arXiv subjects

Yulin Chang

Publications and source records attributed to Yulin Chang.

6 recordsLinked to original sources

arXiv2025

An exact Ore-degree condition for Hamilton cycles in oriented graphs

An oriented graph is a digraph that contains no 2-cycles, i.e., there is at most one arc between any two vertices. We show that every oriented graph $G$ of sufficiently large order $n$ with $\mathrm{deg}^+(x) +\mathrm{deg}^{-}(y)\geq (3n-3)/4$ whenever $G$ does not have an edge from $x$ to $y$ contains a Hamilton cycle. This is best possible and solves a problem of K\"uhn and Osthus from 2012. Our result generalizes the result of Keevash, K\"uhn, and Osthus and improves the asymptotic bound obtained by Kelly, K\"uhn, and Osthus.

math.CO↗

arXiv2023

The threshold for powers of tight Hamilton cycles in random hypergraphs

We investigate the occurrence of powers of tight Hamilton cycles in random hypergraphs. For every $r\ge 3$ and $k\ge 1$, we show that there exists a constant $C > 0$ such that if $p=p(n) \ge Cn^{-1/\binom{k+r-2}{r-1}}$ then asymptotically almost surely the random hypergraph $H^{(r)}(n,p)$ contains the $k$th power of a tight Hamilton cycle. This improves on a result of Parczyk and Person, who proved the same result under the assumption $p=\omega\left(n^{-1/\binom{k+r-2}{r-1}}\right)$ using a second moment argument.

math.CO↗

arXiv2022

On powers of tight Hamilton cycles in randomly perturbed hypergraphs

For integers $k \geq 3$ and $r\geq 2$, we show that for every $α> 0$, there exists $\varepsilon > 0$ such that the union of $k$-uniform hypergraph on $n$ vertices with minimum codegree at least $αn$ and a binomial random $k$-uniform hypergraph $G^{(k)}(n,p)$ with $p\geq n^{-{\binom{k+r-2}{k-1}}^{-1}-\varepsilon}$ on the same vertex set contains the $r^{th}$ power of a tight Hamilton cycle with high probability. Moreover, a construction shows that one cannot take $\varepsilon > Cα$, where $C=C(k,r)$ is a constant. Thus the bound on $p$ is optimal up to the value of $\varepsilon$ and this answers a question of Bedenknecht, Han, Kohayakawa, and Mota.

math.CO↗

arXiv2022

Matching of given sizes in hypergraphs

For all integers $k,d$ such that $k \geq 3$ and $k/2\leq d \leq k-1$, let $n$ be a sufficiently large integer {\rm(}which may not be divisible by $k${\rm)} and let $s\le \lfloor n/k\rfloor-1$. We show that if $H$ is a $k$-uniform hypergraph on $n$ vertices with $δ_{d}(H)>\binom{n-d}{k-d}-\binom{n-d-s+1}{k-d}$, then $H$ contains a matching of size $s$. This improves a recent result of Lu, Yu, and Yuan and also answers a question of Kühn, Osthus, and Townsend. In many cases, our result can be strengthened to $s\leq \lfloor n/k\rfloor$, which then covers the entire possible range of $s$. On the other hand, there are examples showing that the result does not hold for certain $n, k, d$ and $s= \lfloor n/k\rfloor$.

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arXiv2021

Factors in randomly perturbed hypergraphs

We determine, up to a multiplicative constant, the optimal number of random edges that need to be added to a $k$-graph $H$ with minimum vertex degree $Ω(n^{k-1})$ to ensure an $F$-factor with high probability, for any $F$ that belongs to a certain class $\mathcal{F}$ of $k$-graphs, which includes, e.g., all $k$-partite $k$-graphs, $K_4^{(3)-}$ and the Fano plane. In particular, taking $F$ to be a single edge, this settles a problem of Krivelevich, Kwan and Sudakov [Combin. Probab. Comput. 25 (2016), 909--927]. We also address the case in which the host graph $H$ is not dense, indicating that starting from certain such $H$ is essentially the same as starting from an empty graph (namely, the purely random model).

math.CO↗

arXiv2018

List strong edge-coloring of graphs with maximum degree 4

A strong edge-coloring of a graph $G$ is an edge-coloring such that any two edges on a path of length three receive distinct colors. We denote the strong chromatic index by $χ_{s}'(G)$ which is the minimum number of colors that allow a strong edge-coloring of $G$. Erdős and Nešetřil conjectured in 1985 that the upper bound of $χ_{s}'(G)$ is $\frac{5}{4}Δ^{2}$ when $Δ$ is even and $\frac{1}{4}(5Δ^{2}-2Δ+1)$ when $Δ$ is odd, where $Δ$ is the maximum degree of $G$. The conjecture is proved right when $Δ\leq3$. The best known upper bound for $Δ=4$ is 22 due to Cranston previously. In this paper we extend the result of Cranston to list strong edge-coloring, that is to say, we prove that when $Δ=4$ the upper bound of list strong chromatic index is 22.

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