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Sean English

Publications and source records attributed to Sean English.

At least 19 recordsLinked to original sources
arXiv2026

Helly Theorems for Generalized Tur\'an Problems

Given a graph $T$ and a family of graphs $\mathcal{F}$, the generalized Tur\'an number $\mathrm{ex}(n,T,\mathcal{F})$ is the maximum number of copies of $T$ in an $n$-vertex $\mathcal{F}$-free graph. We prove a general theorem which states that for any tree $T$, any family $\mathcal{F}$, and any integer $k$, either $\mathrm{ex}(n,T,\mathcal{F})$ is at least $\Omega(n^{k+1})$ or at most $O(\mathrm{ex}(n,\mathcal{F})^{k})$, from which we derive a number of consequences. Our proofs rely on new variants of the classical Helly Theorem for trees which may be of independent interest. As far as we are aware, this is the first known application of Helly theorems for Tur\'an type problems.

math.CO↗
arXiv2026

On the Domatic Game

The domatic game with pallete size $k$ is a $2$-player game played on a graph $G$ recently introduced by Hartnell and Rall. Players Alice and Bob take turns choosing an uncolored vertex from $G$, and coloring it a color from $\{1,2,\dots,k\}$. The game ends once all vertices in $G$ have been assigned a color. Alice wins if all $k$ colors induce a dominating set of $G$, and otherwise Bob wins. The domatic game number, $\operatorname{dom_g}(G,X)$ is the the largest pallete size $k$ such that Alice wins the domatic game when player $X$ goes first (where $X$ is either Alice or Bob). We prove for any graph $G$ of order $n$, \[ \operatorname{dom_g}(G,X)=\Omega\left(\frac{\delta(G)}{\log n}\right). \] In addition, we show that for any $k$ there exists a graph $G$ with minimum degree $\delta(G)=k$ and $\operatorname{dom_g}(G,X)=1$, and there exists a graph $G'$ with $\operatorname{dom_g}(G',X)=1$ while having (non-game) domatic number $\operatorname{dom}(G')=k$. We explore how the domatic game number changes when changing who goes first, and when considering subgraphs of $G$. We also introduce a score variant of the domatic game, and use this to get bounds on the original domatic game.

math.CO↗
arXiv2026

Cost-Benefit Analysis for PMU Placement in Power Grids

Power domination is a graph-theoretic model for the observance of a power grid using phasor measurement units (PMUs). There are many costs associated with the installation of a PMU, but also costs associated with not observing the entire power grid. In this work, we propose and study a power domination cost function, which balances these two costs. Given a graph $G$, a set of sensor locations $S$, and a parameter $\beta$ (which is the ratio of the cost of a PMU to the cost of non-observance of any given vertex), we define the cost function \[ \mathrm{C}(G;S,\beta)=|S|+\beta\cdot (|V(G)|-|\mathrm{Obs}(G;S)|) \] where $|\mathrm{Obs}(G;S)|$ is the number of vertices observed by sensors placed at $S\subseteq V(G)$ in the power domination process. We explore the values of $k$ for which there is a set $S$ of size $k$ that minimizes this cost function, and explore which values of $\beta$ guarantee that it is optimal to observe the entire power grid to minimize cost. We also introduce notions of marginal cost and marginal observance, providing tools to analyze how many PMUs one should install on a given power grid.

math.CO↗
arXiv2025

On Fragile Power Domination

Power domination is a graph theoretic model which captures how phasor measurement units (PMUs) can be used to monitor a power grid. Fragile power domination takes into account the fact that PMUs may break or otherwise fail. In this model, each sensor fails independently with probability $q\in [0,1]$ and the surviving sensors monitor the grid according to classical power domination. We study the expected number of observed nodes under the fragile power domination model. We give a characterization for when two networks and initial sensor placements will behave the same according to this expectation. We also show how to control the behavior of this expectation by adding structure to a network.

math.CO↗
arXiv2025

Rational Exponents for General Graphs

A rational number $r$ is a \textbf{realizable exponent} for a graph $H$ if there exists a finite family of graphs $\mathcal{F}$ such that $\mathrm{ex}(n,H,\mathcal{F})=\Theta(n^r)$, where $\mathrm{ex}(n,H,\mathcal{F})$ denotes the maximum number of copies of $H$ that an $n$-vertex $\mathcal{F}$-free graph can have. Results for realizable exponents are currently known only when $H$ is either a star or a clique, with the full resolution of the $H=K_2$ case being a major breakthrough of Bukh and Conlon. In this paper, we establish the first set of results for realizable exponents which hold for arbitrary graphs $H$ by showing that for any graph $H$ with maximum degree $\Delta \ge 1$, every rational in the interval $\left[v(H)-\frac{e(H)}{2\Delta^2},\ v(H)\right]$ is realizable for $H$. We also prove a ``stability'' result for generalized Tur\'an numbers of trees which implies that if $T\ne K_2$ is a tree with $\ell$ leaves, then $T$ has no realizable exponents in $[0,\ell]\setminus \mathbb{Z}$. Our proof of this latter result uses a new variant of the classical Helly theorem for trees, which may be of independent interest.

