arXiv subjects

Jesse Geneson

Publications and source records attributed to Jesse Geneson.

At least 55 records · Page 3Linked to original sources

arXiv2020

Broadcast Dimension of Graphs

In this paper we initiate the study of broadcast dimension, a variant of metric dimension. Let $G$ be a graph with vertex set $V(G)$, and let $d(u,w)$ denote the length of a $u-w$ geodesic in $G$. For $k \ge 1$, let $d_k(x,y)=\min \{d(x,y), k+1\}$. A function $f: V(G) \rightarrow \mathbb{Z}^+ \cup \{0\}$ is called a resolving broadcast of $G$ if, for any distinct $x,y \in V(G)$, there exists a vertex $z \in V(G)$ such that $f(z)=i>0$ and $d_{i}(x,z) \neq d_{i}(y,z)$. The broadcast dimension, $bdim(G)$, of $G$ is the minimum of $c_f(G)=\sum_{v \in V(G)} f(v)$ over all resolving broadcasts of $G$, where $c_f(G)$ can be viewed as the total cost of the transmitters (of various strength) used in resolving the entire network described by the graph $G$. Note that $bdim(G)$ reduces to $adim(G)$ (the adjacency dimension of $G$, introduced by Jannesari and Omoomi in 2012) if the codomain of resolving broadcasts is restricted to $\{0,1\}$. We determine its value for cycles, paths, and other families of graphs. We prove that $bdim(G) = Ω(\log{n})$ for all graphs $G$ of order $n$, and that the result is sharp up to a constant factor. We show that $\frac{adim(G)}{bdim(G)}$ and $\frac{bdim(G)}{dim(G)}$ can both be arbitrarily large, where $dim(G)$ denotes the metric dimension of $G$. We also examine the effect of vertex deletion on the adjacency dimension and the broadcast dimension of graphs.

math.CO↗

arXiv2020

Metric dimension and pattern avoidance in graphs

In this paper, we prove a number of results about pattern avoidance in graphs with bounded metric dimension or edge metric dimension. We show that the maximum possible number of edges in a graph of diameter $D$ and edge metric dimension $k$ is at most $(\lfloor \frac{2D}{3}\rfloor +1)^{k}+k \sum_{i = 1}^{\lceil \frac{D}{3}\rceil } (2i)^{k-1}$, sharpening the bound of $\binom{k}{2}+k D^{k-1}+D^{k}$ from Zubrilina (2018). We also show that the maximum value of $n$ for which some graph of metric dimension $\leq k$ contains the complete graph $K_{n}$ as a subgraph is $n = 2^{k}$. We prove that the maximum value of $n$ for which some graph of metric dimension $\leq k$ contains the complete bipartite graph $K_{n,n}$ as a subgraph is $2^{Θ(k)}$. Furthermore, we show that the maximum value of $n$ for which some graph of edge metric dimension $\leq k$ contains $K_{1,n}$ as a subgraph is $n = 2^{k}$. We also show that the maximum value of $n$ for which some graph of metric dimension $\leq k$ contains $K_{1,n}$ as a subgraph is $3^{k}-O(k)$. In addition, we prove that the $d$-dimensional grids $\prod_{i = 1}^{d} P_{r_{i}}$ have edge metric dimension at most $d$. This generalizes two results of Kelenc et al. (2016), that non-path grids have edge metric dimension $2$ and that $d$-dimensional hypercubes have edge metric dimension at most $d$. We also provide a characterization of $n$-vertex graphs with edge metric dimension $n-2$, answering a question of Zubrilina. As a result of this characterization, we prove that any connected $n$-vertex graph $G$ such that $edim(G) = n-2$ has diameter at most $5$. More generally, we prove that any connected $n$-vertex graph with edge metric dimension $n-k$ has diameter at most $3k-1$.

