Three-dimensional graph products with unbounded stack-number

TL;DR

This paper determines the stack-number behavior for the 3D strong product of paths, proving $sn(P_n\boxtimes P_n\boxtimes P_n) \in \Theta(n^{1/3})$, while showing the corresponding Cartesian products keep bounded stack-number. The authors develop a dual approach: a lower bound derived from Gromov’s Topological Overlap Theorem applied to triangulations of Cartesian products, and a tight upper bound achieved via Hadamard-matrix–based permutations, with a broader bound $sn(G\boxtimes P_n) = O(n^{1/2-\varepsilon})$ for graphs $G$ of bounded stack-number and degree. They extend the framework to triangulations of three-dimensional products of trees, obtaining $sn(G) \in \Omega\left(\left(\dfrac{n}{\Delta_1\Delta_2}\right)^{1/3}\right)$, and show that the least maximum degree for an unbounded stack-number with bounded queue-number lies in $\{6,7\}$. The work connects stack-number to topology, geometry, and minor theory, establishing a first explicit bounded-degree family with unbounded stack-number and posing several natural open questions on minor-closedness, growth, and degree bounds.

Abstract

We prove that the stack-number of the strong product of three $n$-vertex paths is $Θ(n^{1/3})$. 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 $Δ_0$ such that there exist a graph family with unbounded stack-number, bounded queue-number and maximum degree $Δ_0$. We show that $Δ_0\in \{6,7\}$.

Figures (7)

• Figure 1: A $4$-stack layout of the strong product $P_5 \boxtimes P_5$ of two paths.
• Figure 2: (a) $P_4\boxempty P_4\boxempty P_2$, (b) $P_4\boxslash P_4\boxslash P_2$, (c) a triangulation of $P_4\boxempty P_4\boxempty P_2$, (d) $P_4\boxtimes P_4\boxtimes P_2$.
• Figure 3: Construction in the proof of \ref{['InductiveTriangles']}.
• Figure 4: Four permutations of $\{1, \ldots, 8\}$, no two of which have a common subsequence of length greater than $2$.
• Figure 5: The truncated octahedron $Q_0$ inscribed in a cube.

Theorems & Definitions (47)

• Theorem 1
• Theorem 2
• Theorem 3: DEHMW21
• Lemma 4
• Lemma 5
• Corollary 6
• Theorem 7
• Conjecture 8: BO99
• Lemma 9: DujWoo05
• Theorem 10