arXiv subjects

Ruifang Liu

Publications and source records attributed to Ruifang Liu.

At least 19 recordsLinked to original sources

arXiv2026

The exact Tur\'{a}n number of the even wheel $W_{2k+2}$ among non-$3$-partite graphs

Let $\mathrm{ex}(n,H)$ denote the Tur\'{a}n number of $H$. A graph is color-critical if there exists an edge $e\in E(H)$ such that $\chi(H-e)<\chi(H)$. For a color-critical graph $H$ with $\chi(H)=r+1$, Simonovits' chromatic critical edge theorem implies that there exists an $n_0(H)$ such that $\mathrm{ex}(n,H)=e(T_{n,r})$ and the Tur\'{a}n graph $T_{n,r}$ is the only extremal graph provided $n\geq n_0(H).$ Let $W_{2k+2}$ be the even wheel obtained by joining a vertex to a cycle of length $2k+1,$ where $k\geq1$ is an integer. Since $W_{2k+2}$ is color-critical and $\chi(W_{2k+2})=4$, $T_{n,3}$ is the unique extremal graph for $W_{2k+2}$-free graphs of sufficiently large $n.$ Note that the extremal graph $T_{n,3}$ is 3-partite. In this paper, we determine the exact Tur\'{a}n number of $W_{2k+2}$ in non-$3$-partite graphs and characterize all extremal graphs provided $n$ is sufficiently large.

math.CO↗

arXiv2026

Extremal problems on $[a, b]$-covered graphs

A graph $G$ is $[a,b]$-covered if for each edge $e$ of $G$ there is an $[a,b]$-factor containing it. For $a=b=1$, an $[a,b]$-covered graph is a matching covered graph. The structural theory of matching covered graphs constitutes a cornerstone of modern matching theory. Determining whether a given graph is matching covered is a fundamental problem in structural graph theory. Lucchesi et al. [SIAM J. Discrete Math., 2018] showed that a connected graph $G$ is matching covered if and only if every barrier of $G$ is a stable set. In this paper, we completely characterize the extremal graphs that maximize the size or the spectral radius among all non-matching-covered graphs. For $a \leq b$ and $b \geq 2,$ Hao and Li [Electron. J. Combin., 2024] investigated the extremal problems on $[a,b]$-factor graphs: If $G$ contains no $[a,b]$-factors, then $e(G)\leq \binom{n-1}{2}+a-1$ with equality if and only if $G\cong H_{n,a},$ where $H_{n,a} = K_{a-1} \vee (K_{n-a} \cup K_1).$ Moreover, if $G$ contains no $[a,b]$-factors, then $\rho(G)\leq \rho(H_{n,a})$ with equality if and only if $G \cong H_{n,a}.$ Judging from the structral characterization, non-$[a,b]$-covered graphs exhibit highly complex structures, making the associated extremal problems significantly challenging. To overcome this, we develop a novel minimum-degree forcing technique. Combining this technique and spectral-structural analysis, we in this paper provide complete characterizations of the extremal graphs that maximize the size or the spectral radius within the set of non-$[a,b]$-covered graphs. An intriguing phenomenon revealed by our results is that $H_{n,a}$ remains both the size-extremal graph and the spectral extremal graph for this larger set of non-$[a,b]$-covered graphs. Consequently, our results strengthen the results of Hao-Li.

