arXiv subjects

Fengming Dong

Publications and source records attributed to Fengming Dong.

At least 19 recordsLinked to original sources

arXiv2026

When chromatic polynomials coincide with list-color functions: a threshold linear in the maximum degree

Let $G$ be a simple graph with maximum degree $\Delta\ge 3$, 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 $k\ge 23.41\Delta$. 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.

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arXiv2026

Non-persistence of equality between chromatic polynomials and list-color functions

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 $P(H',k+1)>P_{\ell}(H',k+1)$.

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arXiv2026

Real-rooted flow polynomials have only integral roots

Let $G$ be a connected bridgeless graph. In 2011, Kung and Royle showed that all roots of the flow polynomial $F(G,\lambda)$ 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 $F(G,\lambda)$ has real roots only, then $G$ is the dual of a chordal plane graph and each root of $F(G,\lambda)$ is an integer in the set $\{1,2,3\}$.

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arXiv2026

Sufficient conditions for $(K_2 \cup kK_1)$-free graphs to be Hamilton-connected

The toughness of a non-complete graph $G$, denoted $\tau(G)$, is defined as \[ \tau(G) = \min\left\{ \frac{|S|}{\omega(G-S)} : S \subseteq V(G),\ \omega(G-S) \geq 2 \right\}, \] where $\omega(G-S)$ is the number of components of $G - S$. For a complete graph $G$, we define $\tau(G) = \infty$. A graph $G$ is $t$-tough if $\tau(G) \geq t$. 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 $\tau(G) > 1$ 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 $\tau(G) > 1$ and minimum degree $\delta(G) \geq 2k$ is Hamilton-connected. Moreover, by imposing restrictions to the independence number $\alpha(G)$, we prove that every $k$-connected $(K_2 \cup kK_1)$-free graph $G$ of order $n$ with $2k+1 \leq \alpha(G) < \frac{n}{2}$ and $\delta(G) \geq 2k$ is Hamilton-connected, and that the bounds on $\alpha(G)$ are sharp.

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arXiv2026

Tight cuts in matching covered graphs

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.

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arXiv2026

Extremal 1-planar graphs without k-cliques

In 2016, Dowden initiated the study of planar Tur\'an-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 $n \ge 10$.

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arXiv2026

A Recursive Characterization of Laplacian Spectral Radii of Trees with Bounded Maximum Degree

For a positive integer $r$ and a real number $\alpha\ge 2$, let $\mathscr L_r(\alpha)$ be the set of positive real numbers containing $\alpha-1$ and closed under the following operation: if $q_1,\ldots,q_s\in\mathscr L_r(\alpha)$, where $1\leq s\leq r-1$, and $q=\alpha-1-s-\sum\limits_{i=1}^{s}q_i^{-1}>0$, then $q\in\mathscr L_r(\alpha)$. We prove that there exists a tree $T$ with $\Delta(T)\leq r$ and Laplacian spectral radius $\mu(T)=\alpha$ if and only if $(\alpha-1)^{-1}\in\mathscr L_r(\alpha)$. Consequently, the Laplacian spectral radii of nontrivial trees are precisely the real numbers $\alpha\geq2$ satisfying $(\alpha-1)^{-1}\in\mathscr L_{\lfloor\alpha\rfloor-1}(\alpha)$. As an application, we prove that, for integers $k\geq2$ and $r\geq2$, a tree $T$ with $\mu(T)=k^2$ and $\Delta(T)=r$ exists if and only if $(k-1)^2+2\leq r\leq k^2-1$.

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arXiv2026

Every 3-connected $\{K_{1,4},K_{1,4}+e\}$-free split graph of order at least 13 is Hamilton-connected

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. Ryj\'{a}\v{c}ek 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 Ryj\'{a}\v{c}ek et al.'s conjecture holds for split graphs of order at least $13$.

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arXiv2026

DP color functions versus chromatic polynomials for hypergraphs (I)

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 $k\geq N$.

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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.

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arXiv2025

The minimum crossing number and minimum size of maximal 1-plane graphs with given connectivity

A 1-planar graph is a graph which has a drawing on the plane such that each edge is crossed at most once. If a 1-planar graph is drawn in that way, the drawing is called a {\it 1-plane graph}. A graph is maximal 1-plane (or 1-planar) if no additional edge can be added without violating 1-planarity or simplicity. It is known that any maximal 1-plane graph is $k$-connected for some $k$ with $2\le k\le 7$. Recently, Huang et al. proved that any maximal 1-plane graph with $n$ ($\ge 5$) vertices has at least $\lceil\frac{7}{3}n\rceil-3$ edges, which is tight for all integers $n\ge 5$. In this paper, we study $k$-connected maximal 1-plane graphs for each $k$ with $3\le k\le 7$, and establish a lower bound for their crossing numbers and a lower bound for their edge numbers, respectively.

