arXiv subjects

Fan Wei

Publications and source records attributed to Fan Wei.

At least 37 records · Page 2Linked to original sources

arXiv2024

Undecidability of polynomial inequalities in tournaments

Many fundamental problems in extremal combinatorics are equivalent to proving certain polynomial inequalities in graph homomorphism densities. In 2011, a breakthrough result by Hatami and Norine showed that it is undecidable to verify polynomial inequalities in graph homomorphism densities. Recently, Blekherman, Raymond and Wei extended this result by showing that it is also undecidable to determine the validity of polynomial inequalities in homomorphism densities for weighted graphs with edge weights taking real values. These two results resolved a question of Lovász. In this paper, we consider the problem of determining the validity of polynomial inequalities in digraph homomorphism densities for tournaments. We prove that the answer to this problem is also undecidable.

math.CO↗

arXiv2024

Non-bipartite k-common graphs

A graph H is k-common if the number of monochromatic copies of H in a k-edge-coloring of K_n is asymptotically minimized by a random coloring. For every k, we construct a connected non-bipartite k-common graph. This resolves a problem raised by Jagger, Stovicek and Thomason [Combinatorica 16 (1996), 123-141]. We also show that a graph H is k-common for every k if and only if H is Sidorenko and that H is locally k-common for every k if and only if H is locally Sidorenko.

math.CO↗

arXiv2024

The binomial random graph is a bad inducer

For a finite graph $F$ and a value $p \in [0,1]$, let $I(F,p)$ denote the largest $y$ for which there is a sequence of graphs of edge density approaching $p$ so that the induced $F$-density of the sequence approaches $y$. We show that for all $F$ on at least three vertices and all $p \in (0,1)$, the binomial random graph $G(n,p)$ has induced $F$-density strictly less than $I(F,p).$ This provides a negative answer to a problem posed by Liu, Mubayi and Reiher. Our approach is in the limiting setting of graphons, and we in fact show a stronger result: the binomial random graph is never a \emph{local} maximum in the space of graphons of edge density $p$. This is done by finding a sequence of balanced perturbations of arbitrarily small norm that increase the $F$-density.

math.CO↗

arXiv2024

Extremal number of cliques of given orders in graphs with a forbidden clique minor

Alon and Shikhelman initiated the systematic study of a generalization of the extremal function. Motivated by algorithmic applications, the study of the extremal function $\text{ex}(n, K_k, K_t\text{-minor})$, i.e., the number of cliques of order $k$ in $K_t$-minor free graphs on $n$ vertices, has received much attention. In this paper, we determine essentially sharp bounds on the maximum possible number of cliques of order $k$ in a $K_t$-minor free graph on $n$ vertices. More precisely, we determine a function $C(k,t)$ such that for each $k < t$ with $t-k\gg \log_2 t$, every $K_t$-minor free graph on $n$ vertices has at most $ n C(k, t)^{1+o_t(1)}$ cliques of order $k$. We also show this bound is sharp by constructing a $K_t$-minor-free graph on $n$ vertices with $C(k, t) n$ cliques of order $k$. This bound answers a question of Wood and Fox-Wei asymptotically up to $o_t(1)$ in the exponent except the extreme values when $k$ is very close to $t$.