math.CO↗

arXiv2026

Extremal results for graphs with binding number strictly less than $1/r$

The binding number $b(G)$ of a graph, introduced by Woodall [J. Combin. Theory, Ser. B, 1973], is a central topic of both structural and extremal graph theory. It is closely related to fundamental combinatorial and structural properties of graphs. The graphs with $b(G)\geq1$ exhibit strong expansion properties and a highly connected global structure. In contrast, the structure for graphs with $b(G)<1$ remains far less well understood. Kane et al. [J. Graph Theory, 1981] proved that if $b(G)<1$, then every binding set of $G$ is independent. Goddard and Swart [Quaest. Math., 1990] showed that if $b(G)\leq1$, then the toughness $\tau(G)\leq b(G).$ This makes it particularly interesting to investigate extremal problems for graphs with $b(G)<1$. For any integer $r\geq1,$ we completely characterize the unique extremal graph that maximizes the size (spectral radius) among all graphs of order $n$ satisfying $b(G)<\frac{1}{r}.$ For any bipartite graph $G=(X,Y)$ on $n$ vertices, it is readily seen that $b(G)\leq\min\{|X|/|Y|,|Y|/|X|\}\leq1.$ Notably, the complete balanced bipartite graph $K_{\frac{n}{2}, \frac{n}{2}}$ achieves the maximum size (spectral radius) among all bipartite graphs with $b(G)=1$. In this paper, we completely determine the extremal graphs maximizing the size or the spectral radius among all bipartite graphs with $b(G)<\frac{1}{r}$, where $r\geq1$ is an integer.

math.CO↗

arXiv2026

Spectral radius and rainbow $k$-factors in a bipartite graph family

Let $\mathcal{G}=\{G_1, G_2, \ldots , G_{kn}\}$ be a family of balanced bipartite graphs on the same vertex set $[2n]$. A rainbow $k$-factor of $\mathcal{G}$ is defined as a $k$-factor such that any two distinct edges come from different graphs in $\mathcal{G}.$ In this paper, we provide a tight sufficient condition in terms of the spectral radius for a family of balanced bipartite graphs $\mathcal{G}$ to contain a rainbow $k$-factor. Furthermore, we completely characterize the corresponding spectral extremal graph.

math.CO↗

arXiv2026

Spectral radius and rainbow Hamiltonicity in bipartite graphs

Let $\mathcal{G}=\{G_1, G_2, \ldots , G_k\}$ be a family of bipartite graphs on the same vertex set. A rainbow Hamilton path (cycle) in $\mathcal{G}$ is a path (cycle) that visits each vertex precisely once such that any two edges belong to different graphs of $\mathcal{G}.$ In this paper, by adopting the technique of bi-shifting, we present tight sufficient conditions in terms of the spectral radius for a family $\mathcal{G}$ to admit a rainbow Hamilton path and cycle, respectively. Meanwhile, we completely characterize the corresponding spectral extremal graphs.

math.CO↗

arXiv2026

Spectral radius and parity $[a,b]$-factors in graphs

Let $a$, $b$, and $n$ be three integers such that $1\leq a \leq b < n$, $a \equiv b$ (mod $2$), and $na$ is even. A parity $[a,b]$-factor of $G$ is a spanning subgraph $H$ such that for each vertex $v \in V(G)$, $a \leq d_H(v) \leq b$ and $d_H(v) \equiv a \equiv b$ (mod $2$). Recently, O [J. Graph Theory 100 (2022) 458-469] proved eigenvalue conditions for a regular graph to have a parity $[a,b]$-factor. In this paper, we prove a sharp lower bound on the spectral radius for an $n$-vertex graph $G$ to have a parity $[a,b]$-factor as follows: If $G$ is an $n$-vertex connected graph with $\delta(G)\geq a$ and $\rho(G)\geq\rho(G_{n}^{a})$, then $G$ contains a parity $[a,b]$-factor unless $G \cong G_{n}^{a}$, where $2\leq a<b$ and $G_{n}^{a}$ is the graph obtained from $K_{a-1}\vee(K_{n-2a-1}\cup(a+1)K_1)$ by adding a new vertex and adding all possible edges between the added vertex and each vertex in $(a+1)K_1$.

