Circle graphs are quadratically χ‐bounded

Davies, James C.; McCarty, Rose

doi:10.1112/blms.12447

Cited by 25 publications

(25 citation statements)

References 12 publications

(15 reference statements)

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“…Although proving that G has the form given by Theorem 5.1 is not part of the result statement, it seems necessary to generate a straightforward argument. Where before we began by expanding (17) only in x, we now expand in both x and y and then extract coefficients in x to turn the functional equation into the identity…”

Section: A Differential Equation For the Binomial 1-cocycle Propertymentioning

confidence: 99%

“…Permutation diagrams are chord diagram representatives of permutation graphs, which have been extensively studied in the graph theory literature and play a central role in the study of circle graphs (e.g. [17]). It is notable then that they also appear in this context and, in particular, are well-behaved with respect 1-terminality and ψ.…”

Section: Relationship With Connectivitymentioning

confidence: 99%

“…, 2n}. Such objects have also been called matchings [34,11], linked diagrams [56], complete pairings [56], and interval systems [17], and can be viewed as set partitions with every block of size 2; see Figure 1. Here chord diagrams are rooted at the chord containing 1, but the term has also been used to refer to the unrooted object obtained after modding out by cyclic permutations of [2n].…”

Section: Introductionmentioning

confidence: 99%

“…Undirected intersection graphs of diagrams, known as circle graphs, have been well studied as pure graph objects (e.g. [44,17]) and in particular play a major role in vertex minor theory [7].…”

Section: Introductionmentioning

confidence: 99%

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The combinatorics of a tree-like functional equation for connected chord diagrams

Nabergall

2021

Preprint

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We build on recent work of Yeats, Courtiel, and others involving connected chord diagrams. We first derive from a Hopf-algebraic foundation a class of tree-like functional equations and prove that they are solved by weighted generating functions of two different subsets of weighted connected chord diagrams: arbitrary diagrams and diagrams forbidding so-called top cycle subdiagrams. These equations generalize the classic specification for increasing ordered trees and their solution uses a novel decomposition, simplifying and generalizing previous results. The resulting tree perspective on chord diagrams leads to new enumerative insights through the study of novel diagram classes. We present a recursive bijection between connected top-cycle-free diagrams with n chords and triangulations of a disk with n + 1 vertices, thereby counting the former. This connects to combinatorial maps, Catalan intervals, and uniquely sorted permutations, leading to new conjectured bijective relationships between diagram classes defined by forbidding graphical subdiagrams and imposing connectedness properties and a variety of other rich combinatorial objects. We conclude by exhibiting and studying a direct bijection between diagrams of size n with a single terminal chord and diagrams of size n − 1.

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Section: A Differential Equation For the Binomial 1-cocycle Propertymentioning

confidence: 99%

Section: Relationship With Connectivitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The combinatorics of a tree-like functional equation for connected chord diagrams

Nabergall

2021

Preprint

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“…As circle graphs are χ-bounded [19], one can actually bound the number of pages in terms of the largest twist size. The currently best result due to Davies and McCarty [13] states that χ(H) ⩽ 7ω(H) 2 for every circle graph H, which gives the following. Already in 2007, Černý [11] proved that χ(H) ⩽ O(ω(H) • log(|V (H)|) for every circle graph H. As the vertices of H correspond to the edges of G in this application, this gives the following.…”

Section: Related Work Nowakowski and Parkermentioning

confidence: 99%

A Sublinear Bound on the Page Number of Upward Planar Graphs

Jungeblut,

Merker,

Ueckerdt

2021

Preprint

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The page number of a directed acyclic graph G is the minimum k for which there is a topological ordering of G and a k-coloring of the edges such that no two edges of the same color cross, i.e., have alternating endpoints along the topological ordering. We address the long-standing open problem asking for the largest page number among all upward planar graphs. We improve the best known lower bound to 5 and present the first asymptotic improvement over the trivial O(n) upper bound, where n denotes the number of vertices in G. Specifically, we first prove that the page number of every upward planar graph is bounded in terms of its width, as well as its height. We then combine both approaches to show that every n-vertex upward planar graph has page number O(n 2/3 log(n) 2/3 ).

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