David R. Wood
It is proved that there exist graphs of bounded degree with arbitrarily large queue-number. In particular, for all $Δ\geq3$ and for all sufficiently large $n$, there is a simple $Δ$-regular $n$-vertex graph with queue-number at least $c\sqrtΔn^{1/2-1/Δ}$ for some absolute constant $c$.
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Motivated by the question of whether planar graphs have bounded queue-number, we prove that planar graphs with maximum degree $Δ$ have queue-number $O(Δ^{2})$, which improves upon the best previous bound of $O(Δ^6)$. More generally, we prove that graphs with bounded degree and bounded Euler genus have bounded queue-number. In particular graphs with Euler genus $g$ and maximum degree $Δ$ have queue-number $O(g+Δ^{2})$. As a byproduct we prove that if planar graphs have bounded queue-number, then graphs of Euler genus $g$ have queue-number $O(g)$.
Some of the most important open problems for linear layouts of graphs ask for the relation between a graph's queue number and its stack number or mixed number. In such, we seek a vertex order and edge partition of $G$ into parts with pairwise non-crossing edges (a stack) or with pairwise non-nesting edges (a queue). Allowing only stacks, only queues, or both, the minimum number of required parts is the graph's stack number $sn(G)$, queue number $qn(G)$, and mixed number $mn(G)$, respectively. Already in 1992, Heath and Rosenberg asked whether $qn(G)$ is bounded in terms of $sn(G)$, that is, wh…
Dujmović, Joret, Micek, Morin, Ueckerdt and Wood recently in [Planar graphs have bounded queue-number, Journal of the ACM, Volume 67, Issue 4, Article No.: 22, August 2020] showed some attractive graph product structure theorems for planar graphs. By using the product structure, they proved that planar graphs have bounded queue-number $48$; in [Planar graphs have bounded nonrepetitive chromatic number, Advances in Combinatorics, 5, 11 pp, 2020], the authors proved that planar graphs have bounded nonrepetitive chromatic number $768$. In this paper, still by using some product structure theorem,…
We describe a family of graphs with queue-number at most 4 but unbounded stack-number. This resolves open problems of Heath, Leighton and Rosenberg (1992) and Blankenship and Oporowski (1999).
Heath and Pemmaraju conjectured that the queue-number of a poset is bounded by its width and if the poset is planar then also by its height. We show that there are planar posets whose queue-number is larger than their height, refuting the second conjecture. On the other hand, we show that any poset of width $2$ has queue-number at most $2$, thus confirming the first conjecture in the first non-trivial case. Moreover, we improve the previously best known bounds and show that planar posets of width $w$ have queue-number at most $3w-2$ while any planar poset with $0$ and $1$ has queue-number at m…
A queue layout of a graph $G$ consists of a vertex ordering of $G$ and a partition of the edges into so-called queues such that no two edges in the same queue nest, i.e., have their endpoints ordered in an ABBA-pattern. Continuing the research on local ordered covering numbers, we introduce the local queue number of a graph $G$ as the minimum $\ell$ such that $G$ admits a queue layout with each vertex having incident edges in no more than $\ell$ queues. Similarly to the local page number [Merker, Ueckerdt, GD'19], the local queue number is closely related to the graph's density and can be arbi…
A queue layout of a graph consists of a linear order on the vertices and an assignment of the edges to queues, such that no two edges in a single queue are nested. The minimum number of queues needed in a queue layout of a graph is called its queue-number. We show that for each $k\geq1$, graphs with tree-width at most $k$ have queue-number at most $2^k-1$. This improves upon double exponential upper bounds due to Dujmović et al. and Giacomo et al. As a consequence we obtain that these graphs have track-number at most $2^{O(k^2)}$. We complement these results by a construction of $k$-trees that…
We show that planar graphs have bounded queue-number, thus proving a conjecture of Heath, Leighton and Rosenberg from 1992. The key to the proof is a new structural tool called layered partitions, and the result that every planar graph has a vertex-partition and a layering, such that each part has a bounded number of vertices in each layer, and the quotient graph has bounded treewidth. This result generalises for graphs of bounded Euler genus. Moreover, we prove that every graph in a minor-closed class has such a layered partition if and only if the class excludes some apex graph. Building on …
Stack number and queue number of graphs
本文综述了图不变量队列数与栈数，给出了栈数不被队列数所界的简化证明，并调查了队列数上可能的栈数界的已知结果。
The geometric thickness of a graph G is the minimum integer k such that there is a straight line drawing of G with its edge set partitioned into k plane subgraphs. Eppstein [Separating thickness from geometric thickness. In: Towards a Theory of Geometric Graphs, vol. 342 of Contemp. Math., AMS, 2004] asked whether every graph of bounded maximum degree has bounded geometric thickness. We answer this question in the negative, by proving that there exists Delta-regular graphs with arbitrarily large geometric thickness. In particular, for all Delta >= 9 and for all large n, there exists a Delta-re…
