arXiv subjects
Publications and source records attributed to Fengming Dong.
For a connected graph $G$, let $τ(G)$, $α(G)$ and $α^*(G)$ denote its numbers of spanning trees, acyclic orientations and totally cyclic orientations, respectively. We prove that $α(G)α^*(G)\geqτ(G)^2$ for every connected bridgeless chordal graph.
A vertex $v$ of a graph $G$ $ve$-dominates every edge incident with a vertex in its closed neighborhood $N_{G}[v]$. A set of vertices is a vertex-edge dominating set ($ve$-dominating set) if every edge of $G$ is $ve$-dominated by some vertex in the set. The minimum cardinality of such a set is the vertex-edge domination number $γ_{ve}(G)$, also known as the isolation number. In this paper, we study the effect of vertex removal on $γ_{ve}(G)$, establish sharp bounds on $γ_{ve}(G-v)$ and characterize the vertices whose removal decreases, preserves, or increases $γ_{ve}(G)$. Let $V^0(G)$ and $V^+(G)$ denote the sets of vertices whose removal preserves and increases $γ_{ve}(G)$, respectively, and let $V_{\mathrm{fix}}^0(G)$ consist of the vertices in $V^0(G)$ belonging to every minimum $ve$-dominating set. We prove that \[ |V^0(G)|\geq3|V_{\mathrm{fix}}^0(G)|+\sum_{v\in V^+(G)} \bigl(γ_{ve}(G-v)-γ_{ve}(G)+1\bigr). \] In particular, $|V^0(G)|\geq 2|V^+(G)|$. We also prove that every vertex in $V_{\mathrm{fix}}^0(G)$ lies on a cycle and that $V^0(T)\neq\emptyset$ for every tree $T$ of order at least three.
Recent advances in video generation let users create high-quality videos from natural language descriptions, yet obtaining results that match user intent remains time-consuming: individual generations are computationally expensive, and underspecified prompts force repeated cycles of generation, inspection, and revision. Existing video prompt optimization methods improve generation through automatic rewriting, but cannot supply the ideas and preferences that were not in the initial description. We propose ProactGen, a video generation framework that learns proactive clarification, eliciting user intent through questions grounded in generated video drafts. Specifically, we train a vision-language model as a questioning policy with reinforcement learning: observing the initial prompt and a video draft, it learns to ask the questions most valuable to downstream generation, rewarded by the alignment between the final generated video and a target reference. We further find that effective questioning does not require drafts of final-output quality, and exploit this property by generating drafts with fewer denoising steps, keeping the added cost of clarification low. Experiments show that ProactGen improves target alignment over direct generation by at least 1.8x the gain of the strongest baselines. Moreover, a single round of ProactGen already exceeds the alignment that the strongest baseline reaches after four refinement rounds, in 29% of its time.
It is well known that the spectral radius $ρ(T)$ of a tree $T$ with at least $3$ vertices satisfies $\frac 14ρ(T)^2+1<Δ(T)\le ρ(T)^2$, where $Δ(T)$ is the maximum degree of $T$. Let $\mathbb{P}$ denote the set of spectral radii of all non-trivial trees. We ask whether, for every $α\in \mathbb{P}\setminus \{1\}$ and every integer $r$ satisfying $\frac 14α^2+1<r\le α^2$, there exists a tree $T$ such that $Δ(T)=r$ and $ρ(T)=α$. For a positive integer $r$ and a positive real number $α$, let ${\mathscr W}_r(α)$ be the smallest subset of $[0,\infty)$ containing $α$ and closed under the following operation: if $q_1,\ldots,q_s$ are positive elements of ${\mathscr W}_r(α)$, not necessarily distinct, and $q=α-\sum_{i=1}^{s}q_i^{-1}$, then $q\in{\mathscr W}_r(α)$ whenever either $1\le s<r$ and $q\ge0$, or $s=r$ and $q=0$. We prove that $0\in {\mathscr W}_r(α)$ if and only if there exists a tree $T$ with $Δ(T)\le r$ and $ρ(T)=α$. Moreover, there exists a tree $T$ with $Δ(T)=r$ and $ρ(T)=α$ if and only if there are positive elements $q_1,\ldots,q_r\in\mathscr W_r(α)$, not necessarily distinct, satisfying $\sum_{i=1}^{r}q_i^{-1}=α$. Consequently, $\mathbb{P}=\{α\geq1:0\in {\mathscr W}_{\lfloorα^2\rfloor}(α)\}.$ As an application, we show that for integers $k,r\ge 2$, there exists a tree $T$ with $Δ(T)=r$ and $ρ(T)=\sqrt k$ if and only if $\frac 14 k+1<r\le k$.