math.CO↗
arXiv2025

The Saturation Spectrum of Berge Stars

The forbidden subgraph problem is among the oldest in extremal combinatorics -- how many edges can an $n$-vertex $F$-free graph have? The answer to this question is the well-studied extremal number of $F$. Observing that every extremal example must be maximally $F$-free, a natural minimization problem is also studied -- how few edges can an $n$-vertex maximal $F$-free graph have? This leads to the saturation number of $F$. Both of these problems are notoriously difficult to extend to $k$-uniform hypergraphs for any $k\ge 3$. Barefoot et al., in the case of forbidding triangles in graphs, asked a beautiful question -- which numbers of edges, between the saturation number and the extremal number, are actually realized by an $n$-vertex maximal $F$-free graph? Hence named the saturation spectrum of $F$, this has since been determined precisely for several classes of graphs through a large number of papers over the past two decades. In this paper, we extend the notion of the saturation spectrum to the hypergraph context. Given a graph $F$ and a hypergraph $G$ embedded on the same vertex set, we say $G$ is a {\bf{Berge-$F$}} if there exists a bijection $\phi:E(F)\to E(G)$ such that $e\subseteq \phi(e)$ for all $e\in E(F)$. We completely determine the saturation spectrum for $3$-uniform Berge-$K_{1,\ell}$ for $1\leq \ell\leq 4$, and for $\ell=5$ when $5\mid n$. We also determine all but a constant number of values in the spectrum for $3$-uniform Berge-$K_{1,\ell}$ for all $\ell\geq 5$. We note that this is the first result determining the saturation spectrum for any non-trivial hypergraph.

math.CO↗
arXiv2024

Rational exponents for cliques

Let $\mathrm{ex}(n,H,\mathcal{F})$ be the maximum number of copies of $H$ in an $n$-vertex graph which contains no copy of a graph from $\mathcal{F}$. Thinking of $H$ and $\mathcal{F}$ as fixed, we study the asymptotics of $\mathrm{ex}(n,H,\mathcal{F})$ in $n$. We say that a rational number $r$ is \emph{realizable for $H$} if there exists a finite family $\mathcal{F}$ such that $\mathrm{ex}(n,H,\mathcal{F}) = \Theta(n^r)$. Using randomized algebraic constructions, Bukh and Conlon showed that every rational between $1$ and $2$ is realizable for $K_2$. We generalize their result to show that every rational between $1$ and $t$ is realizable for $K_t$, for all $t \geq 2$. We also determine the realizable rationals for stars and note the connection to a related Sidorenko-type supersaturation problem.

math.CO↗
arXiv2024

Understanding High-Order Network Structure using Permissible Walks on Attributed Hypergraphs

Hypergraphs have been a recent focus of study in mathematical data science as a tool to understand complex networks with high-order connections. One question of particular relevance is how to leverage information carried in hypergraph attributions when doing walk-based techniques. In this work, we focus on a new generalization of a walk in a network that recovers previous approaches and allows for a description of permissible walks in hypergraphs. Permissible walk graphs are constructed by intersecting the attributed $s$-line graph of a hypergraph with a relation respecting graph. The attribution of the hypergraph's line graph commonly carries over information from categorical and temporal attributions of the original hypergraph. To demonstrate this approach on a temporally attributed example, we apply our framework to a Reddit data set composed of hyperedges as threads and authors as nodes where post times are tracked.

cs.SI↗
arXiv2023

On the Constructor-Blocker Game

In the Constructor-Blocker game, two players, Constructor and Blocker, alternatively claim unclaimed edges of the complete graph $K_n$. For given graphs $F$ and $H$, Constructor can only claim edges that leave her graph $F$-free, while Blocker has no restrictions. Constructor's goal is to build as many copies of $H$ as she can, while Blocker attempts to stop this. The game ends once there are no more edges that Constructor can claim. The score $g(n,H,F)$ of the game is the number of copies of $H$ in Constructor's graph at the end of the game, when both players play optimally and Constructor plays first. In this paper, we extend results of Patk\'os, Stojakovi\'c and Vizer on $g(n, H, F)$ to many pairs of $H$ and $F$: We determine $g(n, H, F)$ when $H=K_r$ and $\chi(F)>r$, also when both $H$ and $F$ are odd cycles, using Szemer\'edi's Regularity Lemma. We also obtain bounds of $g(n, H, F)$ when $H=K_3$ and $F=K_{2,2}$.

