arXiv subjects

Lihua You

Publications and source records attributed to Lihua You.

At least 19 recordsLinked to original sources

arXiv2026

Rainbow Hamiltonian paths in graph collections under edge-size conditions

Let $\mathbf{G}=\{G_1,\dots,G_{n-1}\}$ be a collection of not necessarily distinct graphs of order $n$ on the same vertex set $V$. A path $P$ with $V(P)\subseteq V$ is called \emph{rainbow} in $\mathbf{G}$, if there exists an injection $ϕ\colon E(P)\to [n-1]$ such that $e\in E(G_{ϕ(e)})$ for each $e\in E(P)$. In this paper, we show that if $n\ge3$, $e(G_i)> \binom{n-2}{2}+2$ for each $i\in[n-1]$ and $G_j$ is connected for some $j\in[n-1]$, then there exists a rainbow Hamiltonian path in $\mathbf{G}$ unless $n=6$ with $G_1=G_2=\dots=G_{5}\cong K_2\vee 4K_1$, or $n=8$ with $G_1=G_2=\dots=G_{7}\cong K_3\vee 5K_1$. This corrects Theorem 6 in [Graphs Combin., \textbf{41}(6)(2025), 119] and relaxes its hypothesis from requiring all $G_i$ to be connected to only requiring that at least one $G_i$ is connected, where $i\in[n-1]$. Moreover, we show that if $n\ge2$ and $e(G_i)\ge \binom{n-1}{2}$ for each $i\in[n-1]$, then $\mathbf{G}$ contains a rainbow Hamiltonian path unless $G_1=G_2=\dots=G_{n-1}\cong K_{n-1}\cup K_1$, or $n=4$ with $G_1=G_2=G_3\cong K_{1,3}$. As a consequence, Theorem 2 of [Discrete Math., \textbf{347}(2024), 114128] follows immediately.

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arXiv2026

Saturation numbers of $K_3 \vee P_k$

A graph $G$ is said to be $H$-saturated if $G$ contains no copy of $H$, but $G+e$ contains a copy of $H$ for any edge $e$ of $\overline{G}$. The saturation number of $H$, denoted by $\mathrm{sat}(n, H)$, is the minimum number of edges among all $H$-saturated graphs of order $n$. In this paper, we investigate the saturation number of $K_3 \vee P_k$ for $k \ge 3$ and show that $\mathrm{sat}(n, K_3 \vee P_k) = 3n - 6 + \mathrm{sat}(n-3, P_k)$ holds for all $n \ge a_k + 3$ and $k \ge 3$, which verifies the conjecture [Discrete Appl. Math, 394(2026) 62-80] on $\mathrm{sat}(n, K_s \vee P_k)$ is true for the case $s=3$, where $a_k = k$ for $3 \le k \le 5$, $a_k = 3 \cdot 2^{t-1} - 2$ for $k = 2t \ge 6$, and $a_k = 2^{t+1} - 2$ for $k = 2t + 1 \ge 7$.

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arXiv2026

Edge and spectral conditions for rainbow bipancyclicity in bipartite graph collections

Let $\mathbf{G}=\{G_1,\dots,G_{2n}\}$ be a collection of not necessarily distinct balanced bipartite graphs of order $2n$ with a common bipartition $V=(X,Y)$. A cycle $C$ with $V(C)\subseteq V$ and $|E(C)|\leq 2n$ is called \emph{rainbow} in $\mathbf{G}$, if there exists an injection $ϕ\colon E(C)\to [2n]$ such that $e\in E(G_{ϕ(e)})$ for each $e\in E(C)$. The graph collection $\mathbf{G}$ is said to be \emph{rainbow bipancyclic} if it contains a rainbow cycle of every even length from 4 to $2n$. In this paper, we show that if $e(G_i)\ge n^2-n+1$ for each $i\in[2n]$, then $\mathbf{G}$ is rainbow bipancyclic unless $G_1=G_2=\dots=G_{2n}\cong B_{n}^1$, where $B_{n}^1$ is the graph obtained from $K_{n,n}$ by deleting all edges in its one subgraph $K_{n-1,1}$. As a corollary, we obtain that if $ρ(G_i)\ge ρ(B_{n}^1)>\sqrt{n(n-1)}$ for each $i\in[2n]$, then $\mathbf{G}$ is rainbow bipancyclic unless $G_1=G_2=\dots=G_{2n}\cong B_{n}^1$, which strengthens Theorem $5$ of [Electron. J. Combin., \textbf{33}(3)(2026), \#P3.29] and Theorem $1.6$ of [Linear Algebra Appl., \textbf{743}(2026), 250--270] from rainbow Hamiltonicity to rainbow bipancyclicity. We also characterize all bipartite graph collections that are not rainbow bipancyclic under the condition $ρ(G_i)\ge \sqrt{n(n-1)}$ for each $i\in[2n]$

