arXiv subjects

Po-Shen Loh

Publications and source records attributed to Po-Shen Loh.

At least 19 recordsLinked to original sources

arXiv2025

A cute proof that makes $e$ natural

The number $e$ has rich connections throughout mathematics, and has the honor of being the base of the natural logarithm. However, most students finish secondary school (and even university) without suitably memorable intuition for why $e$'s various mathematical properties are related. This article presents a solution. Various proofs for all of the mathematical facts in this article have been well-known for years. This exposition contributes a short, conceptual, intuitive, and visual proof (comprehensible to Pre-Calculus students) of the equivalence of two of the most commonly-known properties of $e$, connecting the continuously-compounded-interest limit $\big(1 + \frac{1}{n}\big)^n$ to the fact that $e^x$ is its own derivative. The exposition further deduces a host of commonly-taught properties of $e$, while minimizing pre-requisite knowledge, so that this article can be practically used for developing secondary school curricula. Since $e$ is such a well-trodden concept, it is hard to imagine that our visual proof is new, but it certainly is not widely known. The author checked 100 books across 7 countries, as well as YouTube videos totaling over 25 million views, and still has not found this method taught anywhere. This article seeks to popularize the 3-page explanation of $e$, while providing a unified, practical, and open-access reference for teaching about $e$.

math.HO↗

arXiv2020

Flipping the Perspective in Contact Tracing

We introduce a fundamentally different paradigm for contact tracing: for each positive case, do not only ask direct contacts to quarantine; instead, tell everyone how many relationships away the disease just struck (so, "2" is a close physical contact of a close physical contact). This new approach, which has already been deployed in a publicly downloadable app, brings a new tool to bear on pandemic control, powered by network theory. Like a weather satellite providing early warning of incoming hurricanes, it empowers individuals to see transmission approaching from far away, and incites behavior change to directly avoid exposure. This flipped perspective engages natural self-interested instincts of self-preservation, reducing reliance on altruism, and the resulting caution reduces pandemic spread in the social vicinity of each infection. Consequently, our new system solves the behavior coordination problem which has hampered many other app-based interventions to date. We also provide a heuristic mathematical analysis that shows how our system already achieves critical mass from the user perspective at very low adoption thresholds (likely below 10% in some common types of communities as indicated empirically in the first practical deployment); after that point, the design of our system naturally accelerates further adoption, while also alerting even non-users of the app. This article seeks to lay the theoretical foundation for our approach, and to open the area for further research along many dimensions.

cs.CY↗

arXiv2020

Large rainbow matchings in edge-colored graphs

A subgraph of an edge-colored graph is called \emph{rainbow} if all of its edges have distinct colors. There has been much research on the topic of finding a large rainbow matching in a properly edge-colored graph, where a proper edge-coloring is a coloring of the edge set such that no same-colored edges are incident. Gao, Ramadurai, Wanless, and Wormald proved that in every proper edge-coloring of a graph with $n$ colors where each color appears at least $n+o(n)$ times, there is always a rainbow matching using every color. We strengthen this result by simultaneously relaxing three conditions: (i) we lift the condition on the number of colors and allow any finite number of colors and instead, put a weaker condition requiring the maximum degree of the graph to be at most $n$, (ii) we relax the proper coloring condition and require that the graph induced by each of the colors have maximum degree $o(n)$, and (iii) we work in a more general setting of multigraphs allowing edge multiplicities to be $o(n)$. As an application of this result, we show that for every proper edge-coloring of a graph with $2n+o(n)$ colors where each color appears at least $n$ times, there is always a rainbow matching of size $n$. Aharoni and Berger conjectured that $2n+o(n)$ can be replaced by $n+1$ in this statement. We dispute this conjecture with an explicit construction.

math.CO↗

arXiv2020

Minimizing the numbers of cliques and cycles of fixed size in an $F$-saturated graph