math.CO↗

arXiv2024

Common graphs with arbitrary chromatic number

Ramsey's Theorem guarantees for every graph H that any 2-edge-coloring of a sufficiently large complete graph contains a monochromatic copy of H. In 1962, Erdos conjectured that the random 2-edge-coloring minimizes the number of monochromatic copies of K_k, and the conjecture was extended by Burr and Rosta to all graphs. In the late 1980s, the conjectures were disproved by Thomason and Sidorenko, respectively. A classification of graphs whose number of monochromatic copies is minimized by the random 2-edge-coloring, which are referred to as common graphs, remains a challenging open problem. If Sidorenko's Conjecture, one of the most significant open problems in extremal graph theory, is true, then every 2-chromatic graph is common, and in fact, no 2-chromatic common graph unsettled for Sidorenko's Conjecture is known. While examples of 3-chromatic common graphs were known for a long time, the existence of a 4-chromatic common graph was open until 2012, and no common graph with a larger chromatic number is known. We construct connected k-chromatic common graphs for every k. This answers a question posed by Hatami, Hladky, Kral, Norine and Razborov [Combin. Probab. Comput. 21 (2012), 734-742], and a problem listed by Conlon, Fox and Sudakov [London Math. Soc. Lecture Note Ser. 424 (2015), 49-118, Problem 2.28]. This also answers in a stronger form the question raised by Jagger, Stovicek and Thomason [Combinatorica 16, (1996), 123-131] whether there exists a common graph with chromatic number at least four.

math.CO↗

arXiv2024

An Asymptotically Sharp Bound on the Maximum Number of Independent Transversals

Let $G$ be a multipartite graph with partition $V_1, V_2,\ldots, V_k$ of $V(G)$. Let $d_{i,j}$ denote the edge density of the pair $(V_i, V_j)$. An independent transversal is an independent set of $G$ with exactly one vertex in each $V_i$. In this paper, we prove an asymptotically sharp upper bound on the maximum number of independent transversals given the $d_{i,j}$'s.

math.CO↗

arXiv2023

The limit points of the top and bottom eigenvalues of regular graphs

We prove that for each $d \geq 3$ the set of all limit points of the second largest eigenvalue of growing sequences of $d$-regular graphs is $[2\sqrt{d-1},d]$. A similar argument shows that the set of all limit points of the smallest eigenvalue of growing sequences of $d$-regular graphs with growing (odd) girth is $[-d, -2 \sqrt{d-1}]$. The more general question of identifying all vectors which are limit points of the vectors of the top $k$ eigenvalues of sequences of $d$-regular graphs is considered as well. As a by product, in the study of discrete counterpart of the "scarring" phenomenon observed in the investigation of quantum ergodicity on manifolds, our technique provides a method to construct $d$-regular almost Ramanujan graphs with large girth and localized eigenvectors corresponding to eigenvalues larger than $2\sqrt{d-1}$, strengthening a result of Alon, Ganguly, and Srivastava.

math.CO↗

arXiv2023

Phase Transition of Degeneracy in Minor-Closed Families

Given an infinite family ${\mathcal G}$ of graphs and a monotone property ${\mathcal P}$, an (upper) threshold for ${\mathcal G}$ and ${\mathcal P}$ is a "fastest growing" function $p: \mathbb{N} \to [0,1]$ such that $\lim_{n \to \infty} \Pr(G_n(p(n)) \in {\mathcal P})= 1$ for any sequence $(G_n)_{n \in \mathbb{N}}$ over ${\mathcal G}$ with $\lim_{n \to \infty}\lvert V(G_n) \rvert = \infty$, where $G_n(p(n))$ is the random subgraph of $G_n$ such that each edge remains independently with probability $p(n)$. In this paper we study the upper threshold for the family of $H$-minor free graphs and for the graph property of being $(r-1)$-degenerate, which is one fundamental graph property with many applications. Even a constant factor approximation for the upper threshold for all pairs $(r,H)$ is expected to be very difficult by its close connection to a major open question in extremal graph theory. We determine asymptotically the thresholds (up to a constant factor) for being $(r-1)$-degenerate for a large class of pairs $(r,H)$, including all graphs $H$ of minimum degree at least $r$ and all graphs $H$ with no vertex-cover of size at most $r$, and provide lower bounds for the rest of the pairs of $(r,H)$. The results generalize to arbitrary proper minor-closed families and the properties of being $r$-colorable, being $r$-choosable, or containing an $r$-regular subgraph, respectively.