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arXiv2026

Edge and spectral conditions for rainbow pancyclicity in graph collections

Let $\mathbf{G}=\{G_1,\dots,G_{n}\}$ be a collection of not necessarily distinct $n$-vertex graphs with a common vertex set $V$. A cycle $C$ with $V(C)\subseteq V$ and $|E(C)|\leq n$ is called \emph{rainbow} in $\mathbf{G}$, if there exists an injection $ϕ\colon E(C)\to [n]$ such that $e\in E(G_{ϕ(e)})$ for each $e\in E(C)$. The graph collection $\mathbf{G}$ is said to be \emph{rainbow pancyclic} if it contains a rainbow cycle of every length from 3 to $n$. In this paper, we show that if $e(G_i)\ge \binom{n-1}{2}+1$ for each $i\in[n]$ with $n\ge 3$, then $\mathbf{G}$ is rainbow pancyclic, apart from three explicitly described exceptional graph collections. This answers Problem $1$ of [Discrete Math., \textbf {348}(2025), 114600] and strengthens the result from rainbow Hamiltonicity to rainbow pancyclicity. As a consequence, we obtain that if $ρ(G_i)>n-2$ for each $i\in[n]$, then $\mathbf{G}$ is rainbow pancyclic unless $G_1=G_2=\dots=G_n\cong K_1\vee(K_{n-2}\cup K_1)$, which improves Theorem $5$ of [Discrete Math., \textbf {348}(2025), 114600]. We also characterize all graph collections that are not rainbow pancyclic under the condition $ρ(G_i)\ge n-2$ for each $i\in[n]$.

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arXiv2026

Two problems on booksize and triangular edges in Nosal graphs

A graph $G$ with $m$ edges is said to be a Nosal graph if $ρ(G)>\sqrt{m}$. For a graph $G$, we write $bk(G)$ for its maximum book size and $τ(G)$ for the number of edges contained in triangles. Li, Liu and Zhang [J. Combin. Theory Ser. B 179 (2026) 219--249] proved that every $m$-edge Nosal graph satisfies $bk(G)> \frac{1}{24}\sqrt{m}$ and $τ(G) > \frac{1}{12}\sqrt{m}$. Recently, two results on the booksize constant are proved: $\frac{1}{9}$ by Zhai, Li and Lou [arXiv:2601.10163v2], and $\frac{1}{4}$ by Chen, Li and Tang [arXiv:2607.16746v1]. In this paper, we establish the following result: Every $m$-edge graph $G$ with no isolated vertices and $ρ(G)\geq \sqrt{m}$ that is not isomorphic to any complete bipartite graph satisfies $bk(G)\geq\frac{ρ(G)}{3}$ and $τ(G)\geq ρ(G)$. As direct consequences, we answer a question of Li, Liu and Zhang [J. Combin. Theory Ser. B 179 (2026) 219--249] and confirm a conjecture of Li, Feng and Peng [J. Graph Theory 110 (4) (2025) 408--425].

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arXiv2026

The maximum number of cycles of a given length in a nonhamiltonian graph

In 2026, Li and Zhan characterized the nonhamiltonian graphs of order $n$ with the maximum number of paths of length $k$, where $n$ and $k$ are integers satisfying $1\leq k\leq n-1$. This work solves and generalizes a problem proposed by Erdős in 1980. In this paper, we further determine the nonhamiltonian graphs of order $n$ attaining the maximum number of cycles of length $k$ for given integers $n$ and $k$ with $3\leq k\leq n-1$. As a corollary, we determine the generalized Turán number $\mathrm{ex}(n,C_k,C_n)$ for every $3\leq k\leq n-1$.

