arXiv subjects

Huiqing Liu

Publications and source records attributed to Huiqing Liu.

14 recordsLinked to original sources

arXiv2026

Extremal number of edges in graphs without homeomorphically irreducible spanning trees

For integers $k\ge 1$ and $n\ge k+1$, let $\operatorname{ex}^{\mathrm{HIST}}_k(n)$ denote the maximum number of edges in a $k$-connected graph of order $n$ which contains no homeomorphically irreducible spanning tree (or briefly HIST). We determine these extremal numbers for $k=1$ and $k=2$. More precisely, we prove that $\operatorname{ex}^{\mathrm{HIST}}_1(n)=\binom{n-2}{2}+2$ for $n\ge 9$, with $L_n$ as the unique extremal graph, and that $\operatorname{ex}^{\mathrm{HIST}}_2(n)=\binom{n-3}{2}+4$ for $n\ge 13$, with $B_n$ as the unique extremal graph. This provides a Tur\'an-type extremal result for spanning trees with no vertices of degree two.

math.CO↗

arXiv2025

A neighborhood union condition for the existence of a spanning tree without samll degree vertices

For an integer k\ge2, a [2,k]-ST of a connected graph G is a spanning tree of G in which there are no vertices of degree between 2 and k. A [2,k]-ST is a natural extension of a homeomorphically irreducible spanning tree (HIST), which is a spanning tree without vertices of degree 2. In this paper, we give a neighborhood union condition for the existence of a [2,k]-ST in G. We generalize a known degree sum condition that guarantees the existence of a [2,k]-ST in G.

math.CO↗

arXiv2025

Spectral radius and homeomorphically irreducible spanning trees of graphs

For a connected graph $G$, a spanning tree $T$ of $G$ is called a homeomorphically irreducible spanning tree (HIST) if $T$ has no vertices of degree 2. Albertson {\em et al.} proved that it is $NP$-complete to decide whether a graph contains a HIST. In this paper, we provide some spectral conditions that guarantee the existence of a HIST in a connected graph. Furthermore, we also present some sufficient conditions in terms of the order of a graph $G$ to ensure the existence of a HIST in $G$.

math.CO↗

arXiv2025

Planar Tur\'an number of quasi-double stars

Given a graph H, we call a graph $\textit{H-free}$ if it does not contain H as a subgraph. The planar Tur\'an number of a graph H, denoted by $ex_{\mathcal{P}}(n, H)$, is the maximum number of edges in a planar H-free graph on n vertices. A (h,k)-quasi-double star $W_{h,k}$, obtained from a path $P_3=v_1v_2v_3$ by adding h leaves and k leaves to the vertices $v_1$ and $v_3$, respectively, is a subclass of caterpillars. In this paper, we study $ex_{\mathcal{P}}(n,W_{h,k})$ for all $1\le h\le 2\le k\le 5$, and obtain some tight bounds $ex_{\mathcal{P}}(n,W_{h,k})\leq\frac{3(h+k)}{h+k+2}n$ for $3\le h+k\le 5$ with equality holds if $(h+k+2)\mid n$, and $ex_{\mathcal{P}}(n,W_{1,5})\le \frac{5}{2}n$ with equality holds if $12\mid n$. Also we show that $\frac{9}{4}n\le ex_{\mathcal{P}}(n,W_{2,4})\le \frac{5}{2}n$ and $\frac{5}{2}n\le ex_{\mathcal{P}}(n,W_{2,5})\le \frac{17}{6}n$, respectively.

math.CO↗

arXiv2024

A neighborhood union condition for the existence of a spanning tree without degree $2$ vertices

For a connected graph $G$, a spanning tree $T$ of $G$ is called a homeomorphically irreducible spanning tree (HIST) if $T$ has no vertices of degree $2$. In this paper, we show that if $G$ is a graph of order $n\ge 270$ and $|N(u)\cup N(v)|\geq\frac{n-1}{2}$ holds for every pair of nonadjacent vertices $u$ and $v$ in $G$, then $G$ has a HIST, unless $G$ belongs to three exceptional families of graphs or $G$ has a cut-vertex of degree $2$. This result improves the latest conclusion, due to Ito and Tsuchiya, that a HIST in $G$ can be guaranteed if $d(u)+d(v)\geq n-1$ holds for every pair of nonadjacent vertices $u$ and $v$ in $G$.