The DP color function $P_{DP}(\mathcal{H}, k)$ of a hypergraph $\mathcal{H}$ is defined as the minimum possible value of $P_{DP}(\mathcal{H}, \mathcal{F})$ taken over all $k$-fold covers $\mathcal{F}$ of $\mathcal{H}$. It serves as an extension of the classical chromatic polynomial. In this paper, we investigate the DP color function for two constructions: the join of a hypergraph with a complete graph and vertex-gluings of certain hypergraphs. Both constructions extend some results of Becker et al.
For a graph \(F\), the Turán number \(\ex(n,F)\) is the maximum number of edges in an \(F\)-free graph on \(n\) vertices. Let \(q\ge 2\) be an integer and set \(n=q^{2}+q+1\). Brown and Erdős, Rényi and Sós independently proved that $\ex(n,C_{4})\ge \frac12 q(q+1)^{2}$ for every prime power \(q\), and Füredi subsequently established the upper bound $\frac12 q(q+1)^{2}$ for $\ex(n,C_{4})$ whenever \(q\notin\{1, 7,9,11,13\}\). In this article, we prove that every \(C_{4}\)-free graph \(G\) on \(n\) vertices with at most \(\frac12 q(q+1)^{2}\) edges satisfies $τ(G)\le n^{(n-3)/2}$, where \(τ(G)\) denotes the number of spanning trees of \(G\). In particular, for every prime power $q\notin\{7,9,11,13\}$, the above upper bound on $τ(G)$ is attained precisely by the orthogonal polarity graphs, thereby proving London's conjecture for all such $q$.
Introduced by Birkhoff in 1912 to study the four-color conjecture, the chromatic polynomial has become one of the most extensively studied graph polynomials. It is closely connected to the Tutte polynomial through specialization, to the characteristic polynomial of the associated graphic matroid, and to the flow polynomial through planar duality. It has also inspired several related polynomials, including the $σ$-polynomial, the $w$-polynomial, and the $τ$-polynomial. This survey examines these relationships, illustrated in Figure 1, and reviews the main results.
Let $G$ be a simple graph with maximum degree $Δ\ge 3$, and let $P(G,k)$ denote its chromatic polynomial. For each positive integer $k$, the list-color function $P_{\ell}(G,k)$ is the minimum number of $L$-colorings of $G$ over all $k$-assignments $L$. In this paper, we prove that $P_{\ell}(G,k)=P(G,k)$ for every integer $k\ge 23.41Δ$. This gives a threshold for equality that is linear in the maximum degree and independent of the number of vertices or edges. It improves the known sufficient condition $k\ge |E(G)|-1$ for graphs with sufficiently many edges relative to their maximum degree.
For a positive integer $r$ and a real number $α\ge 2$, let $\mathscr L_r(α)$ be the set of positive real numbers containing $α-1$ and closed under the following operation: if $q_1,\ldots,q_s\in\mathscr L_r(α)$, where $1\leq s\leq r-1$, and $q=α-1-s-\sum\limits_{i=1}^{s}q_i^{-1}>0$, then $q\in\mathscr L_r(α)$. We prove that there exists a tree $T$ with $Δ(T)\leq r$ and Laplacian spectral radius $μ(T)=α$ if and only if $(α-1)^{-1}\in\mathscr L_r(α)$. Consequently, the Laplacian spectral radii of nontrivial trees are precisely the real numbers $α\geq2$ satisfying $(α-1)^{-1}\in\mathscr L_{\lfloorα\rfloor-1}(α)$. As an application, we prove that, for integers $k\geq2$ and $r\geq2$, a tree $T$ with $μ(T)=k^2$ and $Δ(T)=r$ exists if and only if $(k-1)^2+2\leq r\leq k^2-1$.
