arXiv subjects

Emily Heath

Publications and source records attributed to Emily Heath.

At least 19 recordsLinked to original sources

arXiv2026

A linear upper bound on the $\mathbb{Z}_p$-Ramsey number of graphs with sufficiently large $2$-packing

Given a positive integer $k$ and graph $G$, the $\mathbb{Z}_k$-Ramsey number $R(G,\mathbb{Z}_k)$ is the least $N$ (if it exists) such that every coloring $f:E(K_N)\rightarrow \mathbb{Z}_k$ contains a copy $G'$ of $G$ such that $\sum_{e\in E(G')}f(e)=0$. Motivated by a question of Caro and Mifsud, we study the $\mathbb{Z}_k$-Ramsey number of graphs with a sufficiently large 2-packing, i.e. a set of vertices $S\subseteq V(G)$ such that $N[u]\cap N[v]=\emptyset$ for all distinct $u,v\in S$. In particular, we prove that $R(G,\mathbb{Z}_p)\leq n+6p-9$ for all $n$-vertex graphs $G$ and all primes $p$ such that $p$ divides $e(G)$, the minimum degree of $G$ is at least $1$, and there exists a $2$-packing of $G$ with size $p-1$. This upper bound improves depending on vertex degrees in the $2$-packing, with equality in certain cases. The result also implies an upper bound of the form $R(G,\mathbb{Z}_p)\leq n+C$ for $n$-vertex graphs $G$ of bounded maximum degree.

math.CO↗

arXiv2026

Generalized Ramsey Numbers in the Hypercube

We study the generalized Ramsey numbers $f(Q_n, C_{k}, q)$, that is, the minimum number of colors needed to edge-color the hypercube $Q_n$ so that every copy of the cycle $C_{k}$ has at least $q$ colors. Our main result is that for any integers $k,q$ satisfying $k \geq 6$ and $3 \leq q \leq k/2+1$, we have $f(Q_n, C_{k}, q)= o\left( n^{\frac{k/2-1}{k-q+1}} \right).$ We also prove a few other upper and lower bounds in the special cases $k=4$ and $k=6$. This continues the line of research initiated by Faudree, Gy\'arf\'as, Lesniak, and Schelp and Mubayi and Stading who studied the case $k=q$, and by Conder who considered the case $k=6$ and $q=2$.

math.CO↗

arXiv2025

Ramsey Numbers in Kneser Graphs

We define the $r\textit{-Kneser Ramsey number}$ $R^{\textrm{KG}}_{r}(s, t)$ as the minimum integer $n$ such that every red/blue edge-coloring of the Kneser graph $\textrm{KG}(n,r)$ contains a red $s$-clique or a blue $t$-clique. We obtain general bounds on the numbers $R^{\textrm{KG}}_{r}(s, t)$, and make progress on two related Ramsey-type problems, one raised by Holmsen, Hrusak, and Rold\'an-Pensado, and the other posted by P\'alv\"olgyi.

math.CO↗

arXiv2025

Universality for transversal powers of Hamilton cycles

Let $k \ge 2$ and let $\bf G = \{G_1, \ldots, G_{m}\}$ be a collection of graphs on a common vertex set of cardinality $n$. We show that if each graph in $\bf G$ has minimum degree at least $(1-\frac{1}{2k} + o(1))n$, then for every edge-colouring $\chi$ of the $k$th power of a Hamilton cycle $C_n^k$ with $m$ colours, there is a copy of $C_n^k$ in $\bf G$ such that $e \in G_{\chi(e)}$ for every edge $e$ in $C_n^k$. This generalises a result of Bowtell, Morris, Pehova, and Staden, who provided asymptotically best possible minimum degree conditions for the Hamilton cycle.

math.CO↗

arXiv2025

Odd Ramsey numbers of multipartite graphs and hypergraphs

Given a hypergraph $G$ and a subhypergraph $H$ of $G$, the \emph{odd Ramsey number} $r_{odd}(G,H)$ is the minimum number of colors needed to edge-color $G$ so that every copy of $H$ intersects some color class in an odd number of edges. Generalizing a result of \cite{BHZ} in two different ways, in this paper we prove $r_{odd} \left(K_{n,n}, K_{2,t} \right)=\frac{n}{t} + o(n)$ for all $t\geq 2$, and $r_{odd} \left(\mathcal{K}^{(k)}_{n,\dots,n}, \mathcal{K}_{1,\dots,1,2,2} \right) = \frac{n}{2} + o(n)$ for all $k\geq 2$. The latter is the first result studying odd Ramsey numbers for hypergraphs.