math.CO↗

arXiv2023

On the maximal Sombor index of quasi-tree graphs

The Sombor index $SO(G)$ of a graph $G$ is the sum of the edge weights $\sqrt{d^2_G(u)+d^2_G(v)}$ of all edges $uv$ of $G$, where $d_G(u)$ denotes the degree of the vertex $u$ in $G$. A connected graph $G = (V ,E)$ is called a quasi-tree, if there exists $u\in V (G)$ such that $G-u$ is a tree. Denote $\mathscr{Q}(n,k)$=\{$G$: $G$ is a quasi-tree graph of order $n$ with $G-u$ being a tree and $d_G(u)=k$\}. In this paper, we determined the maximum, the second maximum and the third maximum Sombor indices of all quasi-tree graphs in $\mathscr{Q}(n,k)$, respectively. Moreover, we characterized their corresponding extremal graphs, respectively.

math.CO↗

arXiv2021

Anti-Ramsey problems in the generalized Petersen graphs for cycles

The anti-Ramsey number $Ar(G,H)$ is the maximum number of colors in an edge-coloring of $G$ with no rainbow copy of $H$. In this paper, we determine the exact anti-Ramsey number in the generalized Petersen graph $P_{n,k}$ for cycles $C_d$, where $1\leq k\leq \lfloor \frac{n-1}{2} \rfloor$ and $5\le d \le 6$. We also give an algorithm to obtain the upper bound or lower bound of anti-Ramsey number.

math.CO↗

arXiv2021

Burning numbers of t-unicyclic graphs

Given a graph $G$, the burning number of $G$ is the smallest integer $k$ for which there are vertices $x_1, x_2,\ldots,x_k$ such that $(x_1,x_2,\ldots,x_k)$ is a burning sequence of $G$. It has been shown that the graph burning problem is NP-complete, even for trees with maximum degree three, or linear forests. A $t$-unicyclic graph is a unicycle graph with exactly one vertex of degree greater than $2$. In this paper, we first present the bounds for the burning number of $t$-unicyclic graphs, and then use the burning numbers of linear forests with at most three components to determine the burning number of all $t$-unicyclic graphs for $t\le 2$.

math.CO↗

arXiv2020

Fractional matching preclusion of fault Hamiltonian graphs

Matching preclusion is a measure of robustness in the event of edge failure in interconnection networks. As a generalization of matching preclusion, the fractional matching preclusion number (FMP number for short) of a graph is the minimum number of edges whose deletion results in a graph that has no fractional perfect matchings, and the fractional strong matching preclusion number (FSMP number for short) of a graph is the minimum number of edges and/or vertices whose deletion leaves a resulting graph with no fractional perfect matchings. A graph $G$ is said to be $f$-fault Hamiltonian if there exists a Hamiltonian cycle in $G-F$ for any set $F$ of vertices and/or edges with $|F|\leq f$. In this paper, we establish the FMP number and FSMP number of $(δ-2)$-fault Hamiltonian graphs with minimum degree $δ\geq 3$. As applications, the FMP number and FSMP number of some well-known networks are determined.

math.CO↗

arXiv2018

The $g$-good neighbor conditional diagnosability of locally exchanged twisted cubes

Connectivity and diagnosability are important parameters in measuring the fault tolerance and reliability of interconnection networks. The $R^g$-vertex-connectivity of a connected graph $G$ is the minimum cardinality of a faulty set $X\subseteq V(G)$ such that $G-X$ is disconnected and every fault-free vertex has at least $g$ fault-free neighbors. The $g$-good-neighbor conditional diagnosability is defined as the maximum cardinality of a $g$-good-neighbor conditional faulty set that the system can guarantee to identify. The interconnection network considered here is the locally exchanged twisted cube $LeTQ(s,t)$. For $1\leq s\leq t$ and $0\leq g\leq s$, we first determine the $R^g$-vertex-connectivity of $LeTQ(s,t)$, then establish the $g$-good neighbor conditional diagnosability of $LeTQ(s,t)$ under the PMC model and MM$^*$ model, respectively.

