arXiv subjects

Jane Tan

Publications and source records attributed to Jane Tan.

24 records · Page 2Linked to original sources

arXiv2022

Product structure of graph classes with bounded treewidth

We show that many graphs with bounded treewidth can be described as subgraphs of the strong product of a graph with smaller treewidth and a bounded-size complete graph. To this end, define the "underlying treewidth" of a graph class $\mathcal{G}$ to be the minimum non-negative integer $c$ such that, for some function $f$, for every graph ${G \in \mathcal{G}}$ there is a graph $H$ with ${\text{tw}(H) \leq c}$ such that $G$ is isomorphic to a subgraph of ${H \boxtimes K_{f(\text{tw}(G))}}$. We introduce disjointed coverings of graphs and show they determine the underlying treewidth of any graph class. Using this result, we prove that the class of planar graphs has underlying treewidth 3; the class of $K_{s,t}$-minor-free graphs has underlying treewidth $s$ (for ${t \geq \max\{s,3\}}$); and the class of $K_t$-minor-free graphs has underlying treewidth ${t-2}$. In general, we prove that a monotone class has bounded underlying treewidth if and only if it excludes some fixed topological minor. We also study the underlying treewidth of graph classes defined by an excluded subgraph or excluded induced subgraph. We show that the class of graphs with no $H$ subgraph has bounded underlying treewidth if and only if every component of $H$ is a subdivided star, and that the class of graphs with no induced $H$ subgraph has bounded underlying treewidth if and only if every component of $H$ is a star.

math.CO↗

arXiv2022

A note on infinite antichain density

Let $\mathcal{F}$ be an antichain of finite subsets of $\mathbb{N}$. How quickly can the quantities $|\mathcal{F}\cap 2^{[n]}|$ grow as $n\to\infty$? We show that for any sequence $(f_n)_{n\ge n_0}$ of positive integers satisfying $\sum_{n=n_0}^\infty f_n/2^n \le 1/4$, $f_{n_0}=1$ and $f_n\le f_{n+1}\le 2f_n$, there exists an infinite antichain $\mathcal{F}$ of finite subsets of $\mathbb{N}$ such that $|\mathcal{F}\cap 2^{[n]}| \geq f_n$ for all $n\ge n_0$. It follows that for any $\varepsilon>0$ there exists an antichain $\mathcal{F}\subseteq 2^\mathbb{N}$ such that $$\liminf_{n \to \infty} |\mathcal{F}\cap 2^{[n]}| \cdot \left(\frac{2^n}{n\log^{1+\varepsilon} n}\right)^{-1} > 0.$$ This resolves a problem of Sudakov, Tomon and Wagner in a strong form, and is essentially tight.

math.CO↗

arXiv2022

On Comparable Box Dimension

Two boxes in $\mathbb{R}^d$ are comparable if one of them is a subset of a translation of the other one. The comparable box dimension of a graph $G$ is the minimum integer $d$ such that $G$ can be represented as a touching graph of comparable axis-aligned boxes in $\mathbb{R}^d$. We show that proper minor-closed classes have bounded comparable box dimensions and explore further properties of this notion.

cs.DM↗

arXiv2021

Graph Pseudometrics from a Topological Point of View

We explore pseudometrics for directed graphs in order to better understand their topological properties. The directed flag complex associated to a directed graph provides a useful bridge between network science and topology. Indeed, it has often been observed that phenomena exhibited by real-world networks reflect the topology of their flag complexes, as measured, for example, by Betti numbers or simplex counts. As it is often computationally expensive (or even unfeasible) to determine such topological features exactly, it would be extremely valuable to have pseudometrics on the set of directed graphs that can both detect the topological differences and be computed efficiently. To facilitate work in this direction, we introduce methods to measure how well a graph pseudometric captures the topology of a directed graph. We then use these methods to evaluate some well-established pseudometrics, using test data drawn from several families of random graphs.

math.AT↗

arXiv2019

Eulerian circuits and path decompositions in quartic planar graphs

A subcycle of an Eulerian circuit is a sequence of edges that are consecutive in the circuit and form a cycle. We characterise the quartic planar graphs that admit Eulerian circuits avoiding 3-cycles and 4-cycles. From this, it follows that a quartic planar graph of order $n$ can be decomposed into $k_1+k_2+k_3+k_4$ many paths with $k_i$ copies of $P_{i+1}$, the path with $i$ edges, if and only if $k_1+2k_2+3k_3+4k_4 = 2n$. In particular, every connected quartic planar graph of even order admits a $P_5$-decomposition.

math.CO↗

arXiv2019

Small 4-regular planar graphs that are not circle representable

A 4-regular planar graph $G$ is said to be circle representable if there exists a collection of circles drawn on the plane such that the touching and crossing points correspond to the vertices of $G$, and the circular arcs between those points correspond to the edges of $G$. Lovász (1970) conjectured that every 4-regular planar graph has a circle representation, but an infinite family of counterexamples was given by Bekos and Raftopoulou (2015). We reduce the order of the smallest known counterexamples among simple graphs from 822 to 68 based on a multigraph counterexample of order 12.

math.CO↗