arXiv subjects

Elizaveta Popova

Publications and source records attributed to Elizaveta Popova.

4 recordsLinked to original sources

arXiv2026

On the Minimum Possible Maximum Degree of Induced Subgraphs of Product Graphs

The following is a natural and fundamental question for a graph $G$: if an induced subgraph $H$ of $G$ has $x$ more vertices than a maximum independent set, what can be said about the maximum degree of $H$ as a function of $x$? The case $x=1$ is already of considerable interest. For example, in his celebrated proof of the sensitivity conjecture, Hao Huang showed that every induced subgraph of the hypercube $Q_n$ on more than $2^{n-1}$ vertices has maximum degree at least $\sqrt{n}$. Chung, Fúredi, Graham, and Seymour proved that this bound is tight. In this paper, we study this question when $G$ is either the $n$-fold Hamming product or the $n$-fold tensor product of a triangle. For both graphs, we determine the exact minimum possible average degree of an induced subgraph of a prescribed size. We also prove that the tensor product exhibits several Huang-like phenomena. For the Hamming product, we show that, for several size densities and large $n$, the minimum possible maximum degree is asymptotically equal to the minimum possible average degree. Finally, we extend several of the results from the triangle to an arbitrary complete graph $K_k$.

math.CO↗

arXiv2026

New Greedy Spanners and Applications

We present a simple greedy procedure to compute an $(α,β)$-spanner for a graph $G$. We then show that this procedure is useful for building fault-tolerant spanners, as well as spanners for weighted graphs. Our first main result is an algorithm that, given a multigraph $G$, outputs an $f$ edge fault-tolerant $(k,k-1)$-spanner $H$ of size $O(fn^{1+\frac1k})$ which is tight. To our knowledge, this is the first tight result concerning the price of fault tolerance in spanners which are not multiplicative, in any model of faults. Our second main result is a new construction of a spanner for weighted graphs. We show that any weighted graph $G$ has a subgraph $H$ with $O(n^{1+\frac{1}{k}})$ edges such that any path $P$ of hop-length $\ell$ in $G$ has a replacement path $P'$ in $H$ of weighted length $\leq w(P)+(2k-2)w^{(1/2)}(P)$ where $w(P)$ is the total edge weight of $P$, and $w^{(1/2)}$ denotes the sum of the largest $\lceil \frac{\ell}{2} \rceil$ edge weights along $P$. Moreover, we show such approximation is optimal for shortest paths of hop-length $2$. To our knowledge, this is the first construction of a spanner for weighted graphs that strictly improves upon the stretch of multiplicative $(2k-1)$-spanners for all non-adjacent vertex pairs, while maintaining the same size bound. Our technique is based on using clustering and ball-growing, which are methods commonly used in designing spanner algorithms, to analyze simple greedy algorithms. This allows us to combine the flexibility of clustering approaches with the unique properties of the greedy algorithm to get improved bounds. In particular, our methods give a very short proof that the parallel greedy spanner adds $O(kn^{1+\frac{1}{k}})$ edges, improving upon known bounds.

cs.DS↗

arXiv2025

Satisfying sequences for rainbow partite matchings

Let $\mathcal F_1,\ldots, \mathcal F_s\subset [n]^k$ be a collection of $s$ families. In this paper, we address the following question: for which sequences $f_1,\ldots, f_s$ the conditions $|\ff_i|>f_i$ imply that the families contain a rainbow matching, that is, there are pairwise disjoint $F_1\in \ff_1,\ldots F_s\in \ff_s$? We call such sequences {\em satisfying}. Kiselev and the first author verified the conjecture of Aharoni and Howard and showed that $f_1 = \ldots = f_s=(s-1)n^{k-1}$ is satisfying for $s>470$. This is the best possible if the restriction is uniform over all families. However, it turns out that much more can be said about asymmetric restrictions. In this paper, we investigate this question in several regimes and in particular answer the questions asked by Kiselev and Kupavskii. We use a variety of methods, including concentration and anticoncentration results, spread approximations, and Combinatorial Nullstellenzats.

math.CO↗

arXiv2024

Tilings of $\mathbb Z$ with multisets of distances

In this paper, we study tilings of $\mathbb Z$, that is, coverings of $\mathbb Z$ by disjoint sets (tiles). Let $T=\{d_1,\ldots, d_s\}$ be a given multiset of distances. Is it always possible to tile $\mathbb Z$ by tiles, for which the multiset of distances between consecutive points is equal to $T$? In this paper, we give a sufficient condition that such a tiling exists. Our result allows multisets of distances to have arbitrarily many distinct values. Our result generalizes most of the previously known results, all of which dealt with the cases of $2$ or $3$ distinct distances.

math.CO↗