Partition of C 4 -Designs into Minimum and Maximum Number of P 3 -Designs Gaetano Quattrocchi 1; and Zsolt Tuza 2;y 1 Dipartimento di Matematica e Informatica, Universita` di Catania, viale A. Doria 6, 95125 Catania, Italia. e-mail: quattrocchi@dmi.unict.it 2 Computer and Automation Institute, Hungarian Academy of Sciences, H-1111 Budapest, Kende u. 13-17, Hungary. e-mail: tuza@sztaki.hu Abstract. Let C be a C 4 -design of order n and index k, on the vertex set V , jV j¼ n. If V 1 [[ V m ¼ V is a partition of the vertex set, such that the intersections of the C 2 C with V i form a P 3 -design of order jV i j and the same index k, for each 1  i  m, then 2  m  log 3 ð2n þ 1Þ. The minimum bound is best possible for every k. The maximum bound is best possible for k ¼ 2, and hence also for every even k. Key words. Cycle system, Path design, Embedding 1. Introduction Let C 4 denote the 4-cycle, and P 3 the path with three vertices and two edges. We study vertex partitions of C 4 -designs where each 4-cycle has three vertices in the same partition class and one vertex in a different class. In this way, all the partition classes induce embedded P 3 -designs [12,11]. To be consistent with this terminology of embedding, we also consider a one-element class (if it occurs) an embedded P 3 - design, despite it is a degenerate case. The assumption on the 4-cycles implies, however, that there may occur at most one degenerate class in the partition. It is well-known that a C 4 -design of order n exists if and only if either n ¼ 1 or n  4 and n  1 (mod 8) if k is odd, k n 2   0 (mod 4) if k  2 is even. Moreover, a P 3 -design of order n and index k exists if and only if n ¼ 1 or n  3 and k n 2  is even (i.e., k is even or n  0 or 1 (mod 4)). There are many variants of the colouring problem of graph designs [3, 5–8, 13, 14]; in particular, colourings of C 4 -designs are studied in [1, 2, 4, 9–11]. The * Supported by MIUR, Italy and CNR-GNSAGA y Also affiliated with the Department of Computer Science, University of Veszpre´m, Hun- gary; supported in part by the Hungarian Scientific Research Fund, grant OTKA T-32969 AMS classification: 05B05 Graphs and Combinatorics (2004) 20:531–540 Digital Object Identifier (DOI) 10.1007/s00373-004-0582-z Graphs and Combinatorics Ó Springer-Verlag 2004