A Note on “An improved upper bound on the queuenumber of the hypercube”

A queue layout of a graph consists of a linear ordering σ of its vertices, and a partition of its edges into sets, called queues, such that in each set no two edges are nested with respect to σ. A queue-number of G is the minimal number of queues in a queue layout of G. We improve previously known upper and lower bounds on the queue-number of the hypercube.

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. In this paper, we show that the $n$-dimensional hypercube $Q_n$ can be laid out using $n-3$ queues for $n⩾8$. Our result improves the previously known result for the case $n⩾8$.

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 $\mathit{qn}(G)$, 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 $n-1$ 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 $2n-2$ queues. Recently, Hasunuma and Hirota [Inform. Process. Lett. 104 (2007) 41–44] improved the upper bound on queuenumber to $n-2$ 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 $Q_n^k$ denote the n-dimensional k-ary cube. This paper contributes three main results as follows:

• (1)

  $\mathit{qn}(Q_n^3)⩽2n-3$ if $n⩾3$.

• (2)

  $\mathit{qn}(Q_n^k)⩽2n-2$ if $n⩾2$ and $4⩽k⩽8$.

• (3)

  $\mathit{qn}(Q_n^k)⩽2n-1$ if $n⩾1$ and $k⩾9$.