Well-mixing vertices and almost expanders

We study regular graphs in which the random walks starting from a positive fraction of vertices have small mixing time. We prove that any such graph is virtually an expander and has no small separator. This answers a question of Pak [SODA, 2002]. As a corollary, it shows that sparse (constant degree) regular graphs with many well-mixing vertices have a long cycle, improving a result of Pak. Furthermore, such cycle can be found in polynomial time. Secondly, we show that if the random walks from a positive fraction of vertices are well-mixing, then the random walks from almost all vertices are well-mixing (with a slightly worse mixing time).

Fractional Helly theorem for Cartesian products of convex sets

Hellys theorem and its variants show that for a family of convex sets in Euclidean space, local intersection patterns influence global intersection patterns. A classical result of Eckhoff in 1988 provided an optimal fractional Helly theorem for axis-aligned boxes, which are Cartesian products of line segments. Answering a question raised by Barany and Kalai, and independently Lew, we generalize Eckhoffs result to Cartesian products of convex sets in all dimensions. In particular, we prove that given $alpha in (1-frac{1}{t^d},1]$ and a finite family $mathcal{F}$ of Cartesian products of convex sets $prod_{iin[t]}A_i$ in $mathbb{R}^{td}$ with $A_isubset mathbb{R}^d$ if at least $alpha$-fraction of the $(d+1)$-tuples in $mathcal{F}$ are intersecting then at least $(1-(t^d(1-alpha))^{1/(d+1)})$-fraction of sets in $mathcal{F}$ are intersecting. This is a special case of a more general result on intersections of $d$-Leray complexes. We also provide a construction showing that our result on $d$-Leray complexes is optimal. Interestingly the extremal example is representable as a family of cartesian products of convex sets, implying the bound $alpha>1-frac{1}{t^d}$ and the fraction $(1-(t^d(1-alpha))^{1/(d+1)})$ above are also best possible. The well-known optimal construction for fractional Helly theorem for convex sets in $mathbb{R}^d$ does not have $(p,d+1)$-condition for sublinear $p$. Inspired by this we give constructions showing that, somewhat surprisingly, imposing additional $(p,d+1)$-condition has negligible effect on improving the quantitative bounds in neither the fractional Helly theorem for convex sets nor Cartesian products of convex sets. Our constructions offer a rich family of distinct extremal configurations for fractional Helly theorem, implying in a sense that the optimal bound is stable.

Bipartite independence number and balanced coloring

In this paper, we establish a couple of results on extremal problems in bipartite graphs. Firstly, we show that every sufficiently large bipartite graph with average degree $Delta$ and with $n$ vertices on each side has a balanced independent set containing $(1-epsilon) frac{log Delta}{Delta} n$ vertices from each side for small $epsilon > 0$. Secondly, we prove that the vertex set of every sufficiently large balanced bipartite graph with maximum degree at most $Delta$ can be partitioned into $(1+epsilon)frac{Delta}{log Delta}$ balanced independent sets. Both of these results are algorithmic and best possible up to a factor of 2, which might be hard to improve as evidenced by the phenomenon known as `algorithmic barrier in the literature. The first result improves a recent theorem of Axenovich, Sereni, Snyder, and Weber in a slightly more general setting. The second result improves a theorem of Feige and Kogan about coloring balanced bipartite graphs.

Majority dynamics on sparse random graphs

Majority dynamics on a graph $G$ is a deterministic process such that every vertex updates its $pm 1$-assignment according to the majority assignment on its neighbor simultaneously at each step. Benjamini, Chan, ODonnel, Tamuz and Tan conjectured that, in the ErdH{o}s--Renyi random graph $G(n,p)$, the random initial $pm 1$-assignment converges to a $99%$-agreement with high probability whenever $p=omega(1/n)$. This conjecture was first confirmed for $pgeqlambda n^{-1/2}$ for a large constant $lambda$ by Fountoulakis, Kang and Makai. Although this result has been reproved recently by Tran and Vu and by Berkowitz and Devlin, it was unknown whether the conjecture holds for $p< lambda n^{-1/2}$. We break this $Omega(n^{-1/2})$-barrier by proving the conjecture for sparser random graphs $G(n,p)$, where $lambda n^{-3/5}log n leq p leq lambda n^{-1/2}$ with a large constant $lambda>0$.