math.CO↗

arXiv2025

Toughness in regular graphs from eigenvalues

The {\it toughness} $\tau(G)=\mathrm{min}\{\frac{|S|}{c(G-S)}: S~\mbox{is a vertex cut in}~G\}$ for $G\ncong K_n,$ which was initially proposed by Chv\'{a}tal in 1973. A graph $G$ is called {\it $t$-tough} if $\tau(G)\geq t.$ Let $\lambda_i(G)$ be the $i$-th largest eigenvalue of the adjacency matrix of a graph $G$. In 1996, Brouwer conjectured that $\tau(G)\geq\frac{d}{\lambda}-1$ for a connected $d$-regular graph $G,$ where $\lambda=\mathrm{max}\{|\lambda_2|, |\lambda_n|\}.$ Gu [SIAM J. Discrete Math. 35 (2021) 948-952] completely confirmed this conjecture. From Brouwer and Gu's result $\tau(G)\geq\frac{d}{\lambda}-1,$ we know that if $G$ is a connected $d$-regular graph and $\lambda\leq\frac{bd}{b+1}$, then $\tau(G)\geq\frac{1}{b}$ for an integer $b\geq1.$ Inspired by the above result and utilizing typical spectral techniques and graph construction methods from Cioab\u{a} et al. [J. Combin. Theory Ser. B 99 (2009) 287-297], we prove that if $G$ is a connected $d$-regular graph and $\lambda_2(G)<\phi(d,b)$, then $\tau(G)\geq\frac{1}{b}$. Meanwhile, we construct graphs implying that the upper bound on $\lambda_2(G)$ is best possible. Our theorem strengthens the result of Chen et al. [Discrete Math. 348 (2025) 114404]. Finally, we also prove an upper bound of $\lambda_{b+1}(G)$ to guarantee a connected $d$-regular graph to be $\frac{1}{b}$-tough.

math.CO↗

arXiv2025

Tur\'{a}n number of books in non-bipartite graphs

Let $\mathrm{ex}(n, H)$ be the Tur\'{a}n number of $H$ for a given graph $H$. A graph is color-critical if it contains an edge whose removal reduces its chromatic number. Simonovits' chromatic critical edge theorem states that if $H$ is color-critical with $\chi(H)=k+1$, then there exists an $n_0(H)$ such that ex$(n, H)=e(T_{n,k})$ and the Tur\'{a}n graph $T_{n,k}$ is the only extremal graph provided $n\geq n_0(H).$ A book graph $B_{r+1}$ is a set of $r+1$ triangles with a common edge, where $r\geq0$ is an integer. Note that $B_{r+1}$ is a color-critical graph with $\chi(B_{r+1})=3$. Simonovits' theorem implies that $T_{n,2}$ is the only extremal graph for $B_{r+1}$-free graphs of sufficiently large order $n$. Furthermore, Edwards and independently Khad\v{z}iivanov and Nikiforov completely confirmed Erd\H{o}s' booksize conjecture and obtained that ex$(n, B_{r+1})=e(T_{n,2})$ for $n\geq n_0(B_{r+1})=6r$. Recently, Zhai and Lin [J. Graph Theory 102 (2023) 502-520] investigated the problem of booksize from a spectral perspective. Note that the extremal graph $T_{n,2}$ is bipartite. Motivated by the above elegant results, we in this paper focus on the Tur\'{a}n problem of non-bipartite $B_{r+1}$-free graphs of order $n$. For $r = 0,$ Erd\H{o}s proved a nice result: If $G$ is a non-bipartite triangle-free graph on $n$ vertices, then $e(G)\leq\big\lfloor\frac{(n-1)^{2}}{4}\big\rfloor+1$. For general $r\geq1,$ we determine the exact value of Tur\'{a}n number of $B_{r+1}$ in non-bipartite graphs and characterize all extremal graphs provided $n$ is sufficiently large. An interesting phenomenon is that the Tur\'{a}n numbers and extremal graphs are completely different for $r=0$ and general $r\geq1.$

math.CO↗

arXiv2025

Sufficient conditions of $k$-leaf-connected graphs and spanning trees with bounded total $k$-excess

