tailieunhanh - Báo cáo toán học: "A density result for random sparse oriented graphs and its relation to a conjecture of Woodall"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: A density result for random sparse oriented graphs and its relation to a conjecture of Woodall | A density result for random sparse oriented graphs and its relation to a conjecture of Woodall Jair Donadelli Departamento de Informatica Universidade Federal do Parana Centro Politecnico 81531-990 Curitiba PR Brazil jair@ Yoshiharu Kohayakawa Instituto de Matematica e Estatistica Universidade de Sao Paulo Rua do Matão 1010 05508-090 Sao Paulo SP Brazil yoshi@ Submitted July 24 2000 Accepted November 16 2002 MR Subject Classifications 05C20 05C38 05C80 Abstract We prove that for all 3 and 3 0 there exists a sparse oriented graph of arbitrarily large order with oriented girth and such that any 1 2 3 proportion of its arcs induces an oriented cycle of length . As a corollary we get that there exist infinitely many oriented graphs with vanishing density of oriented girth such that deleting any 1 -fraction of their edges does not destroy all their oriented cycles. The proof is probabilistic. 1 Introduction We call the pair G V E an oriented graph if the set of vertices V is a finite set and the set of oriented edges E c V X V which we call arcs is such that V V 2 E for any v 2 V and if U V 2 E then V U 2 E. Our notation will basically follow 1 . The main result of this note Theorem 1 is related to a conjecture of Woodall which we now describe. Given an oriented graph G V E we say that a subset B c E of E is an oriented cut in G if there exists a subset W c V of V such that B E G 0 W X W Supported by a CNPq PhD Scholarship Proc. 141633 1998-0 . Research supported in part by FAPESP Proc. 96 04505-2 MCT FINEP CNPq through ProNEx Programme Proc. CNPq 664107 1997-4 and by CNPq Proc. 300334 93-1 and 468516 2000-0 . THE ELECTRONIC JOURNAL OF COMBINATORICS 9 2002 R45 1 and E G n W X W where W V W. A subset F c E of E is a transversal of the family of oriented cuts of G if F n B for all oriented cuts B in G. In 1978 Woodall 7 conjectured that for any oriented graph G a minimum oriented cut in G has cardinality equal to the maximum cardinality of a family of