note on the number of edges guaranteeing a C4 in Eulerian bipartite digraphs Jian Shen Department of Mathematics Southwest Texas State University San Marcos TX 78666 email js48@ Raphael Yuster Department of Mathematics University of Haifa at Oranim Tivon 36006 Israel email raphy@ Submitted November 17 2001 Accepted March 21 2002. MR Subject Classifications 05C20 05C35 05C45 Abstract Let G be an Eulerian bipartite digraph with vertex partition sizes m n. We prove the following Turán-type result If e G 2mn 3 then G contains a directed cycle of length at most 4. The result is sharp. We also show that if e G 2mn 3 and no directed cycle of length at most 4 exists then G must be biregular. We apply this result in order to obtain an improved upper bound for the diameter of interchange graphs. 1 Introduction All graphs considered here are finite directed and contain no parallel edges. For standard graph-theoretic terminology the reader is referred to 1 . In this paper we consider the most basic Turan-type problem in bipartite digraphs namely specifying conditions on the cardinality of the edge-set of the digraph that guarantee the existence of a directed simple cycle of length at most four. As usual in Turan type problems in directed graphs one must impose constraints relating the indegree and outdegree of a vertex in order to avoid trivialities if no such constraints exist then one may not have short directed cycles at all THE ELECTRONIC JOURNAL OF COMBINATORICS 9 2001 N6 1 even if the graph is very dense the extreme case being an acyclic orientation of a complete bipartite graph . The most interesting and natural constraint is the requirement that the digraph be Eulerian namely the indegree of a vertex must be equal to its outdegree. Let b m n denote the maximum integer such that there exists an Eulerian bipartite digraph with vertex partition sizes m n having b m n edges and no directed cycle of length at most 4. A biregular bipartite .