Advertisement

On the pseudoachromatic index of the complete graph II

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

Abstract

Let \(\psi _1(K_n)\) and \(\alpha _1(K_n)\) be the pseudoachromatic index and the achromatic index of the complete graph, respectively. Let \(\gamma \ge 2\) be a positive integer \(q=2^{\gamma }\) and \(m=(q+1)^{2}\). In this paper exhibiting three closely related edge-colorings of the complete graph we prove that for \(a \in \{0,1,2\}\):
$$\begin{aligned} \psi _1(K_{m-a})=\alpha _1(K_{m-a})=q(m-2a). \end{aligned}$$

Keywords

Achromatic Pseudo-achromatic Edge-coloring Line-graph 

Mathematics Subject Classification

05C15 51E15 

References

  1. 1.
    Araujo-Pardo, G., Montellano-Ballesteros, J.J., Strausz, R.: On the pseudoachromatic index of the complete graph. J. Graph Theory 66(2), 89–97 (2011)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Bouchet, A.: Indice achromatique des graphes complets et réguliers. Cahiers Centre Études Rech. Opér. 20(3–4), 331–340 (1978)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Chartrand, G., Zhang, P.: Chromatic Graph Theory, Discrete Mathematics and its Applications (Boca Raton). CRC Press, Boca Raton (2009)Google Scholar
  4. 4.
    Gupta, R.P.: Bounds on the Chromatic and Achromatic Numbers of Complementary Graphs; Recent Progress in Combinatorics (Proc. Third Waterloo Conf. on Comb., 1968), pp. 229–235. Academic Press, New York (1969)Google Scholar
  5. 5.
    Harary, F., Hedetniemi, S., Prins, G.: An interpolation theorem for graphical homomorphisms. Port. Math. 26, 453–462 (1967)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Hell, P., Miller, D.J.: Graph with given achromatic number. Discrete Math. 16(3), 195–207 (1976)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Hor\(\check{\text{ n }}\)ák, M., Pčola, Š., Woźniak, M.: On the achromatic index of \(K_{q^2+q}\) for a prime \(q\). Graphs Combin. 20(2), 191–203 (2004)Google Scholar
  8. 8.
    Jamison, R.E.: On the edge achromatic number of complete graphs. Discrete Math. 74(1–2), 99–115 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Sane, S.S.: Some Improved Lower Bounds for the Edge Achromatic Number of a Complete Graph, Graphs Combinatorics Algorithms and Applications (Krishnankoil, 2004), pp. 149–152. Narosa Publ. House, New Delhi (2005)Google Scholar
  10. 10.
    Turner, C.M., Rowley, R., Jamison, R.E., Laskar, R.: The edge achromatic number of small complete graphs. Congr. Numer. 62, 21–36 (1988)MathSciNetGoogle Scholar

Copyright information

© Sociedad Matemática Mexicana 2014

Authors and Affiliations

  • M. Gabriela Araujo-Pardo
    • 1
  • Juan José Montellano-Ballesteros
    • 1
  • Christian Rubio-Montiel
    • 1
  • Ricardo Strausz
    • 1
  1. 1.Instituto de MatemáticasUniversidad Nacional Autónoma de México, MexicoMexicoMexico

Personalised recommendations