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arXiv2026

The signless Laplacian spectral radius of graphs without disjoint cliques

A graph $G$ is $(t+1)K_{r+1}$-free if it contains no $t+1$ pairwise vertex-disjoint copies of $K_{r+1}$. Moon [Canad. J. Math. 20 (1968) 95-102] and Simonovits [Theory of Graphs (Proc. Colloq., Tihany, 1966)] independently determined that, for sufficiently large $n$, $K_{t}\vee T_{r}(n-t)$ is the unique $n$-vertex $(t+1)K_{r+1}$-free graph with the maximum number of edges. In 2023, Ni, Wang and Kang [Electron. J. Combin. 30 (2023) \#P1.20] showed that the graph $K_{t}\vee T_{r}(n-t)$ is also the unique adjacency spectral extremal graph over all $n$-vertex $(t+1)K_{r+1}$-free graphs for sufficiently large $n$. In this paper, for $r\geq 3$ and $t\geq 0$, we prove that $K_{t}\vee T_r(n-t)$ is the unique graph attaining the maximum signless Laplacian spectral radius among all $(t+1)K_{r+1}$-free graphs of sufficiently large order $n$.

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arXiv2026

Oriented diameter of graphs with diameter $4$ and given edge girth

Let $f(d)$ be the smallest value for which every bridgeless graph $G$ with diameter $d$ admits a strong orientation $\overrightarrow{G}$ such that the diameter of $\overrightarrow{G}$ is at most $f(d)$. Chvátal and Thomassen (JCT-B, 1978) obtained general bounds for $f(d)$ and proved that $f(2)=6$. Kwok et al. (JCT-B, 2010) proved that $9\leq f(3)\leq 11$. Wang and Chen (JCT-B, 2022) determined $f(3)=9$. Babu et al. (DAM, 2021) showed $f(4)\leq 21$. In this paper, we introduce a new approach to studying $f(d)$ via the edge girth of a bridgeless graph $G$, denoted by $g^*(G)=\max\{l_G(e)\mid e\in E(G)\}$, where $l_G(e)$ is the length of the shortest cycle containing $e$ in $G$. Then we define $F(d,g^*)=\max\{\overrightarrow{diam}(G)\mid G\text{ is bridgeless},d(G)=d,g^*(G)=g^*\}$, and show $f(d)=\max\{F(d,g^*)\mid 2\leq g^*\leq 2d+1\}$. As the main result of this paper, we establish $F(4,2)=4$, $F(4,9)=12$, $F(4,3)\le 12$, and $F(4,g^*)\le 13$ for $g^*\in\{6,7,8\}$, and we propose two open problems for further research.

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arXiv2026

On the characterizations of $\mathcal{D}_{k} \backslash \mathcal{D}_{k-2}$

The determinant of a tournament $T$, denoted by $\det(T)$, is defined as the determinant of the skew-adjacency matrix of $T$. It is well-known that $\det(T)$ is equal to $0$ if $n$ is odd, and $\det(T)$ is the square of an odd integer if $n$ is even. For a positive odd integer $k$, let $\mathcal{D}_k$ be the set of tournaments whose all subtournaments have determinant at most $k^2$. Former studies showed that for $k \in \{1,3,5\}$, a tournament $T \in \mathcal{D}_k \backslash \mathcal{D}_{k-2}$ ($T \in \mathcal{D}_1$ when $k=1$) if and only if $T$ is switching equivalent to a transitive blowup of $L_{k+1}$, where $L_{k+1}$ is a tournament of order $k+1$ with a specific structure. For $k \geq 7$, no characterization results are known. A natural problem is to characterize tournaments in $\mathcal{D}_{k} \backslash \mathcal{D}_{k-2}$ that can be switching equivalent to a transitive blowup of $L_{k+1}$ for $k \geq 7$. To address this problem and to further explore the structural properties of tournaments in $\mathcal{D}_{k}$, we introduce CR tournaments, strong CR tournaments, basic tournaments and $Z$-matrices, and investigate their properties. We use these properties to characterize those tournaments $T \in \mathcal{D}_{k} \backslash \mathcal{D}_{k-2}$ where $T$ contains a subtournament switching isomorphic to a basic strong CR tournament in $\mathcal{D}_{k} \backslash \mathcal{D}_{k-2}$. This result implies former characterizations of $\mathcal{D}_3\backslash \mathcal{D}_1$ and $\mathcal{D}_5 \backslash \mathcal{D}_3$. Using $Z$-matrices, we also show that for even $n$, $L_{n}$ is a basic strong CR tournament, and thus solve the open problem posed in [Discrete Math. 349 (2) (2026) 114766].