Colorful Hamilton cycles in random graphs

Given an $n$ vertex graph whose edges have colored from one of $r$ colors $C={c_1,c_2,ldots,c_r}$, we define the Hamilton cycle color profile $hcp(G)$ to be the set of vectors $(m_1,m_2,ldots,m_r)in [0,n]^r$ such that there exists a Hamilton cycle that is the concatenation of $r$ paths $P_1,P_2,ldots,P_r$, where $P_i$ contains $m_i$ edges. We study $hcp(G_{n,p})$ when the edges are randomly colored. We discuss the profile close to the threshold for the existence of a Hamilton cycle and the threshold for when $hcp(G_{n,p})={(m_1,m_2,ldots,m_r)in [0,n]^r:m_1+m_2+cdots+m_r=n}$.

Rainbow matchings in edge-colored simple graphs

There has been much research on the topic of finding a large rainbow matching (with no two edges having the same color) in a properly edge-colored graph, where a proper edge coloring is a coloring of the edge set such that no same-colored edges are incident. Barat, Gyarfas, and Sarkozy conjectured that in every proper edge coloring of a multigraph (with parallel edges allowed, but not loops) with $2q$ colors where each color appears at least $q$ times, there is always a rainbow matching of size $q$. Recently, Aharoni, Berger, Chudnovsky, Howard, and Seymour proved a relaxation of the conjecture with $3q-2$ colors. Our main result proves that $2q + o(q)$ colors are enough if the graph is simple, confirming the conjecture asymptotically for simple graphs. This question restricted to simple graphs was considered before by Aharoni and Berger. We also disprove one of their conjectures regarding the lower bound on the number of colors one needs in the conjecture of Barat, Gyarfas, and Sarkozy for the class of simple graphs. Our methods are inspired by the randomized algorithm proposed by Gao, Ramadurai, Wanless, and Wormald to find a rainbow matching of size $q$ in a graph that is properly edge-colored with $q$ colors, where each color class contains $q + o(q)$ edges. We consider a modified version of their algorithm, with which we are able to prove a generalization of their statement with a slightly better error term in $o(q)$. As a by-product of our techniques, we obtain a new asymptotic version of the Brualdi-Ryser-Stein Conjecture.

Minimizing the number of edges in $K_{s,t}$-saturated bipartite graphs

This paper considers an edge minimization problem in saturated bipartite graphs. An $n$ by $n$ bipartite graph $G$ is $H$-saturated if $G$ does not contain a subgraph isomorphic to $H$ but adding any missing edge to $G$ creates a copy of $H$. More than half a century ago, Wessel and Bollobas independently solved the problem of minimizing the number of edges in $K_{(s,t)}$-saturated graphs, where $K_{(s,t)}$ is the `ordered complete bipartite graph with $s$ vertices from the first color class and $t$ from the second. However, the very natural `unordered analogue of this problem was considered only half a decade ago by Moshkovitz and Shapira. When $s=t$, it can be easily checked that the unordered variant is exactly the same as the ordered case. Later, Gan, Korandi, and Sudakov gave an asymptotically tight bound on the minimum number of edges in $K_{s,t}$-saturated $n$ by $n$ bipartite graphs, which is only smaller than the conjecture of Moshkovitz and Shapira by an additive constant. In this paper, we confirm their conjecture for $s=t-1$ with the classification of the extremal graphs. We also improve the estimates of Gan, Korandi, and Sudakov for general $s$ and $t$, and for all sufficiently large $n$.

Exact results on generalized ErdH{o}s-Gallai problems

Generalized Turan problems have been a central topic of study in extremal combinatorics throughout the last few decades. One such problem is maximizing the number of cliques of size $t$ in a graph of a fixed order that does not contain any path (or cycle) of length at least a given number. Both of the path-free and cycle-free extremal problems were recently considered and asymptotically solved by Luo. We fully resolve these problems by characterizing all possible extremal graphs. We further extend these results by solving the edge-variant of these problems where the number of edges is fixed instead of the number of vertices. We similarly obtain exact characterization of the extremal graphs for these edge variants.

A note on volume thresholds for random polytopes

We study the expected volume of random polytopes generated by taking the convex hull of independent identically distributed points from a given distribution. We show that for log-concave distributions supported on convex bodies, we need at least exponentially many (in dimension) samples for the expected volume to be significant and that super-exponentially many samples suffice for concave measures when their parameter of concavity is positive.

Many cliques with few edges and bounded maximum degree

Generalized Turan problems have been a central topic of study in extremal combinatorics throughout the last few decades. One such problem, maximizing the number of cliques of a fixed order in a graph with fixed number of vertices and bounded maximum degree, was recently completely resolved by Chase. Kirsch and Radcliffe raised a natural variant of this problem where the number of edges is fixed instead of the number of vertices. In this paper, we determine the maximum number of cliques of a fixed order in a graph with fixed number of edges and bounded maximum degree, resolving a conjecture by Kirsch and Radcliffe. We also give a complete characterization of the extremal graphs.