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Timothy Sun

Publications and source records attributed to Timothy Sun.

At least 19 recordsLinked to original sources
arXiv2026

Embeddings of critical graphs near the Heawood bound

Complementing a theorem of \v{S}krekovski, we characterize the $(h-1)$-critical graphs embeddable in surfaces of Euler genus at least $5$, where $h$ denotes the Heawood number of the surface. Outside of a few small cases, the bulk of our proof is determining the genus of the join of a complete graph and the 5-cycle. As a byproduct of our proof, we also provide a simpler solution to the minimum triangulations problem for nonorientable surfaces using the theory of current graphs.

math.CO↗
arXiv2025

Revisiting Cases 2 and 11 of the Map Color Theorem

In 1968, Ringel and Youngs solved the remaining cases of the orientable Map Color Theorem by finding genus embeddings of the complete graphs $K_n$, for sufficiently large $n \equiv 2, 8, 11 \pmod{12}$. Following the approach previously explored by the author for $n \equiv 8 \pmod{12}$, we aim to streamline their constructions for $n \equiv 2, 11 \pmod{12}$ by finding families of current graphs with simpler patterns for the arc labelings.

math.CO↗
arXiv2024

Genus embeddings of complete graphs minus a matching

We show that for all $n \equiv 0 \pmod{6}$, $n \geq 18$, there is an orientable triangular embedding of the octahedral graph on $n$ vertices that can be augmented with handles to produce a genus embedding of the complete graph of the same order. For these values of $n$, the intermediate embeddings of the construction also determine some surface crossing numbers of the complete graph on $n$ vertices and the genus of all graphs on $n$ vertices and minimum degree $n-2$.

math.CO↗
arXiv2024

An optimal construction for complete graph embeddings with duals of low connectivity

We describe a construction for embeddings of complete graphs where the dual has a cutvertex and the genus is close to the minimum genus of the primal graph. When the number of vertices is congruent to 5 modulo 12, we further guarantee that the dual is simple and that the genera of the resulting embeddings match a lower bound of Brinkmann, Noguchi, and Van den Camp, showing that their lower bound is tight infinitely often.

math.CO↗
arXiv2024

On Kainen's conjectures on surface crossing numbers

In 1972, Kainen proved a general lower bound on the crossing number of a graph in a closed surface and conjectured that this bound is tight when the graph is either a complete graph or a complete bipartite graph, and the surface is of genus close to the minimum genus of that graph. Prior to the present work, these conjectures were known to be true only for small cases and when the conjectures predict a crossing number of 0, i.e., when a triangular or quadrangular embedding was already known. We show that Kainen's conjectures are true except for the three graphs $K_9$, $K_{3,5}$, and $K_{5,5}$. We also prove nonorientable analogues of these conjectures, where the only exceptions to the general formulas are $K_7$ and $K_8$.

math.CO↗
arXiv2023

Settling the nonorientable genus of the nearly complete bipartite graphs

A graph is said to be nearly complete bipartite if it can be obtained by deleting a set of independent edges from a complete bipartite graph. The nonorientable genus of such graphs is known except in a few cases where the sizes of the partite classes differ by at most one, and a maximum matching is deleted. We resolve these missing cases using three classic tools for constructing genus embeddings of the complete bipartite graphs: current graphs, diamond sums, and the direct rotation systems of Ringel.

math.CO↗
arXiv2023

Face-simple minimal quadrangulations of surfaces

For each surface besides the sphere, projective plane, and Klein bottle, we construct a face-simple minimal quadrangulation, i.e., a simple quadrangulation on the fewest number of vertices possible, whose dual is also a simple graph. Our result answers a question of Liu, Ellingham, and Ye while providing a simpler proof of their main result. The inductive construction is based on an earlier idea for finding near-quadrangular embeddings of the complete graphs using the diamond sum operation.

math.CO↗
arXiv2022

Index 3 biembeddings of the complete graphs

We show that the complete graphs on $24s+21$ vertices have decompositions into two edge-disjoint subgraphs, each of which triangulates an orientable surface. The special case where the two surfaces are homeomorphic solves a generalized Earth-Moon problem for that surface. Unlike previous constructions, these pairs of triangular embeddings are derived from index 3 current graphs.

math.CO↗
arXiv2021

On the bigenus of the complete graphs

We describe an infinite family of edge-decompositions of complete graphs into two graphs, each of which triangulate the same orientable surface. Previously, such decompositions had only been known for only a few complete graphs. These so-called biembeddings solve a generalization of the Earth-Moon problem for an infinite number of orientable surfaces.

