Bounding the pseudoachromatic index of the complete graph via projective planes

Let q=β2 and n=q2+q+1. Further, let G=L(Kn) be the complete line graph and ψ(G) its pseudoachromatic number. By exhibiting an explicit colouring of E(Kn), we show that ψ(G)⩾q3+q. This result improves the bound ψ(G)⩾q3+1 due to Jamison (1989) [Jamison, R.E.; On the edge achromatic numbers of complete graphs, Discrete. Math. 74 (1989) 99–115].