Chv\'{a}tal and Erd\"{o}s [Discrete Math. 2 (1972) 111-113] stated that, for an $m$-connected graph $G$, if its independence number $\alpha(G)\leq m-1$, then $G$ is Hamilton-connected. Note that $k$-leaf-connectedness is a natural generalization of Hamilton-connectedness of a graph. Ozeki and Yamashita [Graphs Combin. 27 (2011) 1-26] posed an open problem: What is the sufficient condition based on the independence number for an $m$-connected graph to be $k$-leaf-connected? In this paper, we prove that if $\alpha(G)\leq m-k+1,$ then an $m$-connected graph $G$ is $k$-leaf-connected. This not only answers the open problem of Ozeki and Yamashita, but also extends Chv\'{a}tal-Erd\"{o}s Theorem. As applications, we present sufficient spectral conditions for an $m$-connected graph to be $k$-leaf-connected. Let $k\geq 2$ be an integer and $T$ be a spanning tree of a connected graph. The total $k$-excess $te(T,k)$ is the summation of the $k$-excesses of all vertices in $T$, namely, $te(T,k)=\sum_{v\in V(T)}\mbox{max}\{0, d_{T}(v)-k\}.$ One can see that $T$ is a spanning $k$-tree if and only if $te(T,k)=0$. Fan, Goryainov, Huang and Lin [Linear Multilinear Algebra 70 (2022) 7264-7275] presented sufficient spectral conditions for a connected graph to contain a spanning $k$-tree. We in this paper propose sufficient conditions in terms of the spectral radius for a connected graph to contain a spanning tree with $te(T,k)\leq b$, where $b\geq0$ is an integer.

math.SP↗

arXiv2025

Spectral Tur\'{a}n problem of non-bipartite graphs: Forbidden books

A book graph $B_{r+1}$ is a set of $r+1$ triangles with a common edge, where $r\geq0$ is an integer. Zhai and Lin [J. Graph Theory 102 (2023) 502-520] proved that for $n\geq\frac{13}{2}r$, if $G$ is a $B_{r+1}$-free graph of order $n$, then $\rho(G)\leq\rho(T_{n,2})$, with equality if and only if $G\cong T_{n,2}$. Note that the extremal graph $T_{n,2}$ is bipartite. Motivated by the above elegant result, we investigate the spectral Tur\'{a}n problem of non-bipartite $B_{r+1}$-free graphs of order $n$. For general $r\geq1$, let $K_{\lfloor\frac{n-1}{2}\rfloor,\lceil\frac{n-1}{2}\rceil}^{r, r}$ be the graph obtained from $K_{\lceil\frac{n-1}{2}\rceil,\lfloor\frac{n-1}{2}\rfloor}$ by adding a new vertex $v_{0}$ such that $v_{0}$ has exactly $r$ neighbours in each part of $K_{\lceil\frac{n-1}{2}\rceil,\lfloor\frac{n-1}{2}\rfloor}$. By adopting a different technique named the residual index, Chv\'{a}tal-Hanson theorem and typical spectral extremal methods, we in this paper prove that: If $G$ is a non-bipartite $B_{r+1}$-free graph of order $n$, then $\rho(G)\leq\rho\Big(K_{\lfloor\frac{n-1}{2}\rfloor,\lceil\frac{n-1}{2}\rceil}^{r, r}\Big)$ , with equality if and only if $G\cong K_{\lfloor\frac{n-1}{2}\rfloor,\lceil\frac{n-1}{2}\rceil}^{r, r}$. An interesting phenomenon is that the spectral extremal graphs are completely different for $r=0$ and general $r\geq1$.

math.CO↗

arXiv2024

Eliminating the Language Bias for Visual Question Answering with fine-grained Causal Intervention

Despite the remarkable advancements in Visual Question Answering (VQA), the challenge of mitigating the language bias introduced by textual information remains unresolved. Previous approaches capture language bias from a coarse-grained perspective. However, the finer-grained information within a sentence, such as context and keywords, can result in different biases. Due to the ignorance of fine-grained information, most existing methods fail to sufficiently capture language bias. In this paper, we propose a novel causal intervention training scheme named CIBi to eliminate language bias from a finer-grained perspective. Specifically, we divide the language bias into context bias and keyword bias. We employ causal intervention and contrastive learning to eliminate context bias and improve the multi-modal representation. Additionally, we design a new question-only branch based on counterfactual generation to distill and eliminate keyword bias. Experimental results illustrate that CIBi is applicable to various VQA models, yielding competitive performance.