math.CO↗

arXiv2020

Constructing sparse Davenport-Schinzel sequences

For any sequence $u$, the extremal function $Ex(u, j, n)$ is the maximum possible length of a $j$-sparse sequence with $n$ distinct letters that avoids $u$. We prove that if $u$ is an alternating sequence $a b a b \dots$ of length $s$, then $Ex(u, j, n) = Θ(s n^{2})$ for all $j \geq 2$ and $s \geq n$, answering a question of Wellman and Pettie [Lower Bounds on Davenport-Schinzel Sequences via Rectangular Zarankiewicz Matrices, Disc. Math. 341 (2018), 1987--1993] and extending the result of Roselle and Stanton that $Ex(u, 2, n) = Θ(s n^2)$ for any alternation $u$ of length $s \geq n$ [Some properties of Davenport-Schinzel sequences, Acta Arithmetica 17 (1971), 355--362]. Wellman and Pettie also asked how large must $s(n)$ be for there to exist $n$-block $DS(n, s(n))$ sequences of length $Ω(n^{2-o(1)})$. We answer this question by showing that the maximum possible length of an $n$-block $DS(n, s(n))$ sequence is $Ω(n^{2-o(1)})$ if and only if $s(n) = Ω(n^{1-o(1)})$. We also show related results for extremal functions of forbidden 0-1 matrices with any constant number of rows and extremal functions of forbidden sequences with any constant number of distinct letters.

math.CO↗

arXiv2020

A generalization of the Kővári-Sós-Turán theorem

We present a new proof of the Kővári-Sós-Turán theorem that $ex(n, K_{s,t}) = O(n^{2-1/t})$ for $s, t \geq 2$. The new proof is elementary, avoiding the use of convexity. For any $d$-uniform hypergraph $H$, let $ex_d(n,H)$ be the maximum possible number of edges in an $H$-free $d$-uniform hypergraph on $n$ vertices. Let $K_{H, t}$ be the $(d+1)$-uniform hypergraph obtained from $H$ by adding $t$ new vertices $v_1, \dots, v_t$ and replacing every edge $e$ in $E(H)$ with $t$ edges $e \cup \left\{v_1\right\},\dots, e \cup \left\{v_t\right\}$ in $E(K_{H, t})$. If $H$ is the $1$-uniform hypergraph on $s$ vertices with $s$ edges, then $K_{H, t} = K_{s, t}$. We prove that $ex_{d+1}(n,K_{H,t}) = O(ex_d(n, H)^{1/t} n^{d+1-d/t} + t n^d)$ for any $d$-uniform hypergraph $H$ with at least two edges such that $ex_d(n, H) = o(n^d)$. Thus $ex_{d+1}(n,K_{H,t}) = O(n^{d+1-1/t})$ for any $d$-uniform hypergraph $H$ with at least two edges such that $ex_d(n, H) = O(n^{d-1})$, which implies the Kővári-Sós-Turán theorem in the $d = 1$ case. This also implies that $ex_{d+1}(n, K_{H,t}) = O(n^{d+1-1/t})$ when $H$ is a $d$-uniform hypergraph with at least two edges in which all edges are pairwise disjoint, which generalizes an upper bound proved by Mubayi and Verstraëte (JCTA, 2004). We also obtain analogous bounds for 0-1 matrix Turán problems.

math.CO↗

arXiv2019

An algorithm for bounding extremal functions of forbidden sequences

Generalized Davenport-Schinzel sequences are sequences that avoid a forbidden subsequence and have a sparsity requirement on their letters. Upper bounds on the lengths of generalized Davenport-Schinzel sequences have been applied to a number of problems in discrete geometry and extremal combinatorics. Sharp bounds on the maximum lengths of generalized Davenport-Schinzel sequences are known for some families of forbidden subsequences, but in general there are only rough bounds on the maximum lengths of most generalized Davenport-Schinzel sequences. One method that was developed for finding upper bounds on the lengths of generalized Davenport-Schinzel sequences uses a family of sequences called formations. An $(r, s)$-formation is a concatenation of $s$ permutations of $r$ distinct letters. The formation width function $fw(u)$ is defined as the minimum $s$ for which there exists $r$ such that every $(r, s)$-formation contains $u$. The function $fw(u)$ has been used with upper bounds on extremal functions of $(r, s)$-formations to find tight bounds on the maximum possible lengths of many families of generalized Davenport-Schinzel sequences. Algorithms have been found for computing $fw(u)$ for sequences $u$ of length $n$, but they have worst-case run time exponential in $n$, even for sequences $u$ with only three distinct letters. We present an algorithm for computing $fw(u)$ with run time $O(n^{α_r})$, where $r$ is the number of distinct letters in $u$ and $α_r$ is a constant that only depends on $r$. We implement the new algorithm in Python and compare its run time to the next fastest algorithm for computing formation width. We also apply the new algorithm to find sharp upper bounds on the lengths of several families of generalized Davenport-Schinzel sequences with $3$-letter forbidden patterns.

