Searcharxiv⌕ Search

arXiv subjects

Maya Stein

Publications and source records attributed to Maya Stein.

At least 37 records · Page 2Linked to original sources
arXiv2024

Antidirected subgraphs of oriented graphs

We show that for every $η>0$ every sufficiently large $n$-vertex oriented graph D of minimum semidegree exceeding $(1 + η) k/2$ contains every balanced antidirected tree with $k$ edges and bounded maximum degree, if $k \ge ηn$. In particular, this asymptotically confirms a conjecture of the first author for long antidirected paths and dense digraphs. Further, we show that in the same setting, D contains every $k$-edge antidirected subdivision of a sufficiently small complete graph, if the paths of the subdivision that have length 1 or 2 span a forest. As a special case, we can find all antidirected cycles of length at most $k$. Finally, we address a conjecture of Addario-Berry, Havet, Linhares Sales, Reed and Thomassé for antidirected trees in digraphs. We show that this conjecture is asymptotically true in $n$-vertex oriented graphs for all balanced antidirected trees of bounded maximum degree and of size linear in $n$.

math.CO↗
arXiv2023

Towards a hypergraph version of the Pósa-Seymour conjecture

We prove that for fixed $r\ge k\ge 2$, every $k$-uniform hypergraph on $n$ vertices having minimum codegree at least $(1-(\binom{r-1}{k-1}+\binom{r-2}{k-2})^{-1})n+o(n)$ contains the $(r-k+1)$th power of a tight Hamilton cycle. This result may be seen as a step towards a hypergraph version of the Pósa-Seymour conjecture. Moreover, we prove that the same bound on the codegree suffices for finding a copy of every spanning hypergraph of tree-width less than $r$ which admits a tree decomposition where every vertex is in a bounded number of bags.

math.CO↗
arXiv2023

Dirac-type conditions for spanning bounded-degree hypertrees

We prove that for fixed $k$, every $k$-uniform hypergraph on $n$ vertices and of minimum codegree at least $n/2+o(n)$ contains every spanning tight $k$-tree of bounded vertex degree as a sub\-graph. This generalises a well-known result of Komlós, Sárközy and Szemerédi for graphs. Our result is asymptotically sharp. We also prove an extension of our result to hypergraphs that satisfy some weak quasirandomness conditions.

math.CO↗
arXiv2023

Monochromatic paths in $2$-edge coloured graphs and hypergraphs

We answer a question of Gyárfás and Sárközy from 2013 by showing that every 2-edge-coloured complete 3-uniform hypergraph can be partitioned into two monochromatic tight paths of different colours. We also give a lower bound for the number of tight paths needed to partition any 2-edge-coloured complete r-partite r-uniform hypergraph. Finally, we show that any 2-edge-coloured complete bipartite graph has a partition into a monochromatic cycle and a monochromatic path, of different colours, unless the colouring is a split colouring.

math.CO↗
arXiv2022

Spanning Trees in Graphs of High Minimum Degree with a Universal Vertex I: An Asymptotic Result

In this paper and a companion paper, we prove that, if $m$ is sufficiently large, every graph on $m+1$ vertices that has a universal vertex and minimum degree at least $\lfloor \frac{2m}{3} \rfloor$ contains each tree $T$ with $m$ edges as a subgraph. Our result confirms, for large $m$, an important special case of a recent conjecture by Havet, Reed, Stein, and Wood. The present paper already contains an approximate version of the result.

math.CO↗
arXiv2022

Spanning Trees in Graphs of High Minimum Degree with a Universal Vertex II: A Tight Result

We prove that, if $m$ is sufficiently large, every graph on $m+1$ vertices that has a universal vertex and minimum degree at least $\lfloor \frac{2m}{3} \rfloor$ contains each tree $T$ with $m$ edges as a subgraph. Our result confirms, for large $m$, an important special case of a conjecture by Havet, Reed, Stein, and Wood. The present paper builds on the results of a companion paper in which we proved the statement for all trees having a vertex that is adjacent to many leaves.

