arXiv subjects

Qinghai Liu

Publications and source records attributed to Qinghai Liu.

6 recordsLinked to original sources

arXiv2026

Rooted Spider Embeddings and the Erd\H os-Sós Conjecture

Under a local density condition, we prove that every $k$-edge spider embeds at any prescribed center of degree at least $k$, unless all legs are even and the host graph has one of two specified structures. These structures contain complete bipartite subgraphs with prescribed neighborhoods. The proof uses path rerouting and three exchange lemmas that describe equality in neighborhood estimates. As a consequence, we recover the Erd\H os-Sós bound for all spiders.

math.CO↗

arXiv2026

Connectivity keeping pendant extensions of paths in $k$-connected graphs and triangle-free graphs

Motivated by Mader's conjecture on connectivity keeping trees, we study trees obtained from paths by adding one pendant vertex, as well as related problems in triangle-free graphs. For an integer $m$ and $1\leq i\leq m-1$, let $P_m^+(i)$ denote the tree obtained from a path of order $m-1$ by adding one pendant vertex adjacent to its $i$th vertex. We prove that, for positive integers $k,m,1\leq i\leq m-1$, every $k$-connected graph $G$ with $δ(G)\geq \lfloor \frac{3k}{2}\rfloor+m-1$ contains a subgraph $T\cong P_m^+(i)$ such that $κ(G-V(T))\geq k$. This confirms Mader's conjecture for all pendant extensions of paths. For highly connected triangle-free graphs, a connectivity keeping result for paths was obtained in [J. Combin. Theory Ser. B, 174 (2025), 190-206]. Let $(X,Y)$ be the bipartition of $P_m^+(i)$. We further prove that every $k$-connected triangle-free graph $G$ with $δ(G)\geq k+\max\{|X|,|Y|\}+[P_m^+(i)\text{ is bad}]$ contains a subgraph $T\cong P_m^+(i)$ such that $κ(G-V(T))\geq k$, where we use Iverson's convention for $[P_m^+(i)\text{ is bad}]$. This extends the corresponding result for paths to pendant extensions of paths.

math.CO↗

arXiv2026

The Bounded-VC chromatic thresholds of graphs

For a graph $H$, the chromatic threshold $δ_χ(H)$ is the infimum of $c>0$ such that the chromatic number of every $n$-vertex $H$-free graph with minimum degree at least $cn$ is bounded by a constant depending only on $H$ and $c$. Allen, Böttcher, Griffiths, Kohayakawa, and Morris proved that if $χ(H)=r\geq 3$, then $δ_χ(H)\in\{\frac{r-3}{r-2}, \frac{2r-5}{2r-3}, \frac{r-2}{r-1}\}$. Liu, Shangguan, Skokan, and Xu introduced the bounded-VC chromatic threshold $\text{VC}(H)$ by restricting the host graphs to have bounded VC-dimension. We determine this parameter for graph $H$ with $χ(H)\ge 3$. More precisely, let $\mathcal{M}(H)$ be the decomposition family of an $r$-chromatic graph $H$, then \[ \text{VC}(H)= \begin{cases} \dfrac{r-3}{r-2},&\text{if $\mathcal{M}(H)$ contains a forest},\\[4pt] \dfrac{r-2}{r-1},&\text{otherwise}. \end{cases} \]

math.CO↗

arXiv2023

On the edge reconstruction of the characteristic and permanental polynomials of a simple graph

As a variant of the Ulam's vertex reconstruction conjecture and the Harary's edge reconstruction conjecture, Cvetković and Schwenk posed independently the following problem: Can the characteristic polynomial of a simple graph $G$ with vertex set $V$ be reconstructed from the characteristic polynomials of all subgraphs in $\{G-v|v\in V\}$ for $|V|\geq 3$? This problem is still open. A natural problem is: Can the characteristic polynomial of a simple graph $G$ with edge set $E$ be reconstructed from the characteristic polynomials of all subgraphs in $\{G-e|e\in E\}$? In this paper, we prove that if $|V|\neq |E|$, then the characteristic polynomial of $G$ can be reconstructed from the characteristic polynomials of all subgraphs in $\{G-uv, G-u-v|uv\in E\}$, and the similar result holds for the permanental polynomial of $G$. We also prove that the Laplacian (resp. signless Laplacian) characteristic polynomial of $G$ can be reconstructed from the Laplacian (resp. signless Laplacian) characteristic polynomials of all subgraphs in $\{G-e|e\in E\}$ (resp. if $|V|\neq |E|$).

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arXiv2021

Mader's conjecture for graphs with small connectivity

Mader conjectured that for any tree $T$ of order $m$, every $k$-connected graph $G$ with minimum degree at least $\lfloor\frac{3k}{2}\rfloor +m-1$ contains a subtree $T'\cong T$ such that $G-V(T')$ is $k$-connected. In this paper, we give a characterization for a subgraph to contain an embedding of a specified tree avoiding some vertex. As a corollary, we confirm Mader's conjecture for $k\leq3$.

math.CO↗

arXiv2017

Circumference of 3-connected cubic graphs

The circumference of a graph is the length of its longest cycles. Jackson established a conjecture of Bondy by showing that the circumference of a 3-connected cubic graph of order $n$ is $Ω(n^{0.694})$. Bilinski {\it et al.} improved this lower bound to $Ω(n^{0.753})$ by studying large Eulerian subgraphs in 3-edge-connected graphs. In this paper, we further improve this lower bound to $Ω(n^{0.8})$. This is done by considering certain 2-connected cubic graphs, finding cycles through two given edges, and distinguishing the cases whether or not these edges are adjacent.

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