math.CO↗

arXiv2022

Threshold Ramsey multiplicity for paths and even cycles

The Ramsey number $r(H)$ of a graph $H$ is the minimum integer $n$ such that any two-coloring of the edges of the complete graph $K_n$ contains a monochromatic copy of $H$. While this definition only asks for a single monochromatic copy of $H$, it is often the case that every two-edge-coloring of the complete graph on $r(H)$ vertices contains many monochromatic copies of $H$. The minimum number of such copies over all two-colorings of $K_{r(H)}$ will be referred to as the threshold Ramsey multiplicity of $H$. Addressing a problem of Harary and Prins, who were the first to systematically study this quantity, we show that there is a positive constant $c$ such that the threshold Ramsey multiplicity of a path or an even cycle on $k$ vertices is at least $(ck)^k$. This bound is tight up to the constant $c$. We prove a similar result for odd cycles in a companion paper.

math.CO↗

arXiv2022

Undecidability of polynomial inequalities in weighted graph homomorphism densities

Many problems and conjectures in extremal combinatorics concern polynomial inequalities between homomorphism densities of graphs where we allow edges to have real weights. Using the theory of graph limits, we can equivalently evaluate polynomial expressions in homomorphism densities on kernels $W$, i.e., symmetric, bounded, and measurable functions $W$ from $[0,1]^2 \to \mathbb{R}$. In 2011, Hatami and Norin proved a fundamental result that it is undecidable to determine the validity of polynomial inequalities in homomorphism densities for graphons (i.e., the case where the range of $W$ is $[0,1]$, which corresponds to unweighted graphs, or equivalently, to graphs with edge weights between $0$ and $1$). The corresponding problem for more general sets of kernels, e.g., for all kernels or for kernels with range $[-1,1]$, remains open. For any $a > 0$, we show undecidability of polynomial inequalities for any set of kernels which contains all kernels with range $\{0,a\}$. This result also answers a question raised by Lovász about finding computationally effective certificates for the validity of homomorphism density inequalities in kernels.

math.CO↗

arXiv2021

On the asymptotic confirmation of the Faudree-Lehel Conjecture for general graphs

Given a simple graph $G$, the {\it irregularity strength} of $G$, denoted by $s(G)$, is the least positive integer $k$ such that there is a weight assignment on edges $f: E(G) \to \{1,2,\dots, k\}$ attributing distinct weighted degrees: $\tilde f(v):= \sum_{u: \{u,v\}\in E(G)} f(\{u,v\})$ to all vertices $v\in V(G)$. It is straightforward that $s(G) \geq n/d$ for every $d$-regular graph $G$ on $n$ vertices with $d>1$. In 1987, Faudree and Lehel conjectured in turn that there is an absolute constant $c$ such that $s(G) \leq n/d + c$ for all such graphs. Even though the conjecture has remained open in almost all relevant cases, it is more generally believed that there exists a universal constant $c$ such that $s(G) \leq n/δ+ c$ for every graph $G$ on $n$ vertices with minimum degree $δ\geq 1$ which does not contain an isolated edge. In this paper we confirm that the generalized Faudree-Lehel Conjecture holds for graphs with $δ\geq n^β$ where $β$ is any fixed constant larger than $0.8$. Furthermore, we confirm that the conjecture holds in general asymptotically. That is we prove that for any $\varepsilon\in(0,0.25)$ there exist absolute constants $c_1, c_2$ such that for all graphs $G$ on $n$ vertices with minimum degree %at least $δ\geq 1$ and without isolated edges, $s(G) \leq \frac{n}δ(1+\frac{c_1}{δ^{\varepsilon}})+c_2$, thus extending in various aspects and strengthening a recent result of Przybyło, who showed that $s(G) \leq \frac{n}{d}(1+ \frac{1}{\ln^{ε/19}n})=\frac{n}{d}(1+o(1))$ for $d$-regular graphs with $d\in [\ln^{1+ε} n, n/\ln^εn]$, and improving an earlier general upper bound: $s(G)< 6\frac{n}δ+6$ of Kalkowski, Karoński and Pfender.