cs.DM↗

arXiv2019

Throttling numbers for adversaries on connected graphs

In this paper, we answer two open problems from [Breen et al., Throttling for the game of Cops and Robbers on graphs, Discrete Math., 341 (2018) 2418-2430]. The throttling number $th_c(G)$ of a graph $G$ is the minimum possible value of $k + capt_k(G)$ over all positive integers $k$, where $capt_k(G)$ is the number of rounds needed for $k$ cops to capture the robber on $G$. One of the problems from [Breen et al., 2018] was to determine whether there exists a family of trees $T$ of order $n$ for which $th_c(T)$ is asymptotically equal to $2 \sqrt{n}$. We show that such a family cannot exist by improving the upper bound on $\displaystyle \max_{T} th_c(T)$ for all trees $T$ of order $n$ from $2 \sqrt{n}$ to $\frac{\sqrt{14}}{2} \sqrt{n} + O(1)$. We prove this bound by deriving a more general throttling bound for connected graphs that applies to multiple graph adversaries, including the robber and the gambler. This also improves the best known upper bounds on $th_c(G)$ for chordal graphs and unicyclic graphs $G$, as well as throttling numbers for positive semidefinite (PSD) zero forcing on trees. In addition to the results about cop versus robber, we use our general throttling bound to improve previous upper bounds on throttling numbers for the cop versus gambler game on connected graphs. Another open problem from [Breen et al., 2018] was to obtain a bound on $th_c(G)$ for cactus graphs $G$. We prove an $O(\sqrt{n})$ bound for all cactus graphs $G$ of order $n$. Furthermore, we exhibit a family of trees $T$ of order $n$ that have $th_c(T) > 1.4502 \sqrt{n}$ for all $n$ sufficiently large, improving on the previous lower bound of $\lceil \sqrt{2n}-\frac{1}{2} \rceil + 1$ on $\displaystyle \max_{T} th_c(T)$ for trees $T$ of order $n$.

math.CO↗

arXiv2019

Skew throttling

Zero forcing is a process that colors the vertices of a graph blue by starting with some vertices blue and applying a color change rule. Throttling minimizes the sum of the number of initial blue vertices and the time to color the graph. In this paper, we study throttling for skew zero forcing. We characterize the graphs of order $n$ with skew throttling numbers $1, 2, n-1$, and $n$. We find the exact skew throttling numbers of paths, cycles, and balanced spiders with short legs. In addition, we find a sharp lower bound on skew throttling numbers in terms of the diameter.

math.CO↗

arXiv2019

Optimizing the trade-off between number of cops and capture time in Cops and Robbers

The cop throttling number $th_c(G)$ of a graph $G$ for the game of Cops and Robbers is the minimum of $k + capt_k(G)$, where $k$ is the number of cops and $capt_k(G)$ is the minimum number of rounds needed for $k$ cops to capture the robber on $G$ over all possible games in which both players play optimally. In this paper, we construct a family of graphs having $th_c(G)= Ω(n^{2/3})$, establish a sublinear upper bound on the cop throttling number, and show that the cop throttling number of chordal graphs is $O(\sqrt{n})$. We also introduce the product cop throttling number $th_c^{\times}(G)$ as a parameter that minimizes the person-hours used by the cops. This parameter extends the notion of speed-up that has been studied in the context of parallel processing and network decontamination. We establish bounds on the product cop throttling number in terms of the cop throttling number, characterize graphs with low product cop throttling number, and show that for a chordal graph $G$, $th_c^{\times}=1+rad(G)$.

math.CO↗

arXiv2019

Using Markov chains to determine expected propagation time for probabilistic zero forcing

