Colouring of cubic graphs by Steiner triple systems

Let <f>S</f> be a Steiner triple system and <f>G</f> a cubic graph. We say that <f>G</f> is <f>S</f>-colourable if its edges can be coloured so that at each vertex the incident colours form a triple of <f>S</f>. We show that if <f>S</f> is a projective system <f>PG(n,2)</f>, <f>n&ges;2</f>, then <f>G</f> is <f>S</f>-colourable if and only if it is bridgeless, and that every bridgeless cubic graph has an <f>S</f>-colouring for every Steiner triple system of order greater than 3. We establish a condition on a cubic graph with a bridge which ensures that it fails to have an <f>S</f>-colouring if <f>S</f> is an affine system, and we conjecture that this is the only obstruction to colouring any cubic graph with any non-projective system of order greater than 3. [Copyright &y& Elsevier]