Let $G$ be a connected bridgeless graph. In 2011, Kung and Royle showed that all roots of the flow polynomial $F(G,λ)$ of $G$ are integers if and only if $G$ is the dual of a chordal plane graph. In this article, we further prove that if $F(G,λ)$ has real roots only, then $G$ is the dual of a chordal plane graph and each root of $F(G,λ)$ is an integer in the set $\{1,2,3\}$.
For any graph $G$, let $P(G,k)$ and $P_{\ell}(G,k)$ denote the chromatic polynomial and the list-color function of $G$, respectively. It remains an open problem whether, for every graph $G$ and integer $k$, the equality $P(G,k)=P_{\ell}(G,k)>0$ implies that $P(G,k+1)=P_{\ell}(G,k+1)$ also holds. In this paper, we answer this question in the negative. For every integer $k\ge 3$, we construct an infinite family of graphs $G$ such that $P(G,k)=P_{\ell}(G,k)>0$ while $P(G,k+1)>P_{\ell}(G,k+1)$. Moreover, using this infinite family of graphs as attachment gadgets, we further show that any graph $H$ with $P(H,k)=P_{\ell}(H,k)>0$ can be developed into an infinite family of graphs $H'$ with $P(H',k)=P_{\ell}(H',k)>0$ and $P(H',k+1)>P_{\ell}(H',k+1)$.
The study of equimatchable graphs traces back to Sumner (1979), followed by the foundational work of Lesk, Plummer, and Pulleyblank in 1984. Regarding the characterization of equimatchable regular graphs, Akbari et al. and Eiben et al. have characterized almost all such graphs, leaving the class of odd-order equimatchable $r$-regular graphs with independence number at least $3$ and even $r\geq 6$ unclassified. In the present paper, we give a complete structural characterization of graphs in this remaining class.
The toughness of a non-complete graph $G$, denoted $τ(G)$, is defined as \[ τ(G) = \min\left\{ \frac{|S|}{ω(G-S)} : S \subseteq V(G),\ ω(G-S) \geq 2 \right\}, \] where $ω(G-S)$ is the number of components of $G - S$. For a complete graph $G$, we define $τ(G) = \infty$. A graph $G$ is $t$-tough if $τ(G) \geq t$. For a positive integer $k$, a graph $G$ is $(K_2 \cup kK_1)$-free if it contains no induced subgraph isomorphic to $K_2 \cup kK_1$. Recently, Liu \cite{liu} showed that every $2k$-connected $(K_2 \cup kK_1)$-free graph $G$ with $τ(G) > 1$ is Hamilton-connected. In this paper, we strengthen this result by proving that every $(k+1)$-connected $(K_2 \cup kK_1)$-free graph $G$ with $τ(G) > 1$ and minimum degree $δ(G) \geq 2k$ is Hamilton-connected. Moreover, by imposing restrictions to the independence number $α(G)$, we prove that every $k$-connected $(K_2 \cup kK_1)$-free graph $G$ of order $n$ with $2k+1 \leq α(G) < \frac{n}{2}$ and $δ(G) \geq 2k$ is Hamilton-connected, and that the bounds on $α(G)$ are sharp.
