Upper bounds on the queuenumber of k-ary n-cubes

https://doi.org/10.1016/j.ipl.2009.10.006Get rights and content

Abstract

A queue layout of a graph consists of a linear order of its vertices, and a partition of its edges into queues, such that no two edges in the same queue are nested. The minimum number of queues in a queue layout of a graph G, denoted by qn(G), is called the queuenumber of G. Heath and Rosenberg [SIAM J. Comput. 21 (1992) 927–958] showed that boolean n-cube (i.e., the n-dimensional hypercube) can be laid out using at most n1 queues. Heath et al. [SIAM J. Discrete Math. 5 (1992) 398–412] showed that the ternary n-cube can be laid out using at most 2n2 queues. Recently, Hasunuma and Hirota [Inform. Process. Lett. 104 (2007) 41–44] improved the upper bound on queuenumber to n2 for hypercubes. In this paper, we deal with the upper bound on queuenumber of a wider class of graphs called k-ary n-cubes, which contains hypercubes and ternary n-cubes as subclasses. Our result improves the previous bound in the case of ternary n-cubes. Let Qnk denote the n-dimensional k-ary cube. This paper contributes three main results as follows:

  • (1)

    qn(Qn3)2n3 if n3.

  • (2)

    qn(Qnk)2n2 if n2 and 4k8.

  • (3)

    qn(Qnk)2n1 if n1 and k9.

References (26)

  • V. Dujmović et al.

    On linear layouts of graphs

    Discrete Math. Theor. Comput. Sci.

    (2004)
  • J.L. Ganley, Stack and queue layouts of Halin graphs, 1995,...
  • L.S. Heath et al.

    Comparing queues and stacks as mechanisms for laying out graphs

    SIAM J. Discrete Math.

    (1992)
  • Cited by (7)

    • On k-ary n-cubes and isometric words

      2022, Theoretical Computer Science
    • On the queue-number of the hypercube

      2011, Electronic Notes in Discrete Mathematics
    • Quaternary n-cubes and Isometric Words

      2021, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
    • Queue layouts of toroidal grids

      2014, IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences
    View all citing articles on Scopus

    This research was partially supported by National Science Council of Taiwan under contracts NSC-97-2221-E260-007-MY3 and NSC97-2115-M-141-001-MY2.

    View full text