Zero forcing is a coloring game played on a graph where each vertex is initially colored blue or white and the goal is to color all the vertices blue by repeated use of a (deterministic) color change rule starting with as few blue vertices as possible. Probabilistic zero forcing yields a discrete dynamical system governed by a Markov chain. Since in a connected graph any one vertex can eventually color the entire graph blue using probabilistic zero forcing, the expected time to do this studied. Given a Markov transition matrix for a probabilistic zero forcing process, we establish an exact formula for expected propagation time. We apply Markov chains to determine bounds on expected propagation time for various families of graphs.

math.CO↗

arXiv2019

Expected capture time and throttling number for cop versus gambler

We bound expected capture time and throttling number for the cop versus gambler game on a connected graph with $n$ vertices, a variant of the cop versus robber game that is played in darkness, where the adversary hops between vertices using a fixed probability distribution. The paper that originally defined the cop versus gambler game focused on two versions, a known gambler whose distribution the cop knows, and an unknown gambler whose distribution is secret. We define a new version of the gambler where the cop makes a fixed number of observations before the lights go out and the game begins. We show that the strategy that gives the best possible expected capture time of $n$ for the known gambler can also be used to achieve nearly the same expected capture time against the observed gambler when the cop makes a sufficiently large number of observations. We also show that even with only a single observation, the cop is able to achieve an expected capture time of approximately $1.5n$, which is much lower than the expected capture time of the best known strategy against the unknown gambler (approximately $1.95n$).

cs.DM↗

arXiv2019

Ramsey numbers of ordered graphs under graph operations

An ordered graph $\mathcal{G}$ is a simple graph together with a total ordering on its vertices. The (2-color) Ramsey number of $\mathcal{G}$ is the smallest integer $N$ such that every 2-coloring of the edges of the complete ordered graph on $N$ vertices has a monochromatic copy of $\mathcal{G}$ that respects the ordering. In this paper we investigate the effect of various graph operations on the Ramsey number of a given ordered graph, and detail a general framework for applying results on extremal functions of 0-1 matrices to ordered Ramsey problems. We apply this method to give upper bounds on the Ramsey number of ordered matchings arising from sum-decomposable permutations, an alternating ordering of the cycle, and an alternating ordering of the tight hyperpath. We also construct ordered matchings on $n$ vertices whose Ramsey number is $n^{q+o(1)}$ for any given exponent $q\in(1,2)$.

math.CO↗

arXiv2018

Propagation time for probabilistic zero forcing

Zero forcing is a coloring game played on a graph that was introduced more than ten years ago in several different applications. The goal is to color all the vertices blue by repeated use of a (deterministic) color change rule. Probabilistic zero forcing was introduced by Kang and Yi in [Probabilistic zero forcing in graphs, Bull. Inst. Combin. Appl. 67 (2013), 9--16] and yields a discrete dynamical system, which is a better model for some applications. Since in a connected graph any one vertex can eventually color the entire graph blue using probabilistic zero forcing, the expected time to do this is a natural parameter to study. We determine expected propagation time exactly for paths and cycles, establish the asymptotic value for stars, and present asymptotic upper and lower bounds for any graph in terms of its radius and order. We apply these results to obtain values and bounds on $\ell$-round probabilistic zero forcing, throttling number for probabilistic zero forcing, and confidence levels for propagation time.

math.CO↗

arXiv2018

A note on long rainbow arithmetic progressions

Jungić et al (2003) defined $T_{k}$ as the minimal number $t \in \mathbb{N}$ such that there is a rainbow arithmetic progression of length $k$ in every equinumerous $t$-coloring of $[t n]$ for every $n \in \mathbb{N}$. They proved that for every $k \geq 3$, $\lfloor \frac{k^2}{4} \rfloor < T_{k} \leq \frac{k(k-1)^2}{2}$ and conjectured that $T_{k} = Θ(k^2)$. We prove for all $ε> 0$ that $T_{k} = O(k^{5/2+ε})$ using the Kővári-Sós-Turán theorem and Wigert's bound on the divisor function.

