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Yingjie Qian

Publications and source records attributed to Yingjie Qian.

4 recordsLinked to original sources
arXiv2021

Lower bounds for graph bootstrap percolation via properties of polynomials

We introduce a simple method for proving lower bounds for the size of the smallest percolating set in a certain graph bootstrap process. We apply this method to determine the sizes of the smallest percolating sets in multidimensional tori and multidimensional grids (in particular hypercubes). The former answers a question of Morrison and Noel, and the latter provides an alternative and simpler proof for one of their main results.

math.CO↗
arXiv2019

Asymptotic Density of Graphs Excluding Disconnected Minors

For a graph $H$, let $$c_{\infty}(H)= \lim_{n \to \infty}\max\frac{|E(G)|}{n},$$ where the maximum is taken over all graphs $G$ on $n$ vertices not containing $H$ as a minor. Thus $c_{\infty}(H)$ is the asymptotic maximum density of graphs not containing $H$ as a minor. Employing a structural lemma due to Eppstein, we prove new upper bounds on $c_{\infty}(H)$ for disconnected graphs $H$. In particular, we determine $c_{\infty}(H)$ whenever $H$ is union of cycles. Finally, we investigate the behaviour of $c_\infty(sK_r)$ for fixed $r$, where $sK_r$ denotes the union of $s$ disjoint copies of the complete graph on $r$ vertices. Improving on a result of Thomason, we show that $$c_\infty(sK_r)=s(r-1)-1 \mathrm{\; for \;} s =Ω\left(\frac{\log{r}}{\log\log{r}}\right),$$ and $$c_\infty(sK_r)>s(r-1)-1 \mathrm{\; for \;} s ={o}\left(\frac{\log{r}}{\log\log{r}}\right).$$

math.CO↗
arXiv2016

Teaching dimension, VC dimension, and critical sets in Latin squares

A critical set in an $n \times n$ Latin square is a minimal set of entries that uniquely identifies it among all Latin squares of the same size. It is conjectured by Nelder in 1979, and later independently by Mahmoodian, and Bate and van Rees that the size of the smallest critical set is $\lfloor n^2/4\rfloor$. We prove a lower-bound of $n^2/10^4$ for sufficiently large $n$, and thus confirm the quadratic order predicted by the conjecture. We prove a lower-bound of $n^2/10^4$ for sufficiently large $n$, and thus confirm the quadratic order predicted by the conjecture. This improves a recent lower-bound of $Ω(n^{3/2})$ due to Cavenagh and Ramadurai. From the point of view of computational learning theory, the size of the smallest critical set corresponds to the minimum teaching dimension of the set of Latin squares. We study two related notions of dimension from learning theory. We prove a lower-bound of $n^2-(e+o(1))n^{5/3}$ for both of the VC-dimension and the recursive teaching dimension.

math.CO↗