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An e-star is a complete bipartite graph K e 1, . An e-star system of order n S n > 1, ( )chromatic number, colouring, decomposition, star system, unique colouring A M S C L A S S I F I C A T I O N 05B30; 05C51; 05C15 1 1  ∪ ⩽⩽ DARIJANI AND PIKE | 531 How to cite this article: Darijani I, Pike DA. Colourings of star systems. J Combin Des.

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A Pm path in a graph is a path on m vertices. A Pm system of order n > 1 is a partition of the edges of the complete graph Kn into Pm paths. A Pm system is said to be k-colourable if the vertex set of Kn can be partitioned into k sets called colour classes such that no path in the system is monochromatic. The system is k-chromatic if it is k-colourable but is not (k − 1)-colourable. If every k-colouring of a Pm system can be obtained from some k-colouring φ by a permutation of the colours, we say that the system is uniquely k-colourable. In this paper, we first observe that there exists a k-chromatic Pm system for any k 2 and m 4 where m is even. Next, we prove that there exists an equitably 2-chromatic P4 system of order n for each admissible order n. We then show that for all k 3, there exists a k-chromatic P4 system of order n for all sufficiently large admissible n. Finally, we show that there exists a uniquely 2-chromatic P4 system of order n for each admissible n 109.

We show that if C1 and C2 are directed cycles (of length at least two), then the Cartesian product C1 C2 has two arc-disjoint hamiltonian paths. (This answers a question asked by J. A. Gallian in 1985.) The same conclusion also holds for the Cartesian product of any four or more directed cycles (of length at least two), but some cases remain open for the Cartesian product of three directed cycles.We also discuss the existence of arc-disjoint hamiltonian paths in 2-generated Cayley digraphs on (finite or infinite) abelian groups.