arXiv subjects

William T. Trotter

Publications and source records attributed to William T. Trotter.

28 records · Page 2Linked to original sources

arXiv2017

The Graph of Critical Pairs of a Crown

There is a natural way to associate with a poset $P$ a hypergraph $H$, called the hypergraph of critical pairs, so that the dimension of $P$ is exactly equal to the chromatic number of $H$. The edges of $H$ have variable sizes, but it is of interest to consider the graph $G$ formed by the edges of $H$ that have size~2. The chromatic number of $G$ is less than or equal to the dimension of $P$ and the difference between the two values can be arbitrarily large. Nevertheless, there are important instances where the two parameters are the same, and we study one of these in this paper. Our focus is on a family $\{S_n^k:n\ge 3, k\ge 0\}$ of height two posets called crowns. We show that the chromatic number of the graph $G_n^k$ of critical pairs of the crown $S_n^k$ is the same as the dimension of $S_n^k$, which is known to be $\lceil 2(n+k)/(k+2)\rceil$. In fact, this theorem follows as an immediate corollary to the stronger result: The independence number of $G_n^k$ is $(k+1)(k+2)/2$. We obtain this theorem as part of a comprehensive analysis of independent sets in $G_n^k$ including the determination of the second largest size among the maximal independent sets, both the reversible and non-reversible types.

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arXiv2017

Boolean dimension and local dimension

Dimension is a standard and well-studied measure of complexity of posets. Recent research has provided many new upper bounds on the dimension for various structurally restricted classes of posets. Bounded dimension gives a succinct representation of the poset, admitting constant response time for queries of the form "is $x<y$?". This application motivates looking for stronger notions of dimension, possibly leading to succinct representations for more general classes of posets. We focus on two: boolean dimension, introduced in the 1980s and revisited in recent research, and local dimension, a very new one. We determine precisely which values of dimension/boolean dimension/local dimension imply that the two other parameters are bounded.

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arXiv2016

On the dimension of posets with cover graphs of treewidth $2$

In 1977, Trotter and Moore proved that a poset has dimension at most $3$ whenever its cover graph is a forest, or equivalently, has treewidth at most $1$. On the other hand, a well-known construction of Kelly shows that there are posets of arbitrarily large dimension whose cover graphs have treewidth $3$. In this paper we focus on the boundary case of treewidth $2$. It was recently shown that the dimension is bounded if the cover graph is outerplanar (Felsner, Trotter, and Wiechert) or if it has pathwidth $2$ (Biró, Keller, and Young). This can be interpreted as evidence that the dimension should be bounded more generally when the cover graph has treewidth $2$. We show that it is indeed the case: Every such poset has dimension at most $1276$.

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arXiv2015

Forcing Posets with Large Dimension to Contain Large Standard Examples

The dimension of a poset $P$, denoted $\dim(P)$, is the least positive integer $d$ for which $P$ is the intersection of $d$ linear extensions of $P$. The maximum dimension of a poset $P$ with $|P|\le 2n+1$ is $n$, provided $n\ge2$, and this inequality is tight when $P$ contains the standard example $S_n$. However, there are posets with large dimension that do not contain the standard example $S_2$. Moreover, for each fixed $d\ge2$, if $P$ is a poset with $|P|\le 2n+1$ and $P$ does not contain the standard example $S_d$, then $\dim(P)=o(n)$. Also, for large $n$, there is a poset $P$ with $|P|=2n$ and $\dim(P)\ge (1-o(1))n$ such that the largest $d$ so that $P$ contains the standard example $S_d$ is $o(n)$. In this paper, we will show that for every integer $c\ge1$, there is an integer $f(c)=O(c^2)$ so that for large enough $n$, if $P$ is a poset with $|P|\le 2n+1$ and $\dim(P)\ge n-c$, then $P$ contains a standard example $S_d$ with $d\ge n-f(c)$. From below, we show that $f(c)=Ω(c^{4/3})$. On the other hand, we also prove an analogous result for fractional dimension, and in this setting $f(c)$ is linear in $c$. Here the result is best possible up to the value of the multiplicative constant.

