On the pseudoachromatic index of the complete graph III

Let $\Pi_q$ be the projective plane of order $q$, let $\psi(m):=\psi(L(K_m))$ the pseudoachromatic number of the complete line graph of order $m$, let $a\in \{ 3,4,\dots,\tfrac{q}{2}+1 \}$ and $m_a=(q+1)^2-a$. In this paper, we improve the upper bound of $\psi(m)$ given by Araujo-Pardo et al. [J Graph Theory 66 (2011), 89--97] and Jamison [Discrete Math. 74 (1989), 99--115] in the following values: if $x\geq 2$ is an integer and $m\in \{4x^2-x,\dots,4x^2+3x-3\}$ then $\psi(m) \leq 2x(m-x-1)$. On the other hand, if $q$ is even and there exists $\Pi_q$ we give a complete edge-colouring of $K_{m_a}$ with $(m_a-a)q$ colours. Moreover, using this colouring we extend the previous results for $a=\{-1,0,1,2\}$ given by Araujo-Pardo et al. in [J Graph Theory 66 (2011), 89--97] and [Bol. Soc. Mat. Mex. (2014) 20:17--28] proving that $\psi(m_a)=(m_a-a)q$ for $a\in \{3,4,\dots,\left\lceil \frac{1+\sqrt{4q+9}}{2}\right\rceil -1 \}$.