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arXiv2026

An improved upper bound on the oriented diameter of graphs with diameter $4$

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arXiv2026

Rainbow panconnectivity in a graph collection

Let $\mathbf{G}=\{G_1,\dots,G_{n-1}\}$ be a collection of not necessarily distinct $n$-vertex graphs with the same vertex set $V$. A path $P$ with $V(P)\subseteq V$ and $|E(P)|\leq n-1$ is called \emph{rainbow} in $\mathbf{G}$, if there exists an injection $ϕ\colon E(P)\to [n-1]$ such that $e\in E(G_{ϕ(e)})$ for each $e\in E(P)$. The graph collection $\mathbf{G}$ is said to be \emph{rainbow panconnected} if for every pair of vertices $x,y\in V$, there exists a rainbow path of $k$ vertices joining $x$ and $y$ in $\mathbf{G}$ for every integer $k\in \left[d_{\mathbf{G}}(x,y)+1, n\right]$, where $d_{\mathbf{G}}(x,y)$ is the length of a shortest rainbow path between $x$ and $y$ in $\mathbf{G}$. In this paper, we study the rainbow panconnectivity of $\mathbf{G}$ under the minimum degree condition. Our result improves upon the corresponding results of [J. Graph Theory, \textbf{104}(2)(2023), 341--359] and [Electron. J. Combin., \textbf{32}(4)(2025), \#P4.17].

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arXiv2026

The saturation number of $K^s_t$

For a given graph $F$, a graph $G$ is said to be $F$-saturated if $G$ contains no copy of $F$ but for any edge $uv\notin E(G)$, $G+uv$ contains a copy of $F$. The saturation number $sat(n,F)$ is defined as the minimum number of edges among all $n$-vertex $F$-saturated graphs. The virus graph $K^s_t$, where $s\geq0$ and $t\geq \max\{3,s\}$, is a graph of order $s+t$ constructed by attaching $s$ distinct leaves to $s$ different vertices of a complete graph $K_t$. Hua and Peng [Discrete Math. 349 (2026) 114674] determined $sat(n,K^2_3)$ and characterized its corresponding extremal graphs. In this paper, we determine $sat(n,K^3_3)$ and $sat(n,K^2_t)$ with $t\geq 4$, together with the structural descriptions of the related extremal saturated graphs.

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arXiv2026

Transversal and Hamiltonicity in a bipartite graph collection

Let $\mathbf{G}=\{G_1,\dots,G_{s}\}$ be a collection of $s$ bipartite graphs with the same bipartition $V=(X,Y)$. For a path $P$ with $V(P)=V$ and $|E(P)|=s$, if there exists an injection $ϕ$: $E(P)\rightarrow [s]$ such that $e\in E(G_{ϕ(e)})$ for each $e\in E(P)$, then we say that the Hamiltonian path $P$ is a $\mathbf{G}$-transversal. A bipartite graph collection $\mathbf{G}$ is called Hamiltonian connected if for any two vertices $x\in X$ and $y\in Y$, there exists a $\mathbf{G}$-transversal isomorphic to a Hamiltonian path between $x$ and $y$. In this paper, we give the minimum degree conditions that ensure the existence of a $\mathbf{G}$-transversal isomorphic to a Hamiltonian path and the Hamiltonian connectivity of a balanced bipartite graph collection $\mathbf{G}$, which improve the results of [Hu, Li, Li and Xu, Discrete Math., 2024]. Moreover, we also provide a minimum degree condition that guarantees a nearly balanced bipartite graph collection $\mathbf{G}$ contains a $\mathbf{G}$-transversal isomorphic to a Hamiltonian path.

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arXiv2025

Saturation numbers of $K_{2}\vee P_{k}$

A graph $G$ is called $H$-saturated if $G$ contains no copy of $H$, but $G+e$ contains a copy of $H$ for any edge $e\in E(\overline{G})$. The saturation number of $H$ is the minimum number of edges in an $H$-saturated graph of order $n$, denoted by $sat(n,H)$. In this paper, we investigate $sat(n,K_{2}\vee P_{k})$, where $k\geq 3$. Let $a_k$ be an integer, defined as follows: $a_k=k$ for $3\leq k\leq 5$; $a_k=3\cdot 2^{t-1}-2$ for $k=2t\geq 6$; and $a_k=2^{t+1}-2$ for $k=2t+1\geq 7$. We show that $sat(n, K_{2}\vee P_{k})=2n-3+sat(n-2,P_{k})$ for $n\geq a_k+2$ and $k\geq 3$, characterize the $K_{2}\vee P_{k}$-saturated graphs with $sat(n,K_{2}\vee P_{k})$ edges, the $K_{1}\vee P_{k}$-saturated graphs with $sat(n,K_{1}\vee P_{k})$ edges for $3\leq k\leq5$ and the $P_{k}$-saturated graphs with $sat(n, P_{k})$ edges for $3\leq k\leq4$. Furthermore, we propose some questions for further research.