math.CO↗
arXiv2022

Clique immersions and independence number

The analogue of Hadwiger's conjecture for the immersion order states that every graph $G$ contains $K_{χ(G)}$ as an immersion. If true, it would imply that every graph with $n$ vertices and independence number $α$ contains $K_{\lceil \frac nα\rceil}$ as an immersion. The best currently known bound for this conjecture is due to Gauthier, Le and Wollan, who recently proved that every graph $G$ contains an immersion of a clique on $\bigl\lceil \frac{χ(G)-4}{3.54}\bigr\rceil$ vertices. Their result implies that every $n$-vertex graph with independence number $α$ contains an immersion of a clique on $\bigl\lceil \frac{n}{3.54α}-1.13\bigr\rceil$ vertices. We improve on this result for all $α\ge 3$, by showing that every $n$-vertex graph with independence number $α\ge 3$ contains an immersion of a clique on $\bigl\lfloor \frac {n}{2.25 α- f(α)} \bigr\rfloor - 1$ vertices, where $f$ is a nonnegative function.

math.CO↗
arXiv2022

Partitioning a 2-edge-coloured graph of minimum degree $2n/3 + o(n)$ into three monochromatic cycles

Lehel conjectured in the 1970s that every red and blue edge-coloured complete graph can be partitioned into two monochromatic cycles. This was confirmed in 2010 by Bessy and Thomassé. However, the host graph $G$ does not have to be complete. It it suffices to require that $G$ has minimum degree at least $3n/4$, where $n$ is the order of $G$, as was shown recently by Letzter, confirming a conjecture of Balogh, Barát, Gerbner, Gyárfás and Sárközy. This degree condition is asymptotically tight. Here we continue this line of research, by proving that for every red and blue edge-colouring of an $n$-vertex graph of minimum degree at least $2n/3 + o(n)$, there is a partition of the vertex set into three monochromatic cycles. This approximately verifies a conjecture of Pokrovskiy and is essentially tight.

math.CO↗
arXiv2021

Active clustering for labeling training data

Gathering training data is a key step of any supervised learning task, and it is both critical and expensive. Critical, because the quantity and quality of the training data has a high impact on the performance of the learned function. Expensive, because most practical cases rely on humans-in-the-loop to label the data. The process of determining the correct labels is much more expensive than comparing two items to see whether they belong to the same class. Thus motivated, we propose a setting for training data gathering where the human experts perform the comparatively cheap task of answering pairwise queries, and the computer groups the items into classes (which can be labeled cheaply at the very end of the process). Given the items, we consider two random models for the classes: one where the set partition they form is drawn uniformly, the other one where each item chooses its class independently following a fixed distribution. In the first model, we characterize the algorithms that minimize the average number of queries required to cluster the items and analyze their complexity. In the second model, we analyze a specific algorithm family, propose as a conjecture that they reach the minimum average number of queries and compare their performance to a random approach. We also propose solutions to handle errors or inconsistencies in the experts' answers.

cs.DS↗
arXiv2020

Tree containment and degree conditions

We survey results and open problems relating degree conditions with tree containment in graphs, random graphs, digraphs and hypergraphs, and their applications in Ramsey theory.

math.CO↗
arXiv2020

Maximum and minimum degree conditions for embedding trees

We propose the following conjecture: For every fixed $α\in [0,\frac 13)$, each graph of minimum degree at least $(1+α)\frac k2$ and maximum degree at least $2(1-α)k$ contains each tree with $k$ edges as a subgraph. Our main result is an approximate version of the conjecture for bounded degree trees and large dense host graphs. We also show that our conjecture is asymptotically best possible. The proof of the approximate result relies on a second result, which we believe to be interesting on its own. Namely, we can embed any bounded degree tree into host graphs of minimum/maximum degree asymptotically exceeding $\frac k2$ and $\frac 43k$, respectively, as long as the host graph avoids a specific structure.

