The strong chromatic index of a cubic graph is at most 10

At the British Combinatorial Conference in Norwich in 1989, A. Gyarfas gave a lecture on the strong chromatic index of a graph. He mentioned several interesting results and posed the open question: What is the maximum strong chromatic index of a cubic graph? The answer would be either 10 or 11. It is the purpose of the present paper to show that 10 is the correct answer, and to present an algorithm, linear in the number of vertices, which will give a strong edge-colouring with at most 10 colours to any graph with maximum degree at most 3. During the writing up of this paper, I have been informed that P. Horak, H. Quin and W.T. Trotter Jr, also have a proof that the strong chromatic index for a cubic graph is at most 10 (private communication with Zsolt Tuza and Peter Horak). I do not know to what extent their proof is similar to the one in this paper. We begin by giving the necessary definitions. Our graphs may have multiple edges, but no loops. A strong edge-colouring of a graph is a colouring of the edges, which is proper, i.e., no pair of edges incident with the same vertex have the same colour, and which has the additional property that no edge joins two vertices that are end-vertices of edges of the same colour. Put differently, in a