An edge cut C of a graph G is tight if |C \M| = 1 for every perfect matching M of G. Barrier-cuts and 2-separation cuts, also referred to as ELP-cuts, are two important types of tight cuts in matching covered graphs. Edmonds, Lovasz and Pulleyblank [Brick decompositions and the matching rank of graphs, Combinatorica 2(3) (1982) 247-274] proved that if a matching covered graph has a non-trivial tight cut, then it also has a non-trivial ELP-cut. In confirmation of a conjecture proposed by Carvalho, Lucchesi and Murty, Chen et. al. [Laminar tight cuts in matching covered graphs, J. Comb. Theory, Ser. B, 150 (2021) 177-194] showed that if C is a non-trivial tight cut of a matching covered graph G, then G has at least one C-sheltered non-trivial barrier or a 2-separation cut that is laminar with C. In this paper, we present a complete characterization of non-trivial tight cuts in matching covered graphs, from which the result of Chen et. al. can be derived directly. Moreover, we show that the lower bound of the number of C-sheltered non-trivial barrier or a 2-separation cut that is laminar with C in the result of Chen et. al. is sharp.
In 2016, Dowden initiated the study of planar Turán-type problems, which has since attracted considerable attention. Recently, Bekos et al. proved that every $K_3$-free $1$-planar graph on $n\ge 4$ vertices has at most $3n-6$ edges. In this paper, we strengthen this bound to $3n - 8$, which is tight for all even $n \ge 8$. Furthermore, we show that every $K_4$-free $1$-planar graph on $n \ge 3$ vertices has at most $\bigl\lfloor \tfrac{7n}{2} \bigr\rfloor - 7$ edges, and this bound is tight for all integers $n \ge 9$. We also prove that every $K_5$-free $1$-planar graph on $n \ge 3$ vertices has at most $4n - 8$ edges, which is tight for $n = 8$ and for all integers $n \ge 10$.
A graph $G$ is $\{F_{1}, F_{2},\dots,F_{k}\}$-free if $G$ contains no induced subgraph isomorphic to any $F_{i}$ $(1\leq i \leq k)$. A connected graph $G$ is a split graph if its vertex set can be partitioned into a clique and an independent set. Ryjáček et al. [J. Comb. Theory, Ser. B 134 (2019) 239--263] conjectured that every $4$-connected $\{K_{1,4},K_{1,4}+e\}$-free graph with minimum degree at least 6 is Hamiltonian and they confirmed the case with connectivity at least 5, where $K_{1,4}+e$ is the graph obtained from $K_{1,4}$ by adding a new edge. In this paper, we show that every 3-connected $\{K_{1,4},K_{1,4}+e\}$-free split graph of order at least $13$ is Hamilton-connected. It implies that Ryjáček et al.'s conjecture holds for split graphs of order at least $13$.
For a hypergraph $\mathcal{H}$, the DP color function $P_{DP}(\mathcal{H},k)$ of $\mathcal{H}$ is an extension of the chromatic polynomial $P(\mathcal{H},k)$ with the property that $P_{DP}(\mathcal{H},k) \le P(\mathcal{H},k)$ for all positive integers $k$. In this article, we primarily investigate the influence of the minimum cycle length on the DP-coloring function, as well as the relevant properties of the DP-coloring function of $\mathcal{H} \vee K_p$ (i.e., the join of $\mathcal{H}$ and $K_p$). We show that for any linear and uniform hypergraph $\mathcal H$ with even girth, there exists a positive integer $N$ such that $P_{DP} (\mathcal H, k) < P(\mathcal H, k)$ for all integers $k\ge N$, and this conclusion also holds for any hypergraph $\mathcal{H}$ that contains an edge $e$ with the properties that $\mathcal{H}-e$ has exactly $|e|-1$ components and any shortest cycle in $\mathcal{H}$ containing $e$ is an even cycle. For the hypergraph $\mathcal{H}\vee K_p$, we prove that if $\mathcal{H}$ is uniform, then there exist positive integers $p$ and $N$ such that $P_{DP}(\mathcal{H} \vee K_p,k)=P(\mathcal{H} \vee K_p,k)$ holds for all integers $k\geq N$.
In this paper, we show that any maximal IC-plane graph of order $n$ has at least $\left\lceil\frac{7}{3}n-\frac{14}{3}\right\rceil$ edges, and any maximal NIC-plane graph of order $n$ has at least $\left\lceil\frac{11}{5}n-\frac{18}{5}\right\rceil$ edges. Moreover, we show that both results are tight for infinitely many integers $n$.