math.CO↗
arXiv2020

On the Erdős-Sós conjecture for trees with bounded degree

We prove the Erd\H os--Sós conjecture for trees with bounded maximum degree and large dense host graphs. As a corollary, we obtain an upper bound on the multicolour Ramsey number of large trees whose maximum degree is bounded by a constant.

math.CO↗
arXiv2020

$3$-Colouring $P_t$-free graphs without short odd cycles

For any odd $t\ge 9$, we present a polynomial-time algorithm that solves the $3$-colouring problem, and finds a $3$-colouring if one exists, in $P_{t}$-free graphs of odd girth at least $t-2$. In particular, our algorithm works for $(P_9, C_3, C_5)$-free graphs, thus making progress towards determining the complexity of $3$-colouring in $P_t$-free graphs, which is open for $t\ge 8$.

math.CO↗
arXiv2020

Better 3-coloring algorithms: excluding a triangle and a seven vertex path

We present an algorithm to color a graph $G$ with no triangle and no induced $7$-vertex path (i.e., a $\{P_7,C_3\}$-free graph), where every vertex is assigned a list of possible colors which is a subset of $\{1,2,3\}$. While this is a special case of the problem solved in [Combinatorica 38(4):779--801, 2018], that does not require the absence of triangles, the algorithm here is both faster and conceptually simpler. The complexity of the algorithm is $O(|V(G)|^5(|V(G)|+|E(G)|))$, and if $G$ is bipartite, it improves to $O(|V(G)|^2(|V(G)|+|E(G)|))$. Moreover, we prove that there are finitely many minimal obstructions to list 3-coloring $\{P_t,C_3\}$-free graphs if and only if $t \leq 7$. This implies the existence of a polynomial time certifying algorithm for list 3-coloring in $\{P_7,C_3\}$-free graphs. We furthermore determine other cases of $t, \ell$, and $k$ such that the family of minimal obstructions to list $k$-coloring in $\{P_t,C_{\ell}\}$-free graphs is finite.

math.CO↗
arXiv2019

Kalai's conjecture in $r$-partite $r$-graphs

Kalai conjectured that every $n$-vertex $r$-uniform hypergraph with more than $\frac{t-1}{r} {n \choose r-1}$ edges contains all tight $r$-trees of some fixed size $t$. We prove Kalai's conjecture for $r$-partite $r$-uniform hypergraphs. Our result is asymptotically best possible up to replacing the term $\frac{t-1}{r}$ with the term $\frac{t-r+1}{r}$. We apply our main result in graphs to show an upper bound for the Turán number of trees.

math.CO↗
arXiv2019

A Variant of the Erdős-Sós Conjecture

A well-known conjecture of Erdős and Sós states that every graph with average degree exceeding $m-1$ contains every tree with $m$ edges as a subgraph. We propose a variant of this conjecture, which states that every graph of maximum degree exceeding $m$ and minimum degree at least $\lfloor \frac{2m}{3}\rfloor$ contains every tree with $m$ edges. As evidence for our conjecture we show (i) for every $m$ there is a $g(m)$ such that the weakening of the conjecture obtained by replacing $m$ by $g(m)$ holds, and (ii) there is a $γ>0$ such that the weakening of the conjecture obtained by replacing $\lfloor \frac{2m}{3}\rfloor$ by $(1-γ)m$ holds.

math.CO↗
arXiv2019

Regularity inheritance in pseudorandom graphs

Advancing the sparse regularity method, we prove one-sided and two-sided regularity inheritance lemmas for subgraphs of bijumbled graphs, improving on results of Conlon, Fox and Zhao [Adv. Math. 256 (2014), 206--290]. These inheritance lemmas also imply improved $H$-counting lemmas for subgraphs of bijumbled graphs, for some $H$.

math.CO↗