math.CO↗

arXiv2021

Short proof of the asymptotic confirmation of the Faudree-Lehel Conjecture

Given a simple graph $G$, the {\it irregularity strength} of $G$, denoted $s(G)$, is the least positive integer $k$ such that there is a weight assignment on edges $f: E(G) \to \{1,2,\dots, k\}$ for which each vertex weight $f^V(v):= \sum_{u: \{u,v\}\in E(G)} f(\{u,v\})$ is unique amongst all $v\in V(G)$. In 1987, Faudree and Lehel conjectured that there is a constant $c$ such that $s(G) \leq n/d + c$ for all $d$-regular graphs $G$ on $n$ vertices with $d>1$, whereas it is trivial that $s(G) \geq n/d$. In this short note we prove that the Faudree-Lehel Conjecture holds when $d \geq n^{0.8+ε}$ for any fixed $ε>0$, with a small additive constant $c=28$ for $d$ large enough. Furthermore, we confirm the conjecture asymptotically by proving that for any fixed $β\in(0,1/4)$ there is a constant $C$ such that for all $d$-regular graphs $G$, $s(G) \leq \frac{n}{d}(1+\frac{C}{d^β})+28$, extending and improving a recent result of Przybyło that $s(G) \leq \frac{n}{d}(1+ \frac{1}{\ln^{ε/19}n})$ whenever $d\in [\ln^{1+ε} n, n/\ln^εn]$ and $d$ is large enough.

math.CO↗

arXiv2021

Threshold Ramsey multiplicity for odd cycles

The Ramsey number $r(H)$ of a graph $H$ is the minimum $n$ such that any two-coloring of the edges of the complete graph $K_n$ contains a monochromatic copy of $H$. The threshold Ramsey multiplicity $m(H)$ is then the minimum number of monochromatic copies of $H$ taken over all two-edge-colorings of $K_{r(H)}$. The study of this concept was first proposed by Harary and Prins almost fifty years ago. In a companion paper, the authors have shown that there is a positive constant $c$ such that the threshold Ramsey multiplicity for a path or even cycle with $k$ vertices is at least $(ck)^k$, which is tight up to the value of $c$. Here, using different methods, we show that the same result also holds for odd cycles with $k$ vertices.

math.CO↗

arXiv2021

Irregular Subgraphs

We suggest two related conjectures dealing with the existence of spanning irregular subgraphs of graphs. The first asserts that any $d$-regular graph on $n$ vertices contains a spanning subgraph in which the number of vertices of each degree between $0$ and $d$ deviates from $\frac{n}{d+1}$ by at most $2$. The second is that every graph on $n$ vertices with minimum degree $δ$ contains a spanning subgraph in which the number of vertices of each degree does not exceed $\frac{n}{δ+1}+2$. Both conjectures remain open, but we prove several asymptotic relaxations for graphs with a large number of vertices $n$. In particular we show that if $d^3 \log n \leq o(n)$ then every $d$-regular graph with $n$ vertices contains a spanning subgraph in which the number of vertices of each degree between $0$ and $d$ is $(1+o(1))\frac{n}{d+1}$. We also prove that any graph with $n$ vertices and minimum degree $δ$ contains a spanning subgraph in which no degree is repeated more than $(1+o(1))\frac{n}{δ+1}+2$ times.

math.CO↗

arXiv2021

On the inducibility problem for random Cayley graphs of abelian groups with a few deleted vertices

Given a $k$-vertex graph $H$ and an integer $n$, what are the $n$-vertex graphs with the maximum number of induced copies of $H$? This question is closely related to the inducibility problem introduced by Pippenger and Golumbic in 1975, which asks for the maximum possible fraction of $k$-vertex subsets of an $n$-vertex graph that induce a copy of $H$. Huang, Lee and the first author proved that for a random $k$-vertex graph $H$, almost surely the $n$-vertex graphs maximizing the number of induced copies of $H$ are the balanced iterated blow-ups of $H$. In this paper, we consider the case where the graph $H$ is obtained by deleting a small number of vertices from a random Cayley graph $\widetilde{H}$ of an abelian group. We prove that in this case, almost surely all $n$-vertex graphs maximizing the number of induced copies of $H$ are balanced iterated blow-ups of $\widetilde{H}$.

