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Stack and Queue Numbers of Graphs Revisited

Petr Hliněný, Adam Straka

Abstract

A long-standing question of the mutual relation between the stack and queue numbers of a graph, explicitly emphasized by Dujmović and Wood in 2005, was "half-answered" by Dujmović, Eppstein, Hickingbotham, Morin and Wood in 2022; they proved the existence of a graph family with the queue number at most 4 but unbounded stack number. We give an alternative very short, and still elementary, proof of the same fact.

Stack and Queue Numbers of Graphs Revisited

Abstract

A long-standing question of the mutual relation between the stack and queue numbers of a graph, explicitly emphasized by Dujmović and Wood in 2005, was "half-answered" by Dujmović, Eppstein, Hickingbotham, Morin and Wood in 2022; they proved the existence of a graph family with the queue number at most 4 but unbounded stack number. We give an alternative very short, and still elementary, proof of the same fact.
Paper Structure (5 sections, 8 theorems, 2 figures)

This paper contains 5 sections, 8 theorems, 2 figures.

Table of Contents

  1. Introduction
  2. Stack Number.
  3. Queue Number.
  4. Mutual relations.
  5. Proof of \ref{['thm:main']}

Key Result

Theorem 1

For every integer $s$, and for $a,n>0$ which are sufficiently large with respect to $s$, the Cartesian product $G:= S_a \square H_n$ is of stack number at least $s$.

Figures (2)

  • Figure 1: Edges $xx'$ and $yy'$ that (a) $\prec$-cross, and (b) $\prec$-nest.
  • Figure 2: (a) The star $S_5$, (b) the graph $H_3$, and (c) their Cartesian product $S_5\square H_3$. The four edge colours illustrate a queue layout for $S_5\square H_3$.

Theorems & Definitions (8)

  • Theorem 1: Dujmović et al. dujmovic2021stack
  • Proposition 2: Ramsey ramsey
  • Proposition 3: Erdős--Szekeres erdos
  • Proposition 4: Gale hex
  • Lemma 5
  • Corollary 6
  • Lemma 7
  • Corollary 8