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arXiv2015

Tree-width and dimension

Over the last 30 years, researchers have investigated connections between dimension for posets and planarity for graphs. Here we extend this line of research to the structural graph theory parameter tree-width by proving that the dimension of a finite poset is bounded in terms of its height and the tree-width of its cover graph.

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arXiv2014

Triangle-free geometric intersection graphs with large chromatic number

Several classical constructions illustrate the fact that the chromatic number of a graph can be arbitrarily large compared to its clique number. However, until very recently, no such construction was known for intersection graphs of geometric objects in the plane. We provide a general construction that for any arc-connected compact set $X$ in $\mathbb{R}^2$ that is not an axis-aligned rectangle and for any positive integer $k$ produces a family $\mathcal{F}$ of sets, each obtained by an independent horizontal and vertical scaling and translation of $X$, such that no three sets in $\mathcal{F}$ pairwise intersect and $χ(\mathcal{F})>k$. This provides a negative answer to a question of Gyarfas and Lehel for L-shapes. With extra conditions, we also show how to construct a triangle-free family of homothetic (uniformly scaled) copies of a set with arbitrarily large chromatic number. This applies to many common shapes, like circles, square boundaries, and equilateral L-shapes. Additionally, we reveal a surprising connection between coloring geometric objects in the plane and on-line coloring of intervals on the line.

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arXiv2014

Triangle-free intersection graphs of line segments with large chromatic number

In the 1970s, Erdos asked whether the chromatic number of intersection graphs of line segments in the plane is bounded by a function of their clique number. We show the answer is no. Specifically, for each positive integer $k$, we construct a triangle-free family of line segments in the plane with chromatic number greater than $k$. Our construction disproves a conjecture of Scott that graphs excluding induced subdivisions of any fixed graph have chromatic number bounded by a function of their clique number.

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arXiv2014

An extremal problem on crossing vectors

For positive integers $w$ and $k$, two vectors $A$ and $B$ from $\mathbb{Z}^w$ are called $k$-crossing if there are two coordinates $i$ and $j$ such that $A[i]-B[i]\geq k$ and $B[j]-A[j]\geq k$. What is the maximum size of a family of pairwise $1$-crossing and pairwise non-$k$-crossing vectors in $\mathbb{Z}^w$? We state a conjecture that the answer is $k^{w-1}$. We prove the conjecture for $w\leq 3$ and provide weaker upper bounds for $w\geq 4$. Also, for all $k$ and $w$, we construct several quite different examples of families of desired size $k^{w-1}$. This research is motivated by a natural question concerning the width of the lattice of maximum antichains of a partially ordered set.

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arXiv2013

Incidence Posets and Cover Graphs

We prove two theorems concerning incidence posets of graphs, cover graphs of posets and a related graph parameter. First, answering a question of Haxell, we show that the chromatic number of a graph is not bounded in terms of the dimension of its incidence poset, provided the dimension is at least four. Second, answering a question of Kříž and Nešetřil, we show that there are graphs with large girth and large chromatic number among the class of graphs having eye parameter at most two.

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arXiv2008

Intersection Graphs of Pseudosegments: Chordal Graphs

We investigate which chordal graphs have a representation as intersection graphs of pseudosegments. For positive we have a construction which shows that all chordal graphs that can be represented as intersection graph of subpaths on a tree are pseudosegment intersection graphs. We then study the limits of representability. We describe a family of intersection graphs of substars of a star which is not representable as intersection graph of pseudosegments. The degree of the substars in this example, however, has to get large. A more intricate analysis involving a Ramsey argument shows that even in the class of intersection graphs of substars of degree three of a star there are graphs that are not representable as intersection graph of pseudosegments. Motivated by representability questions for chordal graphs we consider how many combinatorially different k-segments, i.e., curves crossing k distinct lines, an arrangement of n pseudolines can host. We show that for fixed k this number is in O(n^2). This result is based on a k-zone theorem for arrangements of pseudolines that should be of independent interest.

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