arXiv subjects

Xing Peng

Publications and source records attributed to Xing Peng.

29 records · Page 2Linked to original sources

arXiv2015

On the decomposition of random hypergraphs

For an $r$-uniform hypergraph $H$, let $f(H)$ be the minimum number of complete $r$-partite $r$-uniform subhypergraphs of $H$ whose edge sets partition the edge set of $H$. For a graph $G$, $f(G)$ is the bipartition number of $G$ which was introduced by Graham and Pollak in 1971. In 1988, Erdős conjectured that if $G \in G(n,1/2)$, then with high probability $f(G)=n-α(G)$, where $α(G)$ is the independence number of $G$. This conjecture and related problems have received a lot of attention recently. In this paper, we study the value of $f(H)$ for a typical $r$-uniform hypergraph $H$. More precisely, we prove that if $(\log n)^{2.001}/n \leq p \leq 1/2$ and $H \in H^{(r)}(n,p)$, then with high probability $f(H)=(1-π(K^{(r-1)}_r)+o(1))\binom{n}{r-1}$, where $π(K^{(r-1)}_r)$ is the Turán density of $K^{(r-1)}_r$.

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arXiv2015

The Ramsey number of generalized loose paths in uniform Hypergrpahs

Let $H=(V,E)$ be an $r$-uniform hypergraph. For each $1 \leq s \leq r-1$, an $s$-path ${\mathcal P}^{r,s}_n$ of length $n$ in $H$ is a sequence of distinct vertices $v_1,v_2,\ldots,v_{s+n(r-s)}$ such that $\{v_{1+i(r-s)},\ldots, v_{s+(i+1)(r-s)}\}\in E(H)$ for each $0 \leq i \leq n-1$.Recently, the Ramsey number of $1$-paths in uniform hypergraphs has received a lot of attention. In this paper, we consider the Ramsey number of $r/2-$paths for even $r$. Namely, we prove the following exact result: $R({\mathcal P}^{r,r/2}_n,{\mathcal P}^{r,r/2}_3)=R({\mathcal P}^{r,r/2}_n,{\mathcal P}^{r,r/2}_4)=\tfrac{(n+1)r}{2}+1.$

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arXiv2014

On the chromatic number of the Erdős-Rényi orthogonal polarity graph

For a prime power $q$, let $ER_q$ denote the Erdős-Rényi orthogonal polarity graph. We prove that if $q$ is an even power of an odd prime, then $χ( ER_{q}) \leq 2 \sqrt{q} + O ( \sqrt{q} / \log q)$. This upper bound is best possible up to a constant factor of at most 2. If $q$ is an odd power of an odd prime and satisfies some condition on irreducible polynomials, then we improve the best known upper bound for $χ(ER_{q})$ substantially. We also show that for sufficiently large $q$, every $ER_q$ contains a subgraph that is not 3-chromatic and has at most 36 vertices.

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arXiv2013

Bounds for generalized Sidon sets

Let $Γ$ be an abelian group and $g \geq h \geq 2$ be integers. A set $A \subset Γ$ is a $C_h[g]$-set if given any set $X \subset Γ$ with $|X| = k$, and any set $\{ k_1 , \dots , k_g \} \subset Γ$, at least one of the translates $X+ k_i$ is not contained in $A$. For any $g \geq h \geq 2$, we prove that if $A \subset \{1,2, \dots ,n \}$ is a $C_h[g]$-set in $\mathbb{Z}$, then $|A| \leq (g-1)^{1/h} n^{1 - 1/h} + O(n^{1/2 - 1/2h})$. We show that for any integer $n \geq 1$, there is a $C_3 [3]$-set $A \subset \{1,2, \dots , n \}$ with $|A| \geq (4^{-2/3} + o(1)) n^{2/3}$. We also show that for any odd prime $p$, there is a $C_3[3]$-set $A \subset \mathbb{F}_p^3$ with $|A| \geq p^2 - p$, which is asymptotically best possible. Using the projective norm graphs from extremal graph theory, we show that for each integer $h \geq 3$, there is a $C_h[h! +1]$-set $A \subset \{1,2, \dots , n \}$ with $|A| \geq ( c_h +o(1))n^{1-1/h}$. A set $A$ is a \emph{weak $C_h[g]$-set} if we add the condition that the translates $X +k_1, \dots , X + k_g$ are all pairwise disjoint. We use the probabilistic method to construct weak $C_h[g]$-sets in $\{1,2, \dots , n \}$ for any $g \geq h \geq 2$. Lastly we obtain upper bounds on infinite $C_h[g]$-sequences. We prove that for any infinite $C_h[g$]-sequence $A \subset \mathbb{N}$, we have $A(n) = O ( n^{1 - 1/h} ( \log n )^{ - 1/h} )$ for infinitely many $n$, where $A(n) = | A \cap \{1,2, \dots , n \}|$.

