Bounding the pseudoachromatic index of the complete graph via projective planes

Let q = 2 β and n = q 2 + q + 1 . Further, let G = L ( K n ) be the complete line graph and ψ ( G ) its pseudoachromatic number. By exhibiting an explicit colouring of E ( K n ) , we show that ψ ( G ) ⩾ q 3 + q . This result improves the bound ψ ( G ) ⩾ q 3 + 1 due to Jamison (1989) [Jamison, R.E.; On the edge achromatic numbers of complete graphs, Discrete. Math. 74 (1989) 99–115].