Colouring Cubic Graphs by Small Steiner Triple Systems

Given a Steiner triple system , we say that a cubic graph G is -colourable if its edges can be coloured by points of in such way that the colours of any three edges meeting at a vertex form a triple of . We prove that there is Steiner triple system of order 21 which is universal in the sense that every simple cubic graph is -colourable. This improves the result of Grannell et al. [J. Graph Theory 46 (2004), 15–24] who found a similar system of order 381. On the other hand, it is known that any universal Steiner triple system must have order at least 13, and it has been conjectured that this bound is sharp (Holroyd and Škoviera [J. Combin. Theory Ser. B 91 (2004), 57–66]). Research partially supported by APVT, project 51-027604. Research partially supported by VEGA, grant 1/3022/06.