This paper considers two important questions in the well-studied theory of graphs that are $F$-saturated. A graph $G$ is called $F$-saturated if $G$ does not contain a subgraph isomorphic to $F$, but the addition of any edge creates a copy of $F$. We first resolve a fundamental question of minimizing the number of cliques of size $r$ in a $K_s$-saturated graph for all sufficiently large numbers of vertices, confirming a conjecture of Kritschgau, Methuku, Tait, and Timmons. We also go further and prove a corresponding stability result. Next we minimize the number of cycles of length $r$ in a $K_s$-saturated graph for all sufficiently large numbers of vertices, and classify the extremal graphs for most values of $r$, answering another question of Kritschgau, Methuku, Tait, and Timmons for most $r$. We then move on to a central and longstanding conjecture in graph saturation made by Tuza, which states that for every graph $F$, the limit $\lim_{n \rightarrow \infty} \frac{\sat(n, F)}{n}$ exists, where $\sat(n, F)$ denotes the minimum number of edges in an $n$-vertex $F$-saturated graph. Pikhurko made progress in the negative direction by considering families of graphs instead of a single graph, and proved that there exists a graph family $\mathcal{F}$ of size $4$ for which $\lim_{n \rightarrow \infty} \frac{\sat(n, \mathcal{F})}{n}$ does not exist (for a family of graphs $\mathcal{F}$, a graph $G$ is called $\mathcal{F}$-saturated if $G$ does not contain a copy of any graph in $\mathcal{F}$, but the addition of any edge creates a copy of a graph in $\mathcal{F}$, and $\sat(n, \mathcal{F})$ is defined similarly). We make the first improvement in 15 years by showing that there exist infinitely many graph families of size $3$ where this limit does not exist. Our construction also extends to the generalized saturation problem when we minimize the number of fixed-size cliques.

math.CO↗

arXiv2020

Extremal graphs with local covering conditions

We systematically study a natural problem in extremal graph theory, to minimize the number of edges in a graph with a fixed number of vertices, subject to a certain local condition: each vertex must be in a copy of a fixed graph $H$. We completely solve this problem when $H$ is a clique, as well as more generally when $H$ is any regular graph with degree at least about half its number of vertices. We also characterize the extremal graphs when $H$ is an Erdős-Rényi random graph. The extremal structures turn out to have the similar form as the conjectured extremal structures for a well-studied but elusive problem of similar flavor with local constraints: to maximize the number of copies of a fixed clique in graphs in which all degrees have a fixed upper bound.

math.CO↗

arXiv2019

A Simple Proof of the Quadratic Formula

This article provides a simple proof of the quadratic formula, which also produces an efficient and natural method for solving general quadratic equations. The derivation is computationally light and conceptually natural, and has the potential to demystify quadratic equations for students worldwide.

math.HO↗

arXiv2018

The random k-matching-free process

Let $\mathcal{P}$ be a graph property which is preserved by removal of edges, and consider the random graph process that starts with the empty $n$-vertex graph and then adds edges one-by-one, each chosen uniformly at random subject to the constraint that $\mathcal{P}$ is not violated. These types of random processes have been the subject of extensive research over the last 20 years, having striking applications in extremal combinatorics, and leading to the discovery of important probabilistic tools. In this paper we consider the $k$-matching-free process, where $\mathcal{P}$ is the property of not containing a matching of size $k$. We are able to analyse the behaviour of this process for a wide range of values of $k$; in particular we prove that if $k=o(n)$ or if $n-2k=o(\sqrt{n}/\log n)$ then this process is likely to terminate in a $k$-matching-free graph with the maximum possible number of edges, as characterised by Erdős and Gallai. We also show that these bounds on $k$ are essentially best possible, and we make a first step towards understanding the behaviour of the process in the intermediate regime.

math.CO↗

arXiv2018

Packing Hamilton Cycles Online

It is known that w.h.p. the hitting time $τ_{2σ}$ for the random graph process to have minimum degree $2σ$ coincides with the hitting time for $σ$ edge disjoint Hamilton cycles. In this paper we prove an online version of this property. We show that, for a fixed integer $σ\geq 2$, if random edges of $K_n$ are presented one by one then w.h.p. it is possible to color the edges online with $σ$ colors so that at time $τ_{2σ}$, each color class is Hamiltonian.

