arXiv subjects

Zixiang Xu

Publications and source records attributed to Zixiang Xu.

78 records · Page 5Linked to original sources

arXiv2020

Color isomorphic even cycles and a related Ramsey problem

In this paper, we first study a new extremal problem recently posed by Conlon and Tyomkyn~(arXiv: 2002.00921). Given a graph $H$ and an integer $k\geqslant 2$, let $f_{k}(n,H)$ be the smallest number of colors $c$ such that there exists a proper edge-coloring of the complete graph $K_{n}$ with $c$ colors containing no $k$ vertex-disjoint color-isomorphic copies of $H$. Using algebraic properties of polynomials over finite fields, we give an explicit proper edge-coloring of $K_{n}$ and show that $f_{k}(n, C_{4})=Θ(n)$ when $k\geqslant 3$ and $n\rightarrow\infty$. The methods we used in the edge-coloring may be of some independent interest. We also consider a related generalized Ramsey problem. For given graphs $G$ and $H,$ let $r(G,H,q)$ be the minimum number of edge-colors (not necessarily proper) of $G$, such that the edges of every copy of $H\subseteq G$ together receive at least $q$ distinct colors. Establishing the relation to the Turán number of specified bipartite graphs, we obtain some general lower bounds for $r(K_{n,n},K_{s,t},q)$ with a broad range of $q$.

math.CO↗

arXiv2020

Some tight lower bounds for Turán problems via constructions of multi-hypergraphs

Recently, several hypergraph Turán problems were solved by the powerful random algebraic method. However, the random algebraic method usually requires some parameters to be very large, hence we are concerned about how these Turán numbers depend on such large parameters of the forbidden hypergraphs. In this paper, we determine the dependence on such specified large constant for several hypergraph Turán problems. More specifically, for complete $r$-partite $r$-uniform hypergraphs, we show that if $s_{r}$ is sufficiently larger than $s_{1},s_{2},\ldots,s_{r-1},$ then $$\textup{ex}_{r}(n,K_{s_{1},s_{2},\ldots,s_{r}}^{(r)})=Θ(s_{r}^{\frac{1}{s_{1}s_{2}\cdots s_{r-1}}}n^{r-\frac{1}{s_{1}s_{2}\cdots s_{r-1}}}).$$ For complete bipartite $r$-uniform hypergraphs, we prove that if $s$ is sufficiently larger than $t,$ we have $$\textup{ex}_{r}(n,K_{s,t}^{(r)})=Θ(s^{\frac{1}{t}}n^{r-\frac{1}{t}}).$$ In particular, our results imply that the famous Kővári--Sós--Turán's upper bound $\textup{ex}(n,K_{s,t})=O(t^{\frac{1}{s}}n^{2-\frac{1}{s}})$ has the correct dependence on large $t$. The main approach is to construct random multi-hypergraph via a variant of random algebraic method.

math.CO↗

arXiv2020

On the Turán number of 1-subdivision of $K_{3,t}$

For a graph $H$, the 1-subdivision of $H$, denoted by $H'$, is the graph obtained by replacing the edges of $H$ by internally disjoint paths of length 2. Recently, Conlon, Janzer and Lee (arXiv: 1903.10631) asked the following question: For any integer $s\ge2$, estimate the smallest $t$ such that $\textup{ex}(n,K_{s,t}')=Ω(n^{\frac{3}{2}-\frac{1}{2s}})$. In this paper, we consider the case $s=3$. More precisely, we provide an explicit construction giving \begin{align*} \text{ex}(n,K_{3,30}')=Ω(n^{\frac{4}{3}}), \end{align*} which reduces the estimation for the smallest value of $t$ from a magnitude of $10^{56}$ to the number $30$. The construction is algebraic, which is based on some equations over finite fields.

math.CO↗

arXiv2019

3-uniform hypergraphs with few Berge paths of length three between any two vertices

Recently, Berge theta hypergraphs have received special attention due to the similarity with Berge even cycles. Let $r$-uniform Berge theta hypergraph $Θ_{\ell,t}^{B}$ be the $r$-uniform hypergraph consisting of $t$ internally disjoint Berge paths of length $\ell$ with the same pair of endpoints. In this work, we determine the Turán number of $3$-uniform Berge theta hypergraph when $\ell=3$ and $t$ is relatively small. More precisely, we provide an explicit construction giving \begin{align*} \textup{ex}_{3}(n,Θ_{3,217}^{B})=Ω(n^{\frac{4}{3}}). \end{align*} This matches an earlier upper bound by He and Tait up to an absolute constant factor. The construction is algebraic, which is based on some equations over finite fields, and the parameter $t$ in our construction is much smaller than that in random algebraic construction. Our main technique is using the resultant of polynomials, which appears to be a powerful technique to eliminate variables.

math.CO↗

arXiv2019

On vertex-induced weighted Turán problems

Recently, Bennett et al. introduced the vertex-induced weighted Turán problem. In this paper, we consider their open Turán problem under sum-edge-weight function and characterize the extremal structure of $K_{l}$-free graphs. Based on these results, we propose a generalized version of the Erdős-Stone theorem for weighted graphs under two types of vertex-induced weight functions.

math.CO↗

arXiv2018

New Theoretical Bounds and Constructions of Permutation Codes under Block Permutation Metric

Permutation codes under different metrics have been extensively studied due to their potentials in various applications. Generalized Cayley metric is introduced to correct generalized transposition errors, including previously studied metrics such as Kendall's $τ$-metric, Ulam metric and Cayley metric as special cases. Since the generalized Cayley distance between two permutations is not easily computable, Yang et al. introduced a related metric of the same order, named the block permutation metric. Given positive integers $n$ and $d$, let $\mathcal{C}_{B}(n,d)$ denote the maximum size of a permutation code in $S_n$ with minimum block permutation distance $d$. In this paper, we focus on the theoretical bounds of $\mathcal{C}_{B}(n,d)$ and the constructions of permutation codes under block permutation metric. Using a graph theoretic approach, we improve the Gilbert-Varshamov type bound by a factor of $Ω(\log{n})$, when $d$ is fixed and $n$ goes into infinity. We also propose a new encoding scheme based on binary constant weight codes. Moreover, an upper bound beating the sphere-packing type bound is given when $d$ is relatively close to $n$.

cs.IT↗