math.CO↗

arXiv2018

Structure connectivity and substructure connectivity of twisted hypercubes

Let $G$ be a graph and $T$ a certain connected subgraph of $G$. The $T$-structure connectivity $κ(G; T)$ (or resp., $T$-substructure connectivity $κ^{s}(G; T)$) of $G$ is the minimum number of a set of subgraphs $\mathcal{F}=\{T_{1}, T_{2}, \ldots, T_{m}\}$ (or resp., $\mathcal{F}=\{T^{'}_{1}, T^{'}_{2}, \ldots, T^{'}_{m}\}$) such that $T_{i}$ is isomorphic to $T$ (or resp., $T^{'}_{i}$ is a connected subgraph of $T$) for every $1\leq i \leq m$, and $\mathcal{F}$'s removal will disconnect $G$. The twisted hypercube $H_{n}$ is a new variant of hypercubes with asymptotically optimal diameter introduced by X.D. Zhu. In this paper, we will determine both $κ(H_{n}; T)$ and $κ^{s}(H_{n}; T)$ for $T\in\{K_{1,r}, P_{k}\}$, respectively, where $3\leq r\leq 4$ and $1 \leq k \leq n$.

math.CO↗

arXiv2017

Hamiltonicity of edge-chromatic critical graphs

Given a graph $G$, denote by $Δ$ and $χ^\prime$ the maximum degree and the chromatic index of $G$, respectively. A simple graph $G$ is called {\it edge-$Δ$-critical} if $χ^\prime(G)=Δ+1$ and $χ^\prime(H)\leΔ$ for every proper subgraph $H$ of $G$. We proved that every edge chromatic critical graph of order $n$ with maximum degree at least $\frac{2n}{3}+12$ is Hamiltonian.

math.CO↗

arXiv2017

Average degrees of edge-chromatic critical graphs

Given a graph $G$, denote by $Δ$, $\bar{d}$ and $χ^\prime$ the maximum degree, the average degree and the chromatic index of $G$, respectively. A simple graph $G$ is called {\it edge-$Δ$-critical} if $χ^\prime(G)=Δ+1$ and $χ^\prime(H)\leΔ$ for every proper subgraph $H$ of $G$. Vizing in 1968 conjectured that if $G$ is edge-$Δ$-critical, then $\bar{d}\geq Δ-1+ \frac{3}{n}$. We show that $$ \begin{displaystyle} \avd \ge \begin{cases} 0.69241\D-0.15658 \quad\,\: \mbox{ if } Δ\geq 66, 0.69392\D-0.20642\quad\;\,\mbox{ if } Δ=65, \mbox{ and } 0.68706\D+0.19815\quad\! \quad\mbox{if } 56\leq Δ\leq64. \end{cases} \end{displaystyle} $$ This result improves the best known bound $\frac{2}{3}(Δ+2)$ obtained by Woodall in 2007 for $Δ\geq 56$. Additionally, Woodall constructed an infinite family of graphs showing his result cannot be improved by well-known Vizing's Adjacency Lemma and other known edge-coloring techniques. To over come the barrier, we follow the recently developed recoloring technique of Tashkinov trees to expand Vizing fans technique to a larger class of trees.

math.CO↗

arXiv2016

A Spatial and Temporal Non-Local Filter Based Data Fusion

The trade-off in remote sensing instruments that balances the spatial resolution and temporal frequency limits our capacity to monitor spatial and temporal dynamics effectively. The spatiotemporal data fusion technique is considered as a cost-effective way to obtain remote sensing data with both high spatial resolution and high temporal frequency, by blending observations from multiple sensors with different advantages or characteristics. In this paper, we develop the spatial and temporal non-local filter based fusion model (STNLFFM) to enhance the prediction capacity and accuracy, especially for complex changed landscapes. The STNLFFM method provides a new transformation relationship between the fine-resolution reflectance images acquired from the same sensor at different dates with the help of coarse-resolution reflectance data, and makes full use of the high degree of spatiotemporal redundancy in the remote sensing image sequence to produce the final prediction. The proposed method was tested over both the Coleambally Irrigation Area study site and the Lower Gwydir Catchment study site. The results show that the proposed method can provide a more accurate and robust prediction, especially for heterogeneous landscapes and temporally dynamic areas.

cs.CV↗