An official website of the United States government
Abstract We study the asymptotic limit of random pure dimer coverings on rail yard graphs when the mesh sizes of the graphs go to 0. Each pure dimer covering corresponds to a sequence of interlacing partitions starting with an empty partition and ending in an empty partition. Under the assumption that the probability of each dimer covering is proportional to the product of weights of present edges, we obtain the limit shape (law of large numbers) of the rescaled height functions and the convergence of the unrescaled height fluctuations to a diffeomorphic image of the Gaussian free field (Central Limit Theorem), answering a question in [7]. Applications include the limit shape and height fluctuations for pure steep tilings [9] and pyramid partitions [20; 36; 39; 38]. The technique to obtain these results is to analyze a class of Macdonald processes which involve dual partitions as well.
more »
« less
This content will become publicly available on April 1, 2026
The maximum number of odd cycles in a planar graph
Abstract How many copies of a fixed odd cycle, , can a planar graph contain? We answer this question asymptotically for and prove a bound which is tight up to a factor of 3/2 for all other values of . This extends the prior results of Cox and Martin and of Lv, Győri, He, Salia, Tompkins, and Zhu on the analogous question for even cycles. Our bounds result from a reduction to the following maximum likelihood question: which probability mass on the edges of some clique maximizes the probability that edges sampled independently from form either a cycle or a path?
more »
« less
- Award ID(s):
- 1839918
- PAR ID:
- 10579603
- Publisher / Repository:
- Wiley
- Date Published:
- Journal Name:
- Journal of Graph Theory
- Volume:
- 108
- Issue:
- 4
- ISSN:
- 0364-9024
- Page Range / eLocation ID:
- 745 to 780
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Abstract In this note we examine the following random graph model: for an arbitrary graph , with quadratic many edges, construct a graph by randomly adding edges to and randomly coloring the edges of with colors. We show that for a large enough constant and , every pair of vertices in are joined by a rainbow path, that is, israinbow connected, with high probability. This confirms a conjecture of Anastos and Frieze, who proved the statement for and resolved the case when and is a function of .more » « less
-
Abstract We prove that the number of edges of a multigraph with vertices is at most , provided that any two edges cross at most once, parallel edges are noncrossing, and the lens enclosed by every pair of parallel edges in contains at least one vertex. As a consequence, we prove the following extension of the Crossing Lemma of Ajtai, Chvátal, Newborn, Szemerédi, and Leighton, if has edges, in any drawing of with the above property, the number of crossings is . This answers a question of Kaufmann et al. and is tight up to the logarithmic factor.more » « less
-
null (Ed.)We introduce delft, a factoid question answering system which combines the nuance and depth of knowledge graph question answering approaches with the broader coverage of free-text. delft builds a free-text knowledge graph from Wikipedia, with entities as nodes and sentences in which entities co-occur as edges. For each question, delft finds the subgraph linking question entity nodes to candidates using text sentences as edges, creating a dense and high coverage semantic graph. A novel graph neural network reasons over the free-text graph—combining evidence on the nodes via information along edge sentences—to select a final answer. Experiments on three question answering datasets show delft can answer entity-rich questions better than machine reading based models, bert-based answer ranking and memory networks. delft’s advantage comes from both the high coverage of its free-text knowledge graph—more than double that of dbpedia relations—and the novel graph neural network which reasons on the rich but noisy free-text evidence.more » « less
-
Felsner, Stefan; Klein, Karsten (Ed.)Edge crossings in geometric graphs are sometimes undesirable as they could lead to unwanted situations such as collisions in motion planning and inconsistency in VLSI layout. Short geometric structures such as shortest perfect matchings, shortest spanning trees, shortest spanning paths, and shortest spanning cycles on a given point set are inherently noncrossing. However, the longest such structures need not be noncrossing. In fact, it is intuitive to expect many edge crossings in various geometric graphs that are longest. Recently, Álvarez-Rebollar, Cravioto-Lagos, Marín, Solé-Pi, and Urrutia (Graphs and Combinatorics, 2024) constructed a set of points for which the longest perfect matching is noncrossing. They raised several challenging questions in this direction. In particular, they asked whether the longest spanning path, on any finite set of points in the plane, must have a pair of crossing edges. They also conjectured that the longest spanning cycle must have a pair of crossing edges. In this paper, we give a negative answer to the question and also refute the conjecture. We present a framework for constructing arbitrarily large point sets for which the longest perfect matchings, the longest spanning paths, and the longest spanning cycles are noncrossing.more » « less