math.CO↗

arXiv2018

Intersections, circuits, and colorability of line segments

We derive sharp upper and lower bounds on the number of intersection points and closed regions that can occur in sets of line segments with certain structure, in terms of the number of segments. We consider sets of segments whose underlying planar graphs are Halin graphs, cactus graphs, maximal planar graphs, and triangle-free planar graphs, as well as randomly produced segment sets. We also apply these results to a variant of the Erdős-Faber-Lovász (EFL) Conjecture stating that the intersection points of $m$ segments can be colored with $m$ colors so that no segment contains points with the same color. We investigate an optimization problem related to the EFL Conjecture for line segments, determine its complexity, and provide some computational approaches.

math.CO↗

arXiv2018

Fixed points of competitive threshold-linear networks

Threshold-linear networks (TLNs) are models of neural networks that consist of simple, perceptron-like neurons and exhibit nonlinear dynamics that are determined by the network's connectivity. The fixed points of a TLN, including both stable and unstable equilibria, play a critical role in shaping its emergent dynamics. In this work, we provide two novel characterizations for the set of fixed points of a competitive TLN: the first is in terms of a simple sign condition, while the second relies on the concept of domination. We apply these results to a special family of TLNs, called combinatorial threshold-linear networks (CTLNs), whose connectivity matrices are defined from directed graphs. This leads us to prove a series of graph rules that enable one to determine fixed points of a CTLN by analyzing the underlying graph. Additionally, we study larger networks composed of smaller "building block" subnetworks, and prove several theorems relating the fixed points of the full network to those of its components. Our results provide the foundation for a kind of "graphical calculus" to infer features of the dynamics from a network's connectivity.

q-bio.NC↗

arXiv2018

Forbidden formations in 0-1 matrices

Keszegh (2009) proved that the extremal function $ex(n, P)$ of any forbidden light $2$-dimensional 0-1 matrix $P$ is at most quasilinear in $n$, using a reduction to generalized Davenport-Schinzel sequences. We extend this result to multidimensional matrices by proving that any light $d$-dimensional 0-1 matrix $P$ has extremal function $ex(n, P,d) = O(n^{d-1}2^{α(n)^{t}})$ for some constant $t$ that depends on $P$. To prove this result, we introduce a new family of patterns called $(P, s)$-formations, which are a generalization of $(r, s)$-formations, and we prove upper bounds on their extremal functions. In many cases, including permutation matrices $P$ with at least two ones, we are able to show that our $(P, s)$-formation upper bounds are tight.

math.CO↗

arXiv2018

Forbidden arithmetic progressions in permutations of subsets of the integers

Permutations of the positive integers avoiding arithmetic progressions of length $5$ were constructed in (Davis et al, 1977), implying the existence of permutations of the integers avoiding arithmetic progressions of length $7$. We construct a permutation of the integers avoiding arithmetic progressions of length $6$. We also prove a lower bound of $\frac{1}{2}$ on the lower density of subsets of positive integers that can be permuted to avoid arithmetic progressions of length $4$, sharpening the lower bound of $\frac{1}{3}$ from (LeSaulnier and Vijay, 2011). In addition, we generalize several results about forbidden arithmetic progressions to construct permutations avoiding generalized arithmetic progressions.

math.CO↗

arXiv2017

Improved lower bound on generalized Erdos-Ginzburg-Ziv constants

If $G$ is a finite Abelian group, define $s_{k}(G)$ to be the minimal $m$ such that a sequence of $m$ elements in $G$ always contains a $k$-element subsequence which sums to zero. Recently Bitz et al. proved that if $n = exp(G)$, then $s_{2n}(C_{n}^{r}) > \frac{n}{2}[\frac{5}{4}-O(n^{-\frac{3}{2}})]^{r}$ and $s_{k n}(C_{n}^{r}) > \frac{k n}{4} [1+\frac{1}{e k}-O(\frac{1}{n})]^{r}$ for $k > 2$. In this note, we sharpen their general bound by showing that $s_{k n}(C_{n}^{r}) > \frac{k n}{4} [1+\frac{(k-1)^{(k-1)}}{k^k}-O(\frac{1}{n})]^{r}$ for $k > 2$.

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