On the pseudoachromatic index of the complete graph II

• DOI: 10.1007/s40590-014-0007-9
• Publication Date: 2014-03-13T07:03:33Z
• Authors: M. Gabriela Arauj...; Juan José Montell...; Christian Rubio-M...; Ricardo Strausz
• Keywords: Achromatic lens; Complete graph

10 pages, 2 figures<br/>Let $ ��_q $ be the projective plane of order $ q $, let $��(m):=��(L(K_m))$ 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 $ ��(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 $��(m) \leq 2x(m-x-1)$. On the other hand, if $ q $ is even and there exists $ ��_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 $��(m_a)=(m_a-a)q$ for $ a\in \{3,4,\dots,\left\lceil \frac{1+\sqrt{4q+9}}{2}\right\rceil -1 \} $.<br/>