math.CO↗
arXiv2020

Settling the genus of the $n$-prism

In a 1977 paper, Ringel conjectured a formula for the genus of the $n$-prism $K_n\times K_2$ and verified its correctness for about five-sixths of all values $n$. We complete this calculation by showing that, with the exception of $n = 9$, Ringel's conjecture is true for $n \equiv 5, 9\pmod{12}$.

math.CO↗
arXiv2019

Jungerman ladders and index 2 constructions for genus embeddings of dense regular graphs

We construct several families of minimum genus embeddings of dense graphs using index 2 current graphs. In particular, we complete the genus formula for the octahedral graphs, solving a longstanding conjecture of Jungerman and Ringel, and find triangular embeddings of complete graphs minus a Hamiltonian cycle, making partial progress on a problem of White. Index 2 current graphs are also applied to various cases of the genus of the complete graphs, in some cases yielding simpler solutions, e.g., the nonorientable genus of $K_{12s+8}-K_2$. In addition, we give a topological proof of a theorem of Jungerman that shows that a symmetric type of such current graphs might not exist roughly "half of the time."

math.CO↗
arXiv2019

A Lower Bound on Cycle-Finding in Sparse Digraphs

We consider the problem of finding a cycle in a sparse directed graph $G$ that is promised to be far from acyclic, meaning that the smallest feedback arc set in $G$ is large. We prove an information-theoretic lower bound, showing that for $N$-vertex graphs with constant outdegree any algorithm for this problem must make $\tilde{\Omega}(N^{5/9})$ queries to an adjacency list representation of $G$. In the language of property testing, our result is an $\tilde{\Omega}(N^{5/9})$ lower bound on the query complexity of one-sided algorithms for testing whether sparse digraphs with constant outdegree are far from acyclic. This is the first improvement on the $\Omega(\sqrt{N})$ lower bound, implicit in Bender and Ron, which follows from a simple birthday paradox argument.

cs.DS↗
arXiv2019

Simultaneous current graph constructions for minimum triangulations and complete graph embeddings

The problems of the genus of the complete graphs and minimum triangulations for each surface were both solved using the theory of current graphs, and each of them divided into twelve different cases, depending on the residue modulo 12 of the number of vertices. Cases 8 and 11 were of particular difficulty for both problems, with multiple families of current graphs developed to solve these cases. We solve these cases in a unified manner with families of current graphs applicable to both problems. Additionally, we give new constructions to both problems for Cases 6 and 9, which greatly simplify previous constructions by Ringel, Youngs, Guy, and Jungerman. All these new constructions are index 3 current graphs sharing nearly all of the structure of the simple solution for Case 5 of the Map Color Theorem.

math.CO↗
arXiv2018

Revisiting Mayer: Symmetric solutions for sporadic cases of the Map Color Theorem

The original proof of the genus of the complete graphs $K_n$ depended on Mayer's \emph{ad hoc} solutions for $n = 18, 20, 23$. Recently, an improved solution for $K_{20}$ was found by the author. The purpose of this note is to use the theory of current graphs to interpret the aforementioned result and to provide new embeddings of $K_{18}$ and $K_{23}$.

math.CO↗
arXiv2017

Face distributions of embeddings of complete graphs

A longstanding open question of Archdeacon and Craft asks whether every complete graph has a minimum genus embedding with at most one nontriangular face. We exhibit such an embedding for each complete graph except $K_8$, the complete graph on 8 vertices, and we go on to prove that no such embedding can exist for this graph. Our approach also solves a more general problem, giving a complete characterization of the possible face distributions (i.e. the numbers of faces of each length) realizable by minimum genus embeddings of each complete graph. We also tackle analogous questions for nonorientable and maximum genus embeddings.

math.CO↗
arXiv2017

Sample-based high-dimensional convexity testing

In the problem of high-dimensional convexity testing, there is an unknown set $S \subseteq \mathbb{R}^n$ which is promised to be either convex or $\varepsilon$-far from every convex body with respect to the standard multivariate normal distribution $\mathcal{N}(0, 1)^n$. The job of a testing algorithm is then to distinguish between these two cases while making as few inspections of the set $S$ as possible. In this work we consider sample-based testing algorithms, in which the testing algorithm only has access to labeled samples $(\boldsymbol{x},S(\boldsymbol{x}))$ where each $\boldsymbol{x}$ is independently drawn from $\mathcal{N}(0, 1)^n$. We give nearly matching sample complexity upper and lower bounds for both one-sided and two-sided convexity testing algorithms in this framework. For constant $\varepsilon$, our results show that the sample complexity of one-sided convexity testing is $2^{\tilde{\Theta}(n)}$ samples, while for two-sided convexity testing it is $2^{\tilde{\Theta}(\sqrt{n})}$.

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