arXiv subjects

Kalina Petrova

Publications and source records attributed to Kalina Petrova.

21 records · Page 2Linked to original sources

arXiv2023

Size-Ramsey numbers of structurally sparse graphs

Size-Ramsey numbers are a central notion in combinatorics and have been widely studied since their introduction by Erdős, Faudree, Rousseau and Schelp in 1978. Research has mainly focused on the size-Ramsey numbers of $n$-vertex graphs with constant maximum degree $Δ$. For example, graphs which also have constant treewidth are known to have linear size-Ramsey numbers. On the other extreme, the canonical examples of graphs of unbounded treewidth are the grid graphs, for which the best known bound has only very recently been improved from $O(n^{3/2})$ to $O(n^{5/4})$ by Conlon, Nenadov and Trujić. In this paper, we prove a common generalization of these results by establishing new bounds on the size-Ramsey numbers in terms of treewidth (which may grow as a function of $n$). As a special case, this yields a bound of $\tilde{O}(n^{3/2 - 1/2Δ})$ for proper minor-closed classes of graphs. In particular, this bound applies to planar graphs, addressing a question of Kamcev, Liebenau, Wood and Yepremyan. Our proof combines methods from structural graph theory and classic Ramsey-theoretic embedding techniques, taking advantage of the product structure exhibited by graphs with bounded treewidth.

math.CO↗

arXiv2023

On Connectivity in Random Graph Models with Limited Dependencies

For any positive edge density $p$, a random graph in the Erdős-Renyi $G_{n,p}$ model is connected with non-zero probability, since all edges are mutually independent. We consider random graph models in which edges that do not share endpoints are independent while incident edges may be dependent and ask: what is the minimum probability $ρ(n)$, such that for any distribution $\mathcal{G}$ (in this model) on graphs with $n$ vertices in which each potential edge has a marginal probability of being present at least $ρ(n)$, a graph drawn from $\mathcal{G}$ is connected with non-zero probability? As it turns out, the condition ``edges that do not share endpoints are independent'' needs to be clarified and the answer to the question above is sensitive to the specification. In fact, we formalize this intuitive description into a strict hierarchy of five independence conditions, which we show to have at least three different behaviors for the threshold $ρ(n)$. For each condition, we provide upper and lower bounds for $ρ(n)$. In the strongest condition, the coloring model (which includes, e.g., random geometric graphs), we show that $ρ(n)\rightarrow 2-ϕ\approx 0.38$ for $n\rightarrow\infty$, proving a conjecture by Badakhshian, Falgas-Ravry, and Sharifzadeh. This separates the coloring models from the weaker independence conditions we consider, as there we prove that $ρ(n)>0.5-o(n)$. In stark contrast to the coloring model, for our weakest independence condition -- pairwise independence of non-adjacent edges -- we show that $ρ(n)$ lies within $O(1/n^2)$ of the threshold $1-2/n$ for completely arbitrary distributions.

math.CO↗

arXiv2019

Optimal strategies for patrolling fences

A classical multi-agent fence patrolling problem asks: What is the maximum length $L$ of a line that $k$ agents with maximum speeds $v_1,\ldots,v_k$ can patrol if each point on the line needs to be visited at least once every unit of time. It is easy to see that $L = α\sum_{i=1}^k v_i$ for some efficiency $α\in [\frac{1}{2},1)$. After a series of works giving better and better efficiencies, it was conjectured that the best possible efficiency approaches $\frac{2}{3}$. No upper bounds on the efficiency below $1$ were known. We prove the first such upper bounds and tightly bound the optimal efficiency in terms of the minimum ratio of speeds $s = {v_{\max}}/{v_{\min}}$ and the number of agents $k$. Guided by our upper bounds, we construct a scheme whose efficiency approaches $1$, disproving the conjecture of Kawamura and Soejima. Our scheme asymptotically matches our upper bounds in terms of the maximal speed difference and the number of agents used, proving them to be asymptotically tight. A variation of the fence patrolling problem considers a circular fence instead and asks for its circumference to be maximized. We consider the unidirectional case of this variation, where all agents are only allowed to move in one direction, say clockwise. At first, a strategy yielding $L = \max_{r \in [k]} r \cdot v_r$ where $v_1 \geq v_2 \geq \dots \geq v_k$ was conjectured to be optimal by Czyzowicz et al. This was proven not to be the case by giving constructions for only specific numbers of agents with marginal improvements of $L$. We give a general construction that yields $L = \frac{1}{33 \log_e\log_2(k)} \sum_{i=1}^k v_i$ for any set of agents, which in particular for the case $1, 1/2, \dots, 1/k$ diverges as $k \rightarrow \infty$, thus resolving a conjecture by Kawamura and Soejima affirmatively.

cs.DS↗