Graphs and Combinatorics

, Volume 34, Issue 2, pp 277–287 | Cite as

On the Pseudoachromatic Index of the Complete Graph III

  • M. Gabriela Araujo-Pardo
  • Juan José Montellano-Ballesteros
  • Christian Rubio-Montiel
  • Ricardo Strausz
Original Paper
  • 65 Downloads

Abstract

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 \( a\in \{-1,0,\dots ,\left\lceil \frac{1+\sqrt{4q+9}}{2}\right\rceil -1 \} .\)

Keywords

Finite projective plane Complete colourings Complete edge-colouring Pseudoachromatic index Complete graph 

Mathematics Subject Classification

05C15 51E15 

Notes

Acknowledgements

The authors wish to thank the anonymous referees of this paper for their helpful and detailed corrections and remarks. Research partially supported by PAPIIT of Mexico Grant IN104915 and by CONACyT of Mexico Grants 178395 and 166306.

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Copyright information

© Springer Japan KK, part of Springer Nature 2018

Authors and Affiliations

  • M. Gabriela Araujo-Pardo
    • 1
  • Juan José Montellano-Ballesteros
    • 1
  • Christian Rubio-Montiel
    • 2
    • 3
  • Ricardo Strausz
    • 1
  1. 1.Instituto de MatemáticasUniversidad Nacional Autónoma de MéxicoMexico CityMexico
  2. 2.División de Matemáticas e Ingeniería, FES AcatlánUniversidad Nacional Autónoma de MéxicoState of MexicoMexico
  3. 3.UMI LAFMIA 3175 CNRS at CINVESTAV-IPNMexico CityMexico

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