On the Pseudoachromatic Index of the Complete Graph III

An edge colouring of a graph G is complete if for any distinct colours \(c_1\) and \(c_2\) one can find in G adjacent edges coloured with \(c_1\) and \(c_2\), respectively. The pseudoachromatic index of G is the maximum number of colours in a complete edge colouring of G. Let \(\psi (n)\) denote the pseudoachromatic index of \(K_n\). In the paper we proved that if \( x\ge 2 \) is an integer and \(n\in \{4x^2-x,\dots ,4x^2+3x-3\}\), then \(\psi (n) \le 2x(n-x-1)\). Let q be an even integer and let \( m_a=(q+1)^2-a \). If there is a projective plane of order q, a complete edge colouring of \(K_{m_a}\) with \((m_a-a)q\) colours, \( a\in \{-1,0,\dots ,\frac{q}{2}+1\}\), is presented. The main result states that if \(q\ge 4\) is an integer power of 2, then \(\psi (m_a)=(m_a-a)q\) for any \( a\in \{-1,0,\dots ,\left\lceil \frac{1+\sqrt{4q+9}}{2}\right\rceil -1 \} .\)