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arXiv2013

Infinite Turán problems for bipartite graphs

We consider an infinite version of the bipartite Turán problem. Let $G$ be an infinite graph with $V(G) = \mathbb{N}$ and let $G_n$ be the $n$-vertex subgraph of $G$ induced by the vertices $\{1,2, \dots, n \}$. We show that if $G$ is $K_{2,t+1}$-free then for infinitely many $n$, $e(G_n) \leq 0.471 \sqrt{t} n^{3/2}$. Using the $K_{2,t+1}$-free graphs constructed by Füredi, we construct an infinite $K_{2,t+1}$-free graph with $e(G_n) \geq 0.23 \sqrt{t}n^{3/2}$ for all $n \geq n_0$.

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arXiv2012

The Fractional Chromatic Number of Triangle-free Graphs with $Δ\leq 3$

Let $G$ be any triangle-free graph with maximum degree $Δ\leq 3$. Staton proved that the independence number of $G$ is at least 5/14n. Heckman and Thomas conjectured that Staton's result can be strengthened into a bound on the fractional chromatic number of $G$, namely $χ_f(G)\leq 14/5. Recently, Hatami and Zhu proved $χ_f(G) \leq 3 -{3/64}$. In this paper, we prove $χ_f(G) \leq 3- 3/43$.

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arXiv2012

Spectra of edge-independent random graphs

Let $G$ be a random graph on the vertex set $\{1,2,..., n\}$ such that edges in $G$ are determined by independent random indicator variables, while the probability $p_{ij}$ for $\{i,j\}$ being an edge in $G$ is not assumed to be equal. Spectra of the adjacency matrix and the normalized Laplacian matrix of $G$ are recently studied by Oliveira and Chung-Radcliffe. Let $A$ be the adjacency matrix of $G$, $\bar A=\E(A)$, and $Δ$ be the maximum expected degree of $G$. Oliveira first proved that almost surely $\|A-\bar A\|=O(\sqrt{Δ\ln n})$ provided $Δ\geq C \ln n$ for some constant $C$. Chung-Radcliffe improved the hidden constant in the error term using a new Chernoff-type inequality for random matrices. Here we prove that almost surely $\|A-\bar A\|\leq (2+o(1))\sqrtΔ$ with a slightly stronger condition $Δ\gg \ln^4 n$. For the Laplacian $L$ of $G$, Oliveira and Chung-Radcliffe proved similar results $\|L-\bar L|=O(\sqrt{\ln n}/\sqrtδ)$ provided the minimum expected degree $δ\gg \ln n$; we also improve their results by removing the $\sqrt{\ln n}$ multiplicative factor from the error term under some mild conditions. Our results naturally apply to the classic Erdős-Rényi random graphs, random graphs with given expected degree sequences, and bond percolation of general graphs.

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arXiv2011

Loose Laplacian spectra of random hypergraphs

Let $H=(V,E)$ be an $r$-uniform hypergraph with the vertex set $V$ and the edge set $E$. For $1\leq s \leq r/2$, we define a weighted graph $G^{(s)}$ on the vertex set ${V\choose s}$ as follows. Every pair of $s$-sets $I$ and $J$ is associated with a weight $w(I,J)$, which is the number of edges in $H$ passing through $I$ and $J$ if $I\cap J=\emptyset$, and 0 if $I\cap J\not=\emptyset$. The $s$-th Laplacian $Ł^{(s)}$ of $H$ is defined to be the normalized Laplacian of $G^{(s)}$. The eigenvalues of $\mathcal L^{(s)}$ are listed as $λ^{(s)}_0, λ^{(s)}_1,..., λ^{(s)}_{{n\choose s}-1}$ in non-decreasing order. Let $\barλ^{(s)}(H)=\max_{i\not=0}\{|1-λ^{(s)}_i|\}$. The parameters $\barλ^{(s)}(H)$ and $λ^{(s)}_1(H)$, which were introduced in our previous paper, have a number of connections to the mixing rate of high-ordered random walks, the generalized distances/diameters, and the edge expansions. For $0< p<1$, let $H^r(n,p)$ be a random $r$-uniform hypergraph over $[n]:={1,2,..., n}$, where each $r$-set of $[n]$ has probability $p$ to be an edge independently. For $1 \leq s \leq r/2$, $p(1-p)\gg \frac{\log^4 n}{n^{r-s}}$, and $1-p\gg \frac{\log n}{n^2}$, we prove that almost surely $$\barλ^{(s)}(H^r(n,p))\leq \frac{s}{n-s}+ (3+o(1))\sqrt{\frac{1-p}{{n-s\choose r-s}p}}.$$ We also prove that the empirical distribution of the eigenvalues of $Ł^{(s)}$ for $H^r(n,p)$ follows the Semicircle Law if $p(1-p)\gg \frac{\log^{1/3} n}{n^{r-s}}$ and $1-p\gg \frac{\log n}{n^{2+2r-2s}}$.

