Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của tạp chí toán học quốc tế đề tài: Colouring 4-cycle systems with specified block colour patterns: the case of embedding P3-designs. | Colouring 4-cycle systems with specified block colour patterns the case of embedding Fa-designs Gaetano Quattrocchi Dipartimento di Matematica e Informatica Universita di Catania Catania ITALIA quattrocchi@ Submitted January 20 2001 Accepted June 5 2001 Abstract A colouring of a 4-cycle system V B is a surjective mapping ộ V r. The elements of r are colours. If r m we have an m-colouring of V B . For every B 2 B let ộ B fộ x x 2 B . There are seven distinct colouring patterns in which a 4-cycle can be coloured type a x X XX monochromatic type b x X xd two-coloured of pattern 3 1 type c x X two-coloured of pattern 2 2 type d xd X mixed two-coloured type e x X 24 three-coloured of pattern 2 1 1 type f xd X 4 mixed three-coloured type g xd4 four-coloured or polychromatic . Let S be a subset of fa b C d e f g . An m-colouring ộ of V B is said of type S if the type of every 4-cycle of B is in S. A type S colouring is said to be proper if for every type a 2 S there is at least one 4-cycle of B having colour type a. We say that a P V 3 1 W P is embedded in a 4-cycle system of order n V B if every path p ai a2 a3 2 P occurs in a 4-cycle a1 a2 a3 X 2 B such that x 2 W. In this paper we consider the following spectrum problem given an integer m and a set S c fb d f determine the set of integers n such that there exists a 4-cycle system of order n with a proper m-colouring of type S note that each colour class of a such colouration is the point set of a P3-design embedded in the 4-cycle system . We give a complete answer to the above problem except when S fb . In this case the problem is completely solved only for m 2. AMS classification 05B05. Keywords Graph design m-colouring Embedding Path Cycle. Supported by MURST Cofinanziamento Strutture geometriche combinatorie e loro applicazioni and by . . Italy. THE ELECTRONIC JOURNAL OF COMBINATORICS 8 2001 R24 1 1 Introduction Let G be a subgraph of Kv the complete undirected graph on v vertices. A .