math.CO↗

arXiv2020

Uplink Transmission Design for Crowded Correlated Cell-Free Massive MIMO-OFDM Systems

In cell-free massive multiple-input multiple-output (MIMO) orthogonal frequency division multiplexing (OFDM) systems, user equipments (UEs) are served by many distributed access points (APs), where channels are correlated due to finite angle-delay spread in realistic outdoor wireless propagation environments. Meanwhile, the number of UEs is growing rapidly for future fully networked society. In this paper, we focus on the uplink transmission design in crowded correlated cell-free massive MIMO-OFDM systems with limited number of orthogonal pilots. For the pilot transmission phase, we identify active UEs based on non-orthogonal pilot phase shift hopping patterns and non-orthogonal adjustable phase shift pilots (APSP). We derive a closed-form expression of mean square error of channel estimation (MSE-CE) and obtain an optimal condition for minimizing MSE-CE. According to this condition, the APSP set allocation scheme is proposed. Furthermore, for the data transmission, the max-min power control algorithm is devised to maximize the minimum spectral efficiency (SE) lower bound among active UEs. Simulation results indicate significant performance gains in terms of MSE-CE for the proposed APSP set allocation scheme. The proposed power control scheme can further improve the minimum SE among active UEs. Hence, they are crucial for crowded correlated cell-free massive MIMO-OFDM systems.

cs.IT↗

arXiv2020

Computer Analysis of Architecture Using Automatic Image Understanding

In the past few years, computer vision and pattern recognition systems have been becoming increasingly more powerful, expanding the range of automatic tasks enabled by machine vision. Here we show that computer analysis of building images can perform quantitative analysis of architecture, and quantify similarities between city architectural styles in a quantitative fashion. Images of buildings from 18 cities and three countries were acquired using Google StreetView, and were used to train a machine vision system to automatically identify the location of the imaged building based on the image visual content. Experimental results show that the automatic computer analysis can automatically identify the geographical location of the StreetView image. More importantly, the algorithm was able to group the cities and countries and provide a phylogeny of the similarities between architectural styles as captured by StreetView images. These results demonstrate that computer vision and pattern recognition algorithms can perform the complex cognitive task of analyzing images of buildings, and can be used to measure and quantify visual similarities and differences between different styles of architectures. This experiment provides a new paradigm for studying architecture, based on a quantitative approach that can enhance the traditional manual observation and analysis. The source code used for the analysis is open and publicly available.

cs.CV↗

arXiv2020

Towrad 5G Air Interface Technology: Sparse Code Muliple Access

The fifth generation wireless networks focus on the design of low latency, high data rate, high reliability, and massive connectivity communications. Non-orthogonal multiple access (NOMA) is an essential enabling technology to accommodate the wide range of communication requirements. By coordinating the massive devices within the same resource block on power domain, frequency domain or code domain, NOMA is superior to conventional orthogonal multiple access in terms of the network connectivity, the throughputs of system and etc. Sparse code multiple access (SCMA) is a kind of multi-carrier code domain NOMA and has been studied extensively. The challenges for designing a high quality SCMA system is to seek the feasible encoding and decoding schemes to meet the desired requirements. In this article, we present some recent progresses towards the design of multi-dimensional codebooks, the practical low complexity decoder, as well as the Grant-Free multiple access for SCMA system. In particular, we show how the SCMA codebooks construction are motived by the combined design of multi-dimensional constellation and factor graphs. In addition, various low complexity SCMA decoders are also reviewed with a special focus on sphere decoding. Moreover, based on the framework of belief propagation, the SCMA Grant-Free transmission is introduced and the problem of collision resolution is also discussed.

eess.SP↗