math.CO↗

arXiv2017

Induced Turán numbers

The classical Kővári-Sós-Turán theorem states that if $G$ is an $n$-vertex graph with no copy of $K_{s,t}$ as a subgraph, then the number of edges in $G$ is at most $O(n^{2-1/s})$. We prove that if one forbids $K_{s,t}$ as an induced/ subgraph, and also forbids any/ fixed graph $H$ as a (not necessarily induced) subgraph, the same asymptotic upper bound still holds, with different constant factors. This introduces a nontrivial angle from which to generalize Turán theory to induced forbidden subgraphs, which this paper explores. Along the way, we derive a nontrivial upper bound on the number of cliques of fixed order in a $K_r$-free graph with no induced copy of $K_{s,t}$. This result is an induced analog of a recent theorem of Alon and Shikhelman and is of independent interest.

math.CO↗

arXiv2017

Distance-Uniform Graphs with Large Diameter

An $ε$-distance-uniform graph is one in which from every vertex, all but an $ε$-fraction of the remaining vertices are at some fixed distance $d$, called the critical distance. We consider the maximum possible value of $d$ in an $ε$-distance-uniform graph with $n$ vertices. We show that for $\frac1n \le ε\le \frac1{\log n}$, there exist $ε$-distance-uniform graphs with critical distance $2^{Ω(\frac{\log n}{\log ε^{-1}})}$, disproving a conjecture of Alon et al. that $d$ can be at most logarithmic in $n$. We also show that our construction is best possible, in the sense that an upper bound on $d$ of the form $2^{O(\frac{\log n}{\log ε^{-1}})}$ holds for all $ε$ and $n$.

math.CO↗

arXiv2017

Classifying unavoidable Tverberg partitions

Let $T(d,r) = (r-1)(d+1)+1$ be the parameter in Tverberg's theorem, and call a partition $\mathcal I$ of $\{1,2,\ldots,T(d,r)\}$ into $r$ parts a "Tverberg type". We say that $\mathcal I$ "occurs" in an ordered point sequence $P$ if $P$ contains a subsequence $P'$ of $T(d,r)$ points such that the partition of $P'$ that is order-isomorphic to $\mathcal I$ is a Tverberg partition. We say that $\mathcal I$ is "unavoidable" if it occurs in every sufficiently long point sequence. In this paper we study the problem of determining which Tverberg types are unavoidable. We conjecture a complete characterization of the unavoidable Tverberg types, and we prove some cases of our conjecture for $d\le 4$. Along the way, we study the avoidability of many other geometric predicates. Our techniques also yield a large family of $T(d,r)$-point sets for which the number of Tverberg partitions is exactly $(r-1)!^d$. This lends further support for Sierksma's conjecture on the number of Tverberg partitions.

cs.CG↗

arXiv2015

Directed paths: from Ramsey to Ruzsa and Szemerédi

Starting from an innocent Ramsey-theoretic question regarding directed paths in tournaments, we discover a series of rich and surprising connections that lead into the theory around a fundamental problem in Combinatorics: the Ruzsa-Szemerédi induced matching problem. Using these relationships, we prove that every coloring of the edges of the transitive $n$-vertex tournament using three colors contains a directed path of length at least $\sqrt{n} \cdot e^{\log^* n}$ which entirely avoids some color. We also expose connections to a family of constructions for Ramsey tournaments, and introduce and resolve some natural generalizations of the Ruzsa-Szemerédi problem which we encounter through our investigation.

math.CO↗

arXiv2015

Cops and robbers on planar directed graphs

Aigner and Fromme initiated the systematic study of the cop number of a graph by proving the elegant and sharp result that in every connected planar graph, three cops are sufficient to win a natural pursuit game against a single robber. This game, introduced by Nowakowski and Winkler, is commonly known as Cops and Robbers in the combinatorial literature. We extend this study to directed planar graphs, and establish separation from the undirected setting. We exhibit a geometric construction which shows that a more sophisticated robber strategy can indefinitely evade three cops on a particular strongly connected planar directed graph.

