Ryan Cushman

activity

20202026

collaborators

Showing math.COShow all

9 papers · 1 filter

math.CO2026

Almost-perfect packings and Tuza's conjecture in the random geometric graph

Patrick Bennett, Ryan Cushman, Andrzej Dudek +1

The triangle packing number $ν (G)$ of a graph $G$ is the maximum size of a set of edge-disjoint triangles in $G$ . Tuza conjectured that in any graph $G$ there exists a set of at mo…

math.CO2024

The generalized Ramsey number $f (n, 5, 8) = \frac{6}{7} n + o (n)$

Patrick Bennett, Ryan Cushman, Andrzej Dudek

A $(p, q)$ -coloring of $K_{n}$ is a coloring of the edges of $K_{n}$ such that every $p$ -clique has at least $q$ distinct colors among its edges. The generalized Ramsey number $f(n, p,…

math.CO2023

Generalized Ramsey numbers at the linear and quadratic thresholds

Patrick Bennett, Ryan Cushman, Andrzej Dudek

The generalized Ramsey number $f (n, p, q)$ is the smallest number of colors needed to color the edges of the complete graph $K_{n}$ so that every $p$ -clique spans at least $q$ colors…

math.CO2022

A note on non-isomorphic edge-color classes in random graphs

Patrick Bennett, Ryan Cushman, Andrzej Dudek +1

For a graph $G$ , let $τ (G)$ be the maximum number of colors such that there exists an edge-coloring of $G$ with no two color classes being isomorphic. We investigate the behavior o…

math.CO2022

The Localization Game on Directed Graphs

Anthony Bonato, Ryan Cushman, Trent G. Marbach +1

In the Localization game played on graphs, a set of cops uses distance probes to identify the location of an invisible robber. We present an extension of the game and its main para…

math.CO2022

The Erdős-Gyárfás function $f (n, 4, 5) = \frac{5}{6} n + o (n)$ -- so Gyárfás was right

Patrick Bennett, Ryan Cushman, Andrzej Dudek +1

A $(4, 5)$ -coloring of $K_{n}$ is an edge-coloring of $K_{n}$ where every $4$ -clique spans at least five colors. We show that there exist $(4, 5)$ -colorings of $K_{n}$ using $\frac 56 n…

math.CO2022

On Meyniel extremal families of graphs

Anthony Bonato, Ryan Cushman, Trent G. Marbach

We provide new constructions of Meyniel extremal graphs, which are families of graphs with the conjectured largest asymptotic cop number. Using spanning subgraphs, we prove that th…

math.CO2020

Closing the Random Graph Gap in Tuza's Conjecture Through the Online Triangle Packing Process

Patrick Bennett, Ryan Cushman, Andrzej Dudek

A long-standing conjecture of Zsolt Tuza asserts that the triangle covering number $τ (G)$ is at most twice the triangle packing number $ν (G)$ , where the triangle packing number $ν(…

math.CO2020

On the number of alternating paths in random graphs

Patrick Bennett, Ryan Cushman, Andrzej Dudek

In the noisy channel model from coding theory, we wish to detect errors introduced during transmission by optimizing various parameters of the code. Bennett, Dudek, and LaForge fra…

Almost-perfect packings and Tuza's conjecture in the random geometric graph

The generalized Ramsey number f(n,5,8)=76​n+o(n)

Generalized Ramsey numbers at the linear and quadratic thresholds

A note on non-isomorphic edge-color classes in random graphs

The Localization Game on Directed Graphs

The Erdős-Gyárfás function f(n,4,5)=65​n+o(n) -- so Gyárfás was right

On Meyniel extremal families of graphs

Closing the Random Graph Gap in Tuza's Conjecture Through the Online Triangle Packing Process

On the number of alternating paths in random graphs

The generalized Ramsey number $f (n, 5, 8) = \frac{6}{7} n + o (n)$

The Erdős-Gyárfás function $f (n, 4, 5) = \frac{5}{6} n + o (n)$ -- so Gyárfás was right