math.CO↗
arXiv2023

Saturation Numbers for Berge Cliques

Let $F$ be a graph and $\mathcal{H}$ be a hypergraph, both embedded on the same vertex set. We say $\mathcal{H}$ is a Berge-$F$ if there exists a bijection $\phi:E(F)\to E(\mathcal{H})$ such that $e\subseteq \phi(e)$ for all $e\in E(F)$. We say $\mathcal{H}$ is Berge-$F$-saturated if $\mathcal{H}$ does not contain any Berge-$F$, but adding any missing edge to $\mathcal{H}$ creates a copy of a Berge-$F$. The saturation number $\mathrm{sat}_k(n,\text{Berge-}F)$ is the least number of edges in a Berge-$F$-saturated $k$-uniform hypergraph on $n$ vertices. We show \[ \mathrm{sat}_k(n,\text{Berge-}K_\ell)\sim \frac{\ell-2}{k-1}n, \] for all $k,\ell\geq 3$. Furthermore, we provide some sufficient conditions to imply that $\mathrm{sat}_k(n,\text{Berge-}F)=O(n)$ for general graphs $F$.

math.CO↗
arXiv2022

A random coloring process gives improved bounds for the Erd\H{o}s-Gy\'arf\'as problem on generalized Ramsey numbers

The Erd\H{o}s-Gy\'arf\'as number $f(n, p, q)$ is the smallest number of colors needed to color the edges of the complete graph $K_n$ so that all of its $p$-clique spans at least $q$ colors. In this paper we improve the best known upper bound on $f(n, p, q)$ for many fixed values of $p, q$ and large $n$. Our proof uses a randomized coloring process, which we analyze using the so-called differential equation method to establish dynamic concentration.

math.CO↗
arXiv2022

Rainbow Connection for Complete Multipartite Graphs

A path in an edge-colored graph is said to be rainbow if no color repeats on it. An edge-colored graph is said to be rainbow $k$-connected if every pair of vertices is connected by $k$ internally disjoint rainbow paths. The rainbow $k$-connection number $\mathrm{rc}_k(G)$ is the minimum number of colors $\ell$ such that there exists a coloring with $\ell$ colors that makes $G$ rainbow $k$-connected. Let $f(k,t)$ be the minimum integer such that every $t$-partite graph with part sizes at least $f(k,t)$ has $\mathrm{rc}_k(G) \le 4$ if $t=2$ and $\mathrm{rc}_k(G) \le 3$ if $t \ge 3$. Answering a question of Fujita, Liu and Magnant, we show that \[ f(k,t) = \left\lceil \frac{2k}{t-1} \right\rceil \] for all $k\geq 2$, $t\geq 2$. We also give some conditions for which $\mathrm{rc}_k(G) \le 3$ if $t=2$ and $\mathrm{rc}_k(G) \le 2$ if $t \ge 3$.

math.CO↗
arXiv2022

Saturation for the $3$-uniform loose $3$-cycle

Let $F$ and $H$ be $k$-uniform hypergraphs. We say $H$ is $F$-saturated if $H$ does not contain a subgraph isomorphic to $F$, but $H+e$ does for any hyperedge $e\not\in E(H)$. The saturation number of $F$, denoted $\mathrm{sat}_k(n,F)$, is the minimum number of edges in a $F$-saturated $k$-uniform hypergraph $H$ on $n$ vertices. Let $C_3^{(3)}$ denote the $3$-uniform loose cycle on $3$ edges. In this work, we prove that \[ \left(\frac{4}3+o(1)\right)n\leq \mathrm{sat}_3(n,C_3^{(3)})\leq \frac{3}2n+O(1). \] This is the first non-trivial result on the saturation number for a fixed short hypergraph cycle.