math.CO↗

arXiv2014

Judicious partitions of directed graphs

The area of judicious partitioning considers the general family of partitioning problems in which one seeks to optimize several parameters simultaneously, and these problems have been widely studied in various combinatorial contexts. In this paper, we study essentially the most fundamental judicious partitioning problem for directed graphs, which naturally extends the classical Max Cut problem to this setting: we seek bipartitions in which many edges cross in each direction. It is easy to see that a minimum outdegree condition is required in order for the problem to be nontrivial, and we prove that every directed graph with M edges and minimum outdegree at least two admits a bipartition in which at least (1/6 + o(1))M edges cross in each direction. We also prove that if the minimum outdegree is at least three, then the constant can be increased to 1/5. If the minimum outdegree tends to infinity with N, then the constant increases to 1/4. All of these constants are best-possible, and provide asymptotic answers to a question of Alex Scott.

math.CO↗

arXiv2014

The critical window for the classical Ramsey-Turán problem

The first application of Szemerédi's powerful regularity method was the following celebrated Ramsey-Turán result proved by Szemerédi in 1972: any K_4-free graph on N vertices with independence number o(N) has at most (1/8 + o(1)) N^2 edges. Four years later, Bollobás and Erdős gave a surprising geometric construction, utilizing the isoperimetric inequality for the high dimensional sphere, of a K_4-free graph on N vertices with independence number o(N) and (1/8 - o(1)) N^2 edges. Starting with Bollobás and Erdős in 1976, several problems have been asked on estimating the minimum possible independence number in the critical window, when the number of edges is about N^2 / 8. These problems have received considerable attention and remained one of the main open problems in this area. In this paper, we give nearly best-possible bounds, solving the various open problems concerning this critical window.

math.CO↗

arXiv2014

Diameter critical graphs

A graph is called diameter-$k$-critical if its diameter is $k$, and the removal of any edge strictly increases the diameter. In this paper, we prove several results related to a conjecture often attributed to Murty and Simon, regarding the maximum number of edges that any diameter-$k$-critical graph can have. In particular, we disprove a longstanding conjecture of Caccetta and Häggkvist (that in every diameter-2-critical graph, the average edge-degree is at most the number of vertices), which promised to completely solve the extremal problem for diameter-2-critical graphs. On the other hand, we prove that the same claim holds for all higher diameters, and is asymptotically tight, resolving the average edge-degree question in all cases except diameter-2. We also apply our techniques to prove several bounds for the original extremal question, including the correct asymptotic bound for diameter-$k$-critical graphs, and an upper bound of $(\frac{1}{6} + o(1))n^2$ for the number of edges in a diameter-3-critical graph.

math.CO↗

arXiv2014

Packing tree factors in random and pseudo-random graphs

For a fixed graph H with t vertices, an H-factor of a graph G with n vertices, where t divides n, is a collection of vertex disjoint (not necessarily induced) copies of H in G covering all vertices of G. We prove that for a fixed tree T on t vertices and ε> 0, the random graph G_{n,p}, with n a multiple of t, with high probability contains a family of edge-disjoint T-factors covering all but an ε-fraction of its edges, as long as ε^4 n p >> (log n)^2. Assuming stronger divisibility conditions, the edge probability can be taken down to p > (C log n)/n. A similar packing result is proved also for pseudo-random graphs, defined in terms of their degrees and co-degrees.

math.CO↗

arXiv2014

Hamiltonian increasing paths in random edge orderings

If the edges of the complete graph $K_n$ are totally ordered, a simple path whose edges are in ascending order is called increasing. The worst-case length of the longest increasing path has remained an open problem for several decades, with asymptotic bounds between $\sqrt{n}$ (Graham and Kleitman, 1973) and $n/2$ (Calderbank, Chung, and Sturtevant, 1984). We consider the average case, when the ordering is chosen uniformly at random. We discover the surprising result that in the random setting, an increasing path of the maximum possible length of $n-1$ exists with probability at least about $1/e$. We also prove that with probability $1-o(1)$, there is an increasing path of length at least $0.85n$, suggesting that this Hamiltonian (or near-Hamiltonian) phenomenon may hold asymptotically almost surely.

math.CO↗