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Xiangxiang Nie

Publications and source records attributed to Xiangxiang Nie.

5 recordsLinked to original sources
arXiv2026

Covering graphs by isometric trees

A connected subgraph of a graph is isometric if it preserves distances. Recently, graphs admitting a vertex or edge covering by a small number of isometric paths have been studied. In this paper, we consider the analogous problem for isometric trees, focusing on the treewidth of graphs admitting a vertex or edge covering by a small number of such trees. Baste, De Meyer, Giocanti, Objois, and Picavet showed that for coverings by two isometric trees, the treewidth is bounded. We show that already for three isometric trees, the treewidth can be linear in the number of vertices. On the positive side, we show that for graphs of bounded degree coverable by a small number of isometric trees, the treewidth is sublinear in the number of vertices.

math.CO↗
arXiv2026

Upper bounds on the running time of bootstrap percolation

For $k$-graphs $F$ and $H_0$ the $F$-bootstrap percolation process (or $F$-process) starting with $H_0$ is a sequence $(H_i)_{i\geq0}$ of $k$-graphs such that $H_{i+1}$ is obtained from $H_i$ by adding all those $e\in V(H_0)^{(k)}\setminus E(H_i)$ as edges that complete a new copy of $F$. The running time of this $F$-process, denoted by $M_F(H_0)$, is the smallest $i$ with $H_i=H_{i+1}$. Bollobás proposed the problem of determining the maximum running time for $n\in\mathbb{N}$, i.e., $M_F(n)=\max_{\vert V(H_0)\vert=n}M_F(H_0)$. Although this problem has received a lot of attention recently, until now the best known upper bound for $M_{K_t}(n)$, with $t\geq5$, was the trivial bound $\binom{n}{2}$. Here we provide the first non-trivial upper bound for this problem by showing that $$M_{K_t}(n)\leq\Big(\frac{t-3}{t-2}+o(1)\Big)\binom{n}{2}$$ holds for every integer $t\geq 3$. In fact, we prove the following more general result. For every $k\geq2$, every $k$-graph $F$, and every $e\in E(F)$ we have $M_F(n)\leq\big(π(F-e)+o(1)\big)\binom{n}{k}$, where $π$ is the Turán density.

math.CO↗
arXiv2026

Transversal tilings in k-partite graphs without large holes

We show that for any constant $μ>0$ and $k\ge 3$, there exists $α>0$ such that the following holds for sufficiently large $n \in \mathbb{N}$. If $G=(V_{1},\ldots,V_{k},E)$ is a spanning subgraph of the $n$-blow-up of $K_{k}$ with ${δ^*}(G)\geq (\frac{1}{2}+μ) n$ and $α^*_{k-1}(G)<αn$, then $G$ has a transversal $K_{k}$-factor. Moreover, the bound $\frac{1}{2}$ is asymptotically tight for the case \(k=3\). In addition, we show that if $k\ge 4$, $G=(V_{1},\ldots,V_{k},E)$ is a spanning subgraph of the $n$-blow-up of $C_{k}$ with ${δ^*}(G)\ge (\frac{2}{k}+μ) n$, and $α^*_{2}(G)<αn$, then $G$ has a transversal $C_{k}$-factor. This extends a recent result of Han, Hu, Ping, Wang, Wang and Yang.

math.CO↗
arXiv2025

Perfect tilings with the generalised triangle in $k$-graphs

Denote by $T_k$ the generalised triangle, a $k$-uniform hypergraph on vertex set $\{1,2,\dots,2k-1\}$ with three edges $\{1,\dots,k-1,k\}$,$\{1,\dots,k-1,k+1\}$ and $\{k,k+1,\dots,2k-1\}$. Recently, Bowtell, Kathapurkar, Morrison and Mycroft [arXiv: 2505.05606] established the exact minimum codegree threshold for perfect $T_3$-tilings in $3$-graphs. In this paper, we extend their result to all $k \geq 3$, determining the optimal minimum codegree threshold for perfect $T_k$-tilings in $k$-graphs. Our proof uses the lattice-based absorption method, as is usual, but develops a unified and effective approach to build transferrals for all uniformities, which is of independent interest. Additionally, we establish an asymptotically tight minimum codegree threshold for a rainbow variant of the problem.

math.CO↗