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arXiv2025

Trees with Prescribed Maximum Degree and Spectral Radius

It is well known that the spectral radius $\rho(T)$ of a tree $T$ with at least $3$ vertices satisfies $\frac 14\rho(T)^2+1<\Delta(T)\le \rho(T)^2$, where $\Delta(T)$ 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 $\alpha\in \mathbb{P}$ and every integer $r$ satisfying $\frac 14\alpha^2+1<r\le \alpha^2$, there exists a tree $T$ such that $\Delta(T)=r$ and $\rho(T)=\alpha$. For any positive integer $r$ and positive real number $\alpha$, define ${\mathscr W}_r(\alpha)$ recursively as follows. Initially, $\alpha\in {\mathscr W}_r(\alpha)$. Next, for any multiset $\left \{q_i: 1\le i\le s \right \}$ of positive elements of ${\mathscr W}_r(\alpha)$ with $q:=\alpha-\sum\limits_{i=1}^sq_i^{-1}\ge 0$, if either $s<r$ and $q\ge 0$, or $s=r$ and $q=0$, then $q\in {\mathscr W}_r(\alpha)$. We prove that $0\in {\mathscr W}_r(\alpha)$ if and only if there exists a tree $T$ with $\Delta(T)\le r$ and $\rho(T)=\alpha$. Consequently, $\mathbb{P}$ is exactly the set of positive numbers $\alpha$ such that $0\in {\mathscr W}_{\lfloor\alpha^2\rfloor}(\alpha)$. As an application, we show that for integers $k,r\ge 2$, there exists a tree $T$ with $\Delta(T)=r$ and $\rho(T)=\sqrt k$ if and only if $\frac 14 k+1<r\le k$.

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arXiv2025

DP color functions of hypergraphs

In this article, we introduce the DP color function of a hypergraph, based on the DP coloring introduced by Bernshteyn and Kostochka, which is the minimum value where the minimum is taken over all its k-fold covers. It is an extension of its chromatic polynomial. we obtain an upper bound for the DP color functions of hypergraphs when hypergraphs are connected r-uniform hypergraphs for any r greater than one. The upper bound is attained if and only if the hypergraph is a r-uniform hypertree. We also show the cases of the DP color function equal to its chromatic polynomial. These conclusions coincide with the known results of graphs.

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arXiv2025

Determining the minimum size of maximal 1-plane graphs

A 1-plane graph is a graph together with a drawing in the plane in such a way that each edge is crossed at most once. A 1-plane graph is maximal if no edge can be added without violating either 1-planarity or simplicity. Let $m(n)$ denote the minimum size of a maximal $1$-plane graph of order $n$. Brandenburg et al. established that $m(n)\ge 2.1n-\frac{10}{3}$ for all $n\ge 4$, which was improved by Bar\'{a}t and T\'{o}th to $m(n)\ge \frac{20}{9}n-\frac{10}{3}$. In this paper, we confirm that $m(n)=\left\lceil\frac{7}{3}n\right\rceil-3$ for all $n\ge 5$.

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arXiv2025

A study on $T$-equivalent graphs

In his article [J. Comb. Theory Ser. B 16 (1974), 168-174], Tutte called two graphs $T$-equivalent (i.e., codichromatic) if they have the same Tutte polynomial and showed that graphs $G$ and $G'$ are $T$-equivalent if $G'$ is obtained from $G$ by flipping a rotor (i.e., replacing it by its mirror) of order at most $5$, where a rotor of order $k$ in $G$ is an induced subgraph $R$ having an automorphism $\psi$ with a vertex orbit $\{\psi^i(u): i\ge 0\}$ of size $k$ such that every vertex of $R$ is only adjacent to vertices in $R$ unless it is in this vertex orbit. In this article, we first show the above result due to Tutte can be extended to a rotor $R$ of order $k\ge 6$ if the subgraph of $G$ induced by all those edges of $G$ which are not in $R$ satisfies certain conditions. Also, we provide a new method for generating infinitely many non-isomorphic $T$-equivalent pairs of graphs.

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arXiv2025

The density of maximal IC-plane graphs and maximal NIC-plane graphs

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 $n$.

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arXiv2025

Partial domination of middle graphs

For any graph $G=(V,E)$, a subset $S\subseteq V$ is called {\it an isolating set} of $G$ if $V\setminus N_G[S]$ is an independent set of $G$, where $N_G[S]=S\cup N_G(S)$, and {\it the isolation number} of $G$, denoted by $\iota(G)$, is the size of a smallest isolating set of $G$. In this article, we show that the isolation number of the middle graph of $G$ is equal to the size of a smallest maximal matching of $G$.

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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 $G$.

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