math.CO↗
arXiv2021

Chasing the Threshold Bias of the 3-AP Game

In a Maker-Breaker game there are two players, Maker and Breaker, where Maker wins if they create a specified structure while Breaker wins if they prevent Maker from winning indefinitely. A $3$-term arithmetic progression, or $3$-AP, is a sequence of three distinct integers $a, b, c$ such that $b-a = c-b$. The $3$-AP game is a biased Maker-Breaker game played on $[n]$ where every round Breaker selects $q$ unclaimed integers for every Maker's one integer. Maker is trying to select points such that they have a $3$-AP and Breaker is trying to prevent this. The main question of interest is determining the threshold bias $q^*(n)$, that is the minimum value of $q=q(n)$ for which Breaker has a winning strategy. Kusch, Ru\'e, Spiegel and Szab\'o initially asked this question and proved $\sqrt{n/12-1/6}\leq q^*(n)\leq \sqrt{3n}$. We find new strategies for both Maker and Breaker which improve the existing bounds to \[ (1+o(1))\sqrt{\frac{n}{5.6}} \leq q^*(n) \leq \sqrt{2n} +O(1). \]

math.CO↗
arXiv2021

Linear Bounds for Cycle-free Saturation Games

Given a family of graphs $\mathcal{F}$, we define the $\mathcal{F}$-saturation game as follows. Two players alternate adding edges to an initially empty graph on $n$ vertices, with the only constraint being that neither player can add an edge that creates a subgraph in $\mathcal{F}$. The game ends when no more edges can be added to the graph. One of the players wishes to end the game as quickly as possible, while the other wishes to prolong the game. We let $\textrm{sat}_g(n,\mathcal{F})$ denote the number of edges that are in the final graph when both players play optimally. In general there are very few non-trivial bounds on the order of magnitude of $\textrm{sat}_g(n,\mathcal{F})$. In this work, we find collections of infinite families of cycles $\mathcal{C}$ such that $\textrm{sat}_g(n,\mathcal{C})$ has linear growth rate.

math.CO↗
arXiv2021

Low Diameter Monochromatic Covers of Complete Multipartite Graphs

Let the diameter cover number, $D^t_r(G)$, denote the least integer $d$ such that under any $r$-coloring of the edges of the graph $G$, there exists a collection of $t$ monochromatic subgraphs of diameter at most $d$ such that every vertex of $G$ is contained in at least one of the subgraphs. We explore the diameter cover number with two colors and two subgraphs when $G$ is a complete multipartite graph with at least three parts. We determine exactly the value of $D_2^2(G)$ for all complete tripartite graphs $G$, and almost all complete multipartite graphs with more than three parts.

math.CO↗
arXiv2021

Lower bounds on the Erd\H{o}s-Gy\'arf\'as problem via color energy graphs

Given positive integers $p$ and $q$, a $(p,q)$-coloring of the complete graph $K_n$ is an edge-coloring in which every $p$-clique receives at least $q$ colors. Erd\H{o}s and Shelah posed the question of determining $f(n,p,q)$, the minimum number of colors needed for a $(p,q)$-coloring of $K_n$. In this paper, we expand on the color energy technique introduced by Pohoata and Sheffer to prove new lower bounds on this function, making explicit the connection between bounds on extremal numbers and $f(n,p,q)$. Using results on the extremal numbers of subdivided complete graphs, theta graphs, and subdivided complete bipartite graphs, we generalize results of Fish, Pohoata, and Sheffer, giving the first nontrivial lower bounds on $f(n,p,q)$ for some pairs $(p,q)$ and improving previous lower bounds for other pairs.

math.CO↗
arXiv2020

Firefighting on the Hexagonal Grid and on Infinite Trees

The firefighter problem with $k$ firefighters on an infinite graph $G$ is an iterative graph process, defined as follows: Suppose a fire breaks out at a given vertex $v\in V(G)$ on Turn 1. On each subsequent even turn, $k$ firefighters protect $k$ vertices that are not on fire, and on each subsequent odd turn, any vertex that is on fire spreads the fire to all adjacent unprotected vertices. The firefighters' goal is to eventually stop the spread of the fire. If there exists a strategy for $k$ firefighters to eventually stop the spread of the fire, then we say $G$ is $k$-containable. We consider the firefighter problem on the hexagonal grid, which is the graph whose vertices and edges are exactly the vertices and edges of a regular hexagonal tiling of the plane. It is not known if the hexagonal grid is $1$-containable. In arXiv:1305.7076 [math.CO], it was shown that if the firefighters have one firefighter per turn and one extra firefighter on two turns, the firefighters can contain the fire. We improve on this result by showing that even with only one extra firefighter on one turn, the firefighters can still contain the fire. In addition, we explore $k$-containability for birth sequence trees, which are infinite rooted trees that have the property that every vertex at the same level has the same degree. A birth sequence forest is an infinite forest, each component of which is a birth sequence tree. For birth sequence trees and forests, the fire always starts at the root of each tree. We provide a pseudopolynomial time algorithm to decide if all the vertices at a fixed level can be protected or not.

math.CO↗