cs.CV↗

arXiv2024

Fusion Makes Perfection: An Efficient Multi-Grained Matching Approach for Zero-Shot Relation Extraction

Predicting unseen relations that cannot be observed during the training phase is a challenging task in relation extraction. Previous works have made progress by matching the semantics between input instances and label descriptions. However, fine-grained matching often requires laborious manual annotation, and rich interactions between instances and label descriptions come with significant computational overhead. In this work, we propose an efficient multi-grained matching approach that uses virtual entity matching to reduce manual annotation cost, and fuses coarse-grained recall and fine-grained classification for rich interactions with guaranteed inference speed. Experimental results show that our approach outperforms the previous State Of The Art (SOTA) methods, and achieves a balance between inference efficiency and prediction accuracy in zero-shot relation extraction tasks. Our code is available at https://github.com/longls777/EMMA.

cs.CL↗

arXiv2023

Toughness and distance spectral radius in graphs involving minimum degree

The toughness $τ(G)=\mathrm{min}\{\frac{|S|}{c(G-S)}: S~\mbox{is a cut set of vertices in}~G\}$ for $G\ncong K_n.$ The concept of toughness initially proposed by Chv$\mathrm{\acute{a}}$tal in 1973, which serves as a simple way to measure how tightly various pieces of a graph hold together. A graph $G$ is called $t$-tough if $τ(G)\geq t.$ It is very interesting to investigate the relations between toughness and eigenvalues of graphs. Fan, Lin and Lu [European J. Combin. 110 (2023) 103701] provided sufficient conditions in terms of the spectral radius for a graph to be 1-tough with minimum degree $δ$ and $t$-tough with $t\geq 1$ being an integer, respectively. By using some typical distance spectral techniques and structural analysis, we in this paper present sufficient conditions based on the distance spectral radius to guarantee a graph to be 1-tough with minimum degree $δ.$ Moreover, we also prove sufficient conditions with respect to the distance spectral radius for a graph to be $t$-tough, where $t$ or $\frac{1}{t}$ is a positive integer.

math.CO↗

arXiv2023

Sufficient conditions for $k$-factors and spanning trees of graphs

For any integer $k\geq1,$ a graph $G$ has a $k$-factor if it contains a $k$-regular spanning subgraph. In this paper we prove a sufficient condition in terms of the number of $r$-cliques to guarantee the existence of a $k$-factor in a graph with minimum degree at least $δ$, which improves the sufficient condition of O \cite{O2021} based on the number of edges. For any integer $k\geq2,$ a spanning $k$-tree of a connected graph $G$ is a spanning tree in which every vertex has degree at most $k$. Motivated by the technique of Li and Ning \cite{Li2016}, we present a tight spectral condition for an $m$-connected graph to have a spanning $k$-tree, which extends the result of Fan, Goryainov, Huang and Lin \cite{Fan2021} from $m=1$ to general $m$. Let $T$ be a spanning tree of a connected graph. The leaf degree of $T$ is the maximum number of leaves adjacent to $v$ in $T$ for any $v\in V(T)$. We provide a tight spectral condition for the existence of a spanning tree with leaf degree at most $k$ in a connected graph with minimum degree $δ$, where $k\geq1$ is an integer.