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

On the proper rainbow saturation numbers of cliques, paths, and odd cycles

Given a graph $H$, we say a graph $G$ is properly rainbow $H$-saturated if there is a proper edge-coloring of $G$ which contains no rainbow copy of $H$, but adding any edge to $G$ makes such an edge-coloring impossible. The proper rainbow saturation number, denoted $\text{sat}^*(n,H)$, is the minimum number of edges in an $n$-vertex rainbow $H$-saturated graph. We determine the proper rainbow saturation number for paths up to an additive constant and asymptotically determine $\text{sat}^*(n,K_4)$. In addition, we bound $\text{sat}^*(n,H)$ when $H$ is a larger clique, tree of diameter at least 4, or odd cycle.

math.CO↗

arXiv2024

Online Ramsey numbers of ordered graphs

The online ordered Ramsey game is played between two players, Builder and Painter, on an infinite sequence of vertices with ordered graphs $(G_1,G_2)$, which have linear orderings on their vertices. On each turn, Builder first selects an edge before Painter colors it red or blue. Builder's objective is to construct either an ordered red copy of $G_1$ or an ordered blue copy of $G_2$, while Painter's objective is to delay this for as many turns as possible. The online ordered Ramsey number $r_o(G_1,G_2)$ is the number of turns Builder takes to win in the case that both players play optimally. Few lower bounds are known for this quantity. In this paper, we introduce a succinct proof of a new lower bound based on the maximum left- and right-degrees in the ordered graphs. We also upper bound $r_o(G_1,G_2)$ in two cases: when $G_1$ is a cycle and $G_2$ a complete bipartite graph, and when $G_1$ is a tree and $G_2$ a clique.

math.CO↗

arXiv2024

Generalized Ramsey numbers of cycles, paths, and hypergraphs

Given a $k$-uniform hypergraph $G$ and a set of $k$-uniform hypergraphs $\mathcal{H}$, the generalized Ramsey number $f(G,\mathcal{H},q)$ is the minimum number of colors needed to edge-color $G$ so that every copy of every hypergraph $H\in \mathcal{H}$ in $G$ receives at least $q$ different colors. In this note we obtain bounds, some asymptotically sharp, on several generalized Ramsey numbers, when $G=K_n$ or $G=K_{n,n}$ and $\mathcal{H}$ is a set of cycles or paths, and when $G=K_n^k$ and $\mathcal{H}$ contains a clique on $k+2$ vertices or a tight cycle.

math.CO↗

arXiv2024

The forb-flex method for odd coloring and proper conflict-free coloring of planar graphs

We introduce a new tool useful for greedy coloring, which we call the forb-flex method, and apply it to odd coloring and proper conflict-free coloring of planar graphs. The odd chromatic number, denoted $\chi_{\mathsf{o}}(G)$, is the smallest number of colors needed to properly color $G$ such that every non-isolated vertex of $G$ has a color appearing an odd number of times in its neighborhood. The proper conflict-free chromatic number, denoted $\chi_{\mathsf{PCF}}(G)$, is the smallest number of colors needed to properly color $G$ such that every non-isolated vertex of $G$ has a color appearing uniquely in its neighborhood. Our new tool works by carefully counting the structures in the neighborhood of a vertex and determining if a neighbor of a vertex can be recolored at the end of a greedy coloring process to avoid conflicts. Combining this with the discharging method allows us to prove $\chi_{\mathsf{PCF}}(G) \leq 4$ for planar graphs of girth at least 11, and $\chi_{\mathsf{o}}(G) \leq 4$ for planar graphs of girth at least 10. These results improve upon the recent works of Cho, Choi, Kwon, and Park.

math.CO↗

arXiv2023

Asymmetric Ramsey properties of randomly perturbed graphs

In this note, we investigate for various pairs of graphs $(H,G)$ the question of how many random edges must be added to a dense graph to guarantee that any red-blue coloring of the edges contains a red copy of $H$ or a blue copy of $G$. We determine this perturbed Ramsey threshold for many new pairs of graphs and various ranges of densities, obtaining several generalizations of results obtained by Das and Treglown. In particular, we resolve the remaining cases toward determining the perturbed Ramsey threshold for pairs $(K_t,K_s)$ where $t\geq s\geq 5$.

math.CO↗

arXiv2023

New bounds on the generalized Ramsey number $f(n,5,8)$

Let $f(n,p,q)$ denote the minimum number of colors needed to color the edges of $K_n$ so that every copy of $K_p$ receives at least $q$ distinct colors. In this note, we show $\frac{6}{7}(n-1) \leq f(n,5,8) \leq n + o(n)$. The upper bound is proven using the "conflict-free hypergraph matchings method" which was recently used by Mubayi and Joos to prove $f(n,4,5) = \frac{5}{6}n + o(n)$.

math.CO↗

arXiv2023

Edge-coloring a graph $G$ so that every copy of a graph $H$ has an odd color class

