Colourings of path systems

Abstract

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.