Papers by William Trotter
Journal of Graph Theory, 2005
Usually dimension should be an integer valued parameter, we introduce a refined version of dimens... more Usually dimension should be an integer valued parameter, we introduce a refined version of dimension for graphs which can assume a value [ t − 1 t ] which is thought to be between t − 1 and t. We have the following two results: • A graph is outerplanar if and only if its dimension is at most [ 2 3 ]. This characterization of outerplanar graphs is closely related to the celebrated result of W. Schnyder [16] who proved that a graph is planar if and only if its dimension is at most 3. • The largest n for which the dimension of the complete graph Kn is at most [ t − 1 t ] is the number of antichains in the lattice of all subsets of a set of size t − 2. Accordingly, the refined dimension problem for complete graphs is equivalent to the classical combinatorial problem known as Dedekind's problem. This result extends work of Hoşten and Morris [14]. The main results are enriched by background material which links to a line of reserch in extremal graph theory which was stimulated by a problem posed by G. Agnarsson: Find the maximum number of edges in a graph on n nodes with dimension at most t.
arXiv (Cornell University), Jun 12, 2014
In 1977, Trotter and Moore proved that a poset has dimension at most 3 whenever its cover graph i... more 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.
Journal of Graph Algorithms and Applications, 2010
We investigate which chordal graphs have a representation as intersection graphs of pseudosegment... more 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.
Journal of Combinatorial Theory, Series B, Sep 1, 2019
We apply Ramsey theoretic tools to show that there is a family of graphs which have tree-chromati... more We apply Ramsey theoretic tools to show that there is a family of graphs which have tree-chromatic number at most 2 while the path-chromatic number is unbounded. This resolves a problem posed by Seymour.
arXiv (Cornell University), Jun 30, 2019
We show that height h posets that have planar cover graphs have dimension O(h 6). Previously, the... more We show that height h posets that have planar cover graphs have dimension O(h 6). Previously, the best upper bound was 2 O(h 3). Planarity plays a key role in our arguments, since there are posets such that (1) dimension is exponential in height and (2) the cover graph excludes K 5 as a minor.
arXiv (Cornell University), Dec 31, 2017
We investigate the behavior of Boolean dimension with respect to components and blocks. To put ou... more We investigate the behavior of Boolean dimension with respect to components and blocks. To put our results in context, we note that for Dushnik-Miller dimension, we have that if dim(C) ≤ d for every component C of a poset P , then dim(P) ≤ max{2, d}; also if dim(B) ≤ d for every block B of a poset P , then dim(P) ≤ d + 2. By way of constrast, local dimension is well behaved with respect to components, but not for blocks: if ldim(C) ≤ d for every component C of a poset P , then ldim(P) ≤ d + 2; however, for every d ≥ 4, there exists a poset P with ldim(P) = d and dim(B) ≤ 3 for every block B of P. In this paper we show that Boolean dimension behaves like Dushnik-Miller dimension with respect to both components and blocks: if bdim(C) ≤ d for every component C of P , then bdim(P) ≤ 2 + d + 4 • 2 d ; also if bdim(B) ≤ d for every block of P , then bdim(P) ≤ 19 + d + 18 • 2 d .
Order, Jun 2, 2020
Planar posets can have arbitrarily large dimension. However, a planar poset of height h has dimen... more Planar posets can have arbitrarily large dimension. However, a planar poset of height h has dimension at most 192h + 96, while a planar poset with t minimal elements has dimension at most 2t + 1. In particular, a planar poset with a unique minimal element has dimension at most 3. In this paper, we extend this result by showing that a planar poset has dimension at most 6 if it has a plane diagram in which every minimal element is accessible from below.
arXiv (Cornell University), Jun 19, 2019
Planar posets can have arbitrarily large dimension. However, a planar poset of height h has dimen... more Planar posets can have arbitrarily large dimension. However, a planar poset of height h has dimension at most 192h + 96, while a planar poset with t minimal elements has dimension at most 2t + 1. In particular, a planar poset with a unique minimal element has dimension at most 3. In this paper, we extend this result by showing that a planar poset has dimension at most 6 if it has a plane diagram in which every minimal element is accessible from below.
Order, Jun 1, 2016
In 1977, Trotter and Moore proved that a poset has dimension at most 3 whenever its cover graph i... more 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 wellknown 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.
arXiv (Cornell University), Jun 14, 2022
In 1989, Nešetřil and Pudlák posed the following challenging question: Do planar posets have boun... more In 1989, Nešetřil and Pudlák posed the following challenging question: Do planar posets have bounded Boolean dimension? We show that every poset with a planar cover graph and a unique minimal element has Boolean dimension at most 13. As a consequence, we are able to show that there is a reachability labeling scheme with labels consisting of O(log n) bits for planar digraphs with a single source. The best known scheme for general planar digraphs uses labels with O(log 2 n) bits [Thorup JACM 2004], and it remains open to determine whether a scheme using labels with O(log n) bits exists. The Boolean dimension result is proved in tandem with a second result showing that the dimension of a poset with a planar cover graph and a unique minimal element is bounded by a linear function of its standard example number. However, one of the major challenges in dimension theory is to determine whether dimension is bounded in terms of standard example number for all posets with planar cover graphs.