Recently, Alon introduced the notion of an $H$-code for a graph $H$: a collection of graphs on vertex set $[n]$ is an $H$-code if it contains no two members whose symmetric difference is isomorphic to $H$. Let $D_{H}(n)$ denote the maximum possible cardinality of an $H$-code, and let $d_{H}(n)=D_{H}(n)/2^{n \choose 2}$. Alon observed that a lower bound on $d_{H}(n)$ can be obtained by attaining an upper bound on the number of colors needed to edge-color $K_n$ so that every copy of $H$ has an odd color class. Motivated by this observation, we define $g(G,H)$ to be the minimum number of colors needed to edge-color a graph $G$ so that every copy of $H$ has an odd color class. We prove $g(K_n,K_5) \le n^{o(1)}$ and $g(K_{n,n}, C_4)= n/2+o(n)$. The first result shows $d_{K_5}(n) \ge \frac{1}{n^{o(1)}}$ and was obtained independently in arXiv:2306.14682.

math.CO↗

arXiv2023

The maximum number of odd cycles in a planar graph

How many copies of a fixed odd cycle, $C_{2m+1}$, can a planar graph contain? We answer this question asymptotically for $m\in\{2,3,4\}$ and prove a bound which is tight up to a factor of $3/2$ for all other values of $m$. This extends the prior results of Cox--Martin and Lv et al. on the analogous question for even cycles. Our bounds result from a reduction to the following maximum likelihood question: which probability mass $\mu$ on the edges of some clique maximizes the probability that $m$ edges sampled independently from $\mu$ form either a cycle or a path?

math.CO↗

arXiv2023

Validation of 4D Monte Carlo dose calculations using a programmable deformable lung phantom

Purpose: To validate the accuracy of 4D Monte Carlo (4DMC) simulations to calculate dose deliveries to a deforming anatomy in the presence of realistic respiratory motion traces. A previously developed deformable lung phantom comprising an elastic tumor was modified to enable programming of arbitrary motion profiles. 4D simulations of the dose delivered to the phantom were compared with the measurements. Methods: The deformable lung phantom moving with irregular breathing patterns was irradiated using static and VMAT beam deliveries. Using the RADPOS 4D dosimetry system, point doses were measured inside and outside the tumor. Dose profiles were acquired using films along the motion path of the tumor (S-I). In addition to dose measurements, RADPOS was used to record the motion of the tumor during dose deliveries. Dose measurements were then compared against 4DMC simulations with EGSnrc/4DdefDOSXYZnrc using the recorded tumor motion. Results: The agreements between dose profiles from measurements and simulations were determined to be within 2%/2 mm. Point dose agreements were within 2{\sigma} of experimental and/or positional/dose reading uncertainties. 4DMC simulations were shown to accurately predict the sensitivity of delivered dose to the starting phase of breathing motions. We have demonstrated that our 4DMC method, combined with RADPOS, can accurately simulate realistic dose deliveries to a deforming anatomy moving with realistic breathing traces. This 4DMC tool has the potential to be used as a quality assurance tool to verify treatments involving respiratory motion. Adaptive treatment delivery is another area that may benefit from the potential of this 4DMC tool.

physics.med-ph↗

arXiv2023

Embedded graph 3-coloring and flows

A graph drawn in a surface is a near-quadrangulation if the sum of the lengths of the faces different from 4-faces is bounded by a fixed constant. We leverage duality between colorings and flows to design an efficient algorithm for 3-precoloring-extension in near-quadrangulations of orientable surfaces. Furthermore, we use this duality to strengthen previously known sufficient conditions for 3-colorability of triangle-free graphs drawn in orientable surfaces.

math.CO↗

arXiv2022

Online Ramsey numbers of ordered paths and cycles

An ordered graph is a graph with a linear ordering on its vertices. The online Ramsey game for ordered graphs $G$ and $H$ is played on an infinite sequence of vertices; on each turn, Builder draws an edge between two vertices, and Painter colors it red or blue. Builder tries to create a red $G$ or a blue $H$ as quickly as possible, while Painter wants the opposite. The online ordered Ramsey number $r_o(G,H)$ is the number of turns the game lasts with optimal play. In this paper, we consider the behavior of $r_o(G,P_n)$ for fixed $G$, where $P_n$ is the monotone ordered path. We prove an $O(n \log_2n)$ bound on $r_o(G,P_n)$ for all $G$ and an $O(n)$ bound when $G$ is $3$-ichromatic; we partially classify graphs $G$ with $r_o(G,P_n) = n + O(1)$. Many of these results extend to $r_o(G,C_n)$, where $C_n$ is an ordered cycle obtained from $P_n$ by adding one edge.

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↗