Three-Dimensional Graph Products with Unbounded Stack-Number

Abstract

We prove that the stack-number of the strong product of three n-vertex paths is $$\Theta (n^{1/3})$$ . The best previously known upper bound was O(n). No non-trivial lower bound was known. This is the first explicit example of a graph family with bounded maximum degree and unbounded stack-number. The main tool used in our proof of the lower bound is the topological overlap theorem of Gromov. We actually prove a stronger result in terms of so-called triangulations of Cartesian products. We conclude that triangulations of three-dimensional Cartesian products of any sufficiently large connected graphs have large stack-number. The upper bound is a special case of a more general construction based on families of permutations derived from Hadamard matrices. The strong product of three paths is also the first example of a bounded degree graph with bounded queue-number and unbounded stack-number. A natural question that follows from our result is to determine the smallest $$\Delta _0$$ such that there exists a graph family with unbounded stack-number, bounded queue-number and maximum degree $$\Delta _0$$ . We show that $$\Delta _0\in \{6,7\}$$ .