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arXiv2025

Spectral conditions for spanning $k$-trees or $k$-ended-trees of $t$-connected graphs

Let $G$ be a connected graph of order $n$. A spanning $k$-tree of $G$ is a spanning tree with the maximum degree at most $k$, and a spanning $k$-ended-tree of $G$ is a spanning tree at most $k$ leaves, where $k\geq2$ is an integer. This paper establishes some spectral conditions for the existence of spanning $k$-trees or spanning $k$-ended-trees in $t$-connected graphs, which generalize the results of Fan et al. (2022) and Zhou (2010), and improve the results of Fiedler et al. (2010), Ao et al. (2023) and Ao et al. (2025).

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arXiv2025

Enumeration of spanning trees and resistance distances of generalized blow-up graphs

Let $H$ be a graph with vertex set $V(H)=\{v_1, v_2, \cdots, v_k\}$. The generalized blow-up graph $H_{p_1,\ldots,p_k}^{q_1,\ldots,q_k}$ is constructed by replacing each vertex $v_i \in V(H)$ with the graph $G_i = p_iK_t \cup q_iK_1$$(i=1,2,\cdots,k)$, then connecting all vertices between $G_i$ and $G_j$ whenever $v_iv_j \in E(H)$. In this paper, we enumerate the spanning trees in generalized blow-up graphs $H_{p_1, p_2, \cdots, p_k}^{q_1, q_2, \cdots, q_k}$, which extends the results of Ge [Discrete Appl. Math. 305 (2021) 145-153], Cheng, Chen and Yan [Discrete Appl. Math. 320 (2022) 259-269]. Furthermore, we determine the resistance distances and Kirchhoff indices of generalized blow-up graphs $H_{p_1, p_2, \cdots, p_k}^{q_1, q_2, \cdots, q_k}$, which extends the results of Sun, Yang and Xu [Discrete Math. 348 (2025) 114327], Xu and Xu [Discrete Appl. Math. 362 (2025) 18-33], Ni, Pan and Zhou [Discrete Appl. Math. 362 (2025) 100-108].

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arXiv2025

The connectedness of friends-and-strangers graphs about graph parameters and others

Let $X$ and $Y$ be two graphs of order $n$. The friends-and-strangers graph $\textup{FS}(X,Y)$ of $X$ and $Y$ is a graph whose vertex set consists of all bijections $σ: V(X)\rightarrow V(Y)$, in which two bijections $σ$ and $ σ'$ are adjacent if and only if they agree on all but two adjacent vertices of $X$ such that the corresponding images are adjacent in $Y$. The most fundamental question about these friends-and-strangers graphs is whether they are connected. In this paper, we provide a sufficient condition regarding the maximum degree $Δ(X)$ and vertex connectivity $κ(Y)$ that ensures the graph $\textup{FS}(X,Y)$ is $s$-connected. As a corollary, we improve upon a result by Bangachev and partially confirm a conjecture he proposed. Furthermore, we completely characterize the connectedness of $\textup{FS}(X,Y)$, where $X\in\textup{DL}_{n-k,k}$.

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arXiv2025

The existence of a spanning tree with leaf distance at least $d$ and leaf degree at most $k$ via the size or the spectral radius with respect to the minimum degree

Let $k$, $d$ be a positive integer, $G$ be a connected graph of order $n$, $T$ be a tree. The leaf distance of a tree is defined as the minimum distance between any two leaves. For $v\in V(T)$, the leaf degree of $v$ in $T$ is the number of leaves adjacent to $v$, and the leaf degree of $T$ is defined as maximum leaf degree among the vertices of $T$. In this paper, motivated by the conjecture proposed by Kaneko (2001) and its subsequent partial confirmation by Erbes, Molla, Mousley and Santana (2017), we obtain lower bounds in terms of the size and the adjacent spectral radius to guarantee that $G$ contains a spanning tree with leaf distance at least $d$, where $4\leq d \leq n-1$. Furthermore, we obtain some tight conditions in $G$ for its size and spectral radius to ensure that $G$ has a spanning tree with leaf degree at most $k$, which improves the result of Ao, Liu, Yuan, Ng and Cheng (2023).

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