Three-dimensional graph products with unbounded stack-number

TL;DR

This paper establishes that the stack-number of the strong product of three paths grows as rac{1}{3}n^{1/3}, providing the first explicit example of a bounded degree graph with unbounded stack-number, using topological and combinatorial methods.

Contribution

It proves the first explicit example of a bounded degree graph with unbounded stack-number and introduces new bounds using topological and permutation-based constructions.

Findings

Stack-number of the strong product of three paths is rac{1}{3}n^{1/3}

First example of a bounded degree graph with unbounded stack-number

The strong product of three paths has bounded queue-number but 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 exist a graph family with unbounded stack-number, bounded queue-number and maximum degree $\Delta_0$. We show that $\Delta_0\in \{6,7\}$.