In 1977 L.T. Ramsey showed that any sequence in $\mathbb{Z}^2$ with bounded gaps contains arbitrarily many collinear points. Thereafter, in 1980, C. Pomerance provided a density version of this result, relaxing the condition on the sequence from having bounded gaps to having gaps bounded on average. We give a higher dimensional generalization of these results. Our main theorem is the following. Theorem: Let $d\in\mathbb{N}$, let $f:\mathbb{Z}^d\to\mathbb{Z}^{d+1}$ be a Lipschitz map and let $A\subset\mathbb{Z}^d$ have positive upper Banach density. Then $f(A)$ contains arbitrarily many colline…
Forbidden Patterns in Mixed Linear Layouts
本文针对混合线性布局缺乏禁止模式表征的问题，引入类似扭转和彩虹的厚模式，证明有界度图族的混合页数有界当且仅当最大厚模式大小有界，并指出对固定页数k≥2不存在有限禁止模式精确表征。
The planar graph product structure theorem of Dujmović, Joret, Micek, Morin, Ueckerdt, and Wood [J. ACM 2020] states that every planar graph is a subgraph of the strong product of a graph with bounded treewidth and a path. This result has been the key tool to resolve important open problems regarding queue layouts, nonrepetitive colourings, centered colourings, and adjacency labelling schemes. In this paper, we extend this line of research by utilizing shallow minors to prove analogous product structure theorems for several beyond planar graph classes. The key observation that drives our work …
We show that digraphs with no transitive tournament on $3$ vertices and in which every induced directed cycle has length $3$ can have arbitrarily large dichromatic number. This answers to the negative a question of Carbonero, Hompe, Moore, and Spirkl (and extends some of their results).
A linear layout of a graph typically consists of a total vertex order, and a partition of the edges into sets of either non-crossing edges, called stacks, or non-nested edges, called queues. The stack (queue) number of a graph is the minimum number of required stacks (queues) in a linear layout. Mixed linear layouts combine these layouts by allowing each set of edges to form either a stack or a queue. In this work we initiate the study of the mixed page number of a graph which corresponds to the minimum number of such sets. First, we study the edge density of graphs with bounded mixed page num…
It is known that every proper minor-closed class of graphs has bounded stack-number (a.k.a. book thickness and page number). While this includes notable graph families such as planar graphs and graphs of bounded genus, many other graph families are not closed under taking minors. For fixed $g$ and $k$, we show that every $n$-vertex graph that can be embedded on a surface of genus $g$ with at most $k$ crossings per edge has stack-number $\mathcal{O}(\log n)$; this includes $k$-planar graphs. The previously best known bound for the stack-number of these families was $\mathcal{O}(\sqrt{n})$, exce…
A $k$-stack layout (also called a $k$-page book embedding) of a graph consists of a total order of the vertices, and a partition of the edges into $k$ sets of non-crossing edges with respect to the vertex order. The stack number (book thickness, page number) of a graph is the minimum $k$ such that it admits a $k$-stack layout. A $k$-queue layout is defined similarly, except that no two edges in a single set may be nested. It was recently proved that graphs of various non-minor-closed classes are subgraphs of the strong product of a path and a graph with bounded treewidth. Motivated by this dec…
For $t \ge 2$, let us call a digraph $D$ \emph{t-chordal} if all induced directed cycles in $D$ have length equal to $t$. In a previous paper, we asked for which $t$ it is true that $t$-chordal graphs with bounded clique number have bounded dichromatic number. Recently, Aboulker, Bousquet, and de Verclos answered this in the negative for $t=3$, that is, they gave a construction of $3$-chordal digraphs with clique number at most $3$ and arbitrarily large dichromatic number. In this paper, we extend their result, giving for each $t \ge 3$ a construction of digraphs with clique number at most $3$…
The game of Cops and Robbers is a well known pursuit-evasion game played on graphs. It has been proved \cite{bounded_degree} that cubic graphs can have arbitrarily large cop number $c(G)$, but the known constructions show only that the set $\{c(G) \mid G \text{ cubic}\}$ is unbounded. In this paper we prove that there are arbitrarily large subcubic graphs $G$ whose cop number is at least $n^{1/2-o(1)}$ where $n=|V(G)|$. We also show that proving Meyniel's conjecture for graphs of bounded degree implies a weak Meyniel's conjecture for all graphs.
Three-dimensional graph products with unbounded stack-number
本研究证明三条n顶点路径强乘积的栈数为Θ(n^{1/3})，给出首个有界最大度且栈数无界的图族示例，还确定了满足栈数无界、队列数有界的图族的最小最大度Δ₀属于{6,7}。
Graph separators are a ubiquitous tool in graph theory and computer science. However, in some applications, their usefulness is limited by the fact that the separator can be as large as $Ω(\sqrt{n})$ in graphs with $n$ vertices. This is the case for planar graphs, and more generally, for proper minor-closed classes. We study a special type of graph separator, called a "layered separator", which may have linear size in $n$, but has bounded size with respect to a different measure, called the "width". We prove, for example, that planar graphs and graphs of bounded Euler genus admit layered separ…
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