math.CO↗

arXiv2023

Spectral radius, fractional $[a,b]$-factor and ID-factor-critical graphs

Let $G$ be a graph and $h: E(G)\rightarrow [0,1]$ be a function. For any two positive integers $a$ and $b$ with $a\leq b$, a fractional $[a,b]$-factor of $G$ with the indicator function $h$ is a spanning subgraph with vertex set $V(G)$ and edge set $E_h$ such that $a\leq\sum_{e\in E_{G}(v)}h(e)\leq b$ for any vertex $v\in V(G)$, where $E_h = \{e\in E(G)|h(e)>0\}$ and $E_{G}(v)=\{e\in E(G)| e~\mbox{is incident with}~v~\mbox{in}~G\}$. A graph $G$ is ID-factor-critical if for every independent set $I$ of $G$ whose size has the same parity as $|V(G)|$, $G-I$ has a perfect matching. In this paper, we present a tight sufficient condition based on the spectral radius for a graph to contain a fractional $[a,b]$-factor, which extends the result of Wei and Zhang [Discrete Math. 346 (2023) 113269]. Furthermore, we also prove a tight sufficient condition in terms of the spectral radius for a graph with minimum degree $δ$ to be ID-factor-critical.

math.CO↗

arXiv2023

Zero-Shot Rumor Detection with Propagation Structure via Prompt Learning

The spread of rumors along with breaking events seriously hinders the truth in the era of social media. Previous studies reveal that due to the lack of annotated resources, rumors presented in minority languages are hard to be detected. Furthermore, the unforeseen breaking events not involved in yesterday's news exacerbate the scarcity of data resources. In this work, we propose a novel zero-shot framework based on prompt learning to detect rumors falling in different domains or presented in different languages. More specifically, we firstly represent rumor circulated on social media as diverse propagation threads, then design a hierarchical prompt encoding mechanism to learn language-agnostic contextual representations for both prompts and rumor data. To further enhance domain adaptation, we model the domain-invariant structural features from the propagation threads, to incorporate structural position representations of influential community response. In addition, a new virtual response augmentation method is used to improve model training. Extensive experiments conducted on three real-world datasets demonstrate that our proposed model achieves much better performance than state-of-the-art methods and exhibits a superior capacity for detecting rumors at early stages.

cs.CL↗

arXiv2023

Fractional matching, factors and spectral radius in graphs involving minimum degree

A fractional matching of a graph $G$ is a function $f:E(G)\rightarrow [0, 1]$ such that for any $v\in V(G)$, $\sum_{e\in E_{G}(v)}f(e)\leq1$, where $E_{G}(v)=\{e\in E(G): e~ \mbox{is incident with} ~v~\mbox{in}~G\}$.The fractional matching number of $G$ is $μ_{f}(G)=\mathrm{max}\{\sum_{e\in E(G)}f(e):f$ is a fractional matching of $G\}$. Let $k\in (0,n)$ is an integer. In this paper, we prove a tight lower bound of the spectral radius to guarantee $μ_{f}(G)>\frac{n-k}{2}$ in a graph with minimum degree $δ,$ which implies the result on the fractional perfect matching due to Fan et al. [Discrete Math. 345 (2022) 112892]. For a set $\{A, B, C, \ldots\}$ of graphs, an $\{A, B, C, \ldots\}$-factor of a graph $G$ is defined to be a spanning subgraph of $G$ each component of which is isomorphic to one of $\{A, B, C, \ldots\}$.We present a tight sufficient condition in terms of the spectral radius for the existence of a $\{K_2, \{C_k\}\}$-factor in a graph with minimum degree $δ,$ where $k\geq 3$ is an integer. Moreover, we also provide a tight spectral radius condition for the existence of a $\{K_{1, 1}, K_{1, 2}, \ldots , K_{1, k}\}$-factor with $k\geq2$ in a graph with minimum degree $δ,$ which generalizes the result of Miao et al. [Discrete Appl. Math. 326 (2023) 17-32].

math.CO↗

arXiv2022

Maxima of the $Q$-index of non-bipartite graphs: forbidden short odd cycles

Let $G$ be a non-bipartite graph which does not contain any odd cycle of length at most $2k+1$. In this paper, we determine the maximum $Q$-index of $G$ if its order is fixed, and the corresponding extremal graph is uniquely characterized. Moreover, if the size of $G$ is given, the maximum $Q$-index of $G$ and the unique extremal graph are also proved.

math.CO↗