arXiv (Cornell University), Jun 13, 2019
The original notion of dimension for posets was introduced by Dushnik and Miller in 1941 and has ... more The original notion of dimension for posets was introduced by Dushnik and Miller in 1941 and has been studied extensively in the literature. In 1992, Brightwell and Scheinerman developed the notion of fractional dimension as the natural linear programming relaxation of the Dushnik-Miller concept. In 2016, Ueckerdt introduced the concept of local dimension, and in just three years, several research papers studying this new parameter have been published. In this paper, we introduce and study fractional local dimension. As suggested by the terminology, our parameter is a common generalization of fractional dimension and local dimension. For a pair (n, d) with 2 ≤ d < n, we consider the poset P (1, d; n) consisting of all 1-element and d-element subsets of {1,. .. , n} partially ordered by inclusion. This poset has fractional dimension d + 1, but for fixed d ≥ 2, its local dimension goes to infinity with n. On the other hand, we show that as n tends to infinity, the fractional local dimension of P (1, d; n) tends to a value FLD(d) which we will be able to determine exactly. For all d ≥ 2, FLD(d) is strictly less than d + 1, and for large d, FLD(d) ∼ d/(log d − log log d − o(1)). As an immediate corollary, we show that if P is a poset, and d is the maximum degree of a vertex in the comparability graph of P , then the fractional local dimension of P , is at most 2 + FLD(d). Our arguments use both discrete and continuous methods.
Graph Theory in Memory of G.A. Dirac, 1988
49 questions from the theory of ordered sets are stated, and the present knowledge about each is ... more 49 questions from the theory of ordered sets are stated, and the present knowledge about each is surveyed.
Graphs and Order, 1985
The purpose of this paper is to present a concise and relatively self contained treatment of rece... more The purpose of this paper is to present a concise and relatively self contained treatment of recent results linking partially ordered sets with topics more traditionally associated with graph theory and combinatorics: Ramsey theory and chromatic graph theory. In particular, we will present the major theorems of Nesetril and Rodl ([31], [33], and [34]) concerning Ramsey theory for partially ordered sets in a new setting which will allow nonspecialists to appreciate the power and beauty of these results. Other Ramsey theoretic results for partially ordered sets will be discussed briefly and some directions for future research will be indicated.
The College Mathematics Journal, 1989
Acta Mathematica Hungarica, Jun 30, 2020
Previously, Erdős, Kierstead and Trotter [5] investigated the dimension of random height 2 partia... more Previously, Erdős, Kierstead and Trotter [5] investigated the dimension of random height 2 partially ordered sets. Their research was motivated primarily by two goals: (1) analyzing the relative tightness of the Füredi-Kahn upper bounds on dimension in terms of maximum degree; and (2) developing machinery for estimating the expected dimension of a random labeled poset on n points. For these reasons, most of their effort was focused on the case 0 < p ≤ 1/2. While bounds were given for the range 1/2 ≤ p < 1, the relative accuracy of the results in the original paper deteriorated as p approaches 1. Motivated by two extremal problems involving conditions that force a poset to contain a large standard example, we were compelled to revisit this subject, but now with primary emphasis on the range 1/2 ≤ p < 1. Our sharpened analysis shows that as p approaches 1, the expected value of dimension increases and then decreases, answering in the negative a question posed in the original paper. Along the way, we apply inequalities of Talagrand and Janson, establish connections with latin rectangles and the Euler product function, and make progress on both extremal problems.
Order, Aug 26, 2015
In recent years, researchers have shown renewed interest in combinatorial properties of posets de... more In recent years, researchers have shown renewed interest in combinatorial properties of posets determined by geometric properties of its order diagram and topological properties of its cover graph. In most cases, the roots for the problems being studied today can be traced back to the 1970's, and sometimes even earlier. In this paper, we study the problem of bounding the dimension of a planar poset in terms of the number of minimal elements, where the starting point is the 1977 theorem of Trotter and Moore asserting that the dimension of a planar poset with a single minimal element is at most 3. By carefully analyzing and then refining the details of this argument, we are able to show that the dimension of a planar poset with t minimal elements is at most 2t + 1. This bound is tight for t = 1 and t = 2. But for t ≥ 3, we are only able to show that there exist planar posets with t minimal elements having dimension t +3. Our lower bound construction can be modified in ways that have immediate connections to the following challenging conjecture: For every d ≥ 2, there is an integer f (d) so that if P is a planar poset with dim(P) ≥ f (d), then P contains a standard example of dimension d. To date, the best known examples only showed that the function f , if it exists, satisfies f (d) ≥ d + 2. Here, we show that lim d→∞ f (d)/d ≥ 2.
Order, May 14, 2015
We develop some new inequalities for the dimension of a finite poset. These inequalities are then... more We develop some new inequalities for the dimension of a finite poset. These inequalities are then used to bound dimension in terms of the maximum size of matchings. We prove that if the dimension of P is d and d ≥ 3, then there is a matching of size d in the comparability graph of P. There is no analogue of this result for cover graphs, as we show that there is a poset P of dimension d for which the maximum matching in the cover graph of P has size O(log d). On the other hand, there is a dual result in which the role of chains and antichains is reversed, as we show that there is also a matching of size d in the incomparability graph of P. The proof of the result for comparability graphs has elements in common with Perles' proof of Dilworth's theorem. Either result has the following theorem of Hiraguchi as an immediate corollary: dim(P) ≤ |P |/2 when |P | ≥ 4.
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Papers by William Trotter