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arXiv2011

A Fractional Analogue of Brooks' Theorem

Let $Δ(G)$ be the maximum degree of a graph $G$. Brooks' theorem states that the only connected graphs with chromatic number $χ(G)=Δ(G)+1$ are complete graphs and odd cycles. We prove a fractional analogue of Brooks' theorem in this paper. Namely, we classify all connected graphs $G$ such that the fractional chromatic number $χ_f(G)$ is at least $Δ(G)$. These graphs are complete graphs, odd cycles, $C^2_8$, $C_5\boxtimes K_2$, and graphs whose clique number $ω(G)$ equals the maximum degree $Δ(G)$. Among the two sporadic graphs, the graph $C^2_8$ is the square graph of cycle $C_8$ while the other graph $C_5\boxtimes K_2$ is the strong product of $C_5$ and $K_2$. In fact, we prove a stronger result; if a connected graph $G$ with $Δ(G)\geq 4$ is not one of the graphs listed above, then we have $χ_f(G)\leq Δ(G)- 2/67$.

math.CO↗

arXiv2011

Monochromatic 4-term arithmetic progressions in 2-colorings of $\mathbb Z_n$

This paper is motivated by a recent result of Wolf \cite{wolf} on the minimum number of monochromatic 4-term arithmetic progressions(4-APs, for short) in $\Z_p$, where $p$ is a prime number. Wolf proved that there is a 2-coloring of $\Z_p$ with 0.000386% fewer monochromatic 4-APs than random 2-colorings; the proof is probabilistic and non-constructive. In this paper, we present an explicit and simple construction of a 2-coloring with 9.3% fewer monochromatic 4-APs than random 2-colorings. This problem leads us to consider the minimum number of monochromatic 4-APs in $\Z_n$ for general $n$. We obtain both lower bound and upper bound on the minimum number of monochromatic 4-APs in all 2-colorings of $\Z_n$. Wolf proved that any 2-coloring of $\Z_p$ has at least $(1/16+o(1))p^2$ monochromatic 4-APs. We improve this lower bound into $(7/96+o(1))p^2$. Our results on $\Z_n$ naturally apply to the similar problem on $[n]$ (i.e., $\{1,2,..., n\}$). In 2008, Parillo, Robertson, and Saracino \cite{prs} constructed a 2-coloring of $[n]$ with 14.6% fewer monochromatic 3-APs than random 2-colorings. In 2010, Butler, Costello, and Graham \cite{BCG} extended their methods and used an extensive computer search to construct a 2-coloring of $[n]$ with 17.35% fewer monochromatic 4-APs (and 26.8% fewer monochromatic 5-APs) than random 2-colorings. Our construction gives a 2-coloring of $[n]$ with 33.33% fewer monochromatic 4-APs (and 57.89% fewer monochromatic 5-APs) than random 2-colorings.

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arXiv2011

High-ordered Random Walks and Generalized Laplacians on Hypergraphs

Despite of the extreme success of the spectral graph theory, there are relatively few papers applying spectral analysis to hypergraphs. Chung first introduced Laplacians for regular hypergraphs and showed some useful applications. Other researchers treated hypergraphs as weighted graphs and then studied the Laplacians of the corresponding weighted graphs. In this paper, we aim to unify these very different versions of Laplacians for hypergraphs. We introduce a set of Laplacians for hypergraphs through studying high-ordered random walks on hypergraphs. We prove the eigenvalues of these Laplacians can effectively control the mixing rate of high-ordered random walks, the generalized distances/diameters, and the edge expansions.

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