arXiv subjects

Fengming Dong

Publications and source records attributed to Fengming Dong.

At least 19 recordsLinked to original sources

arXiv2026

A Survey of Graph Polynomials Related to the Chromatic Polynomial

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.

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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 $Δ\ge 3$, 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.

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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 $α\ge 2$, 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 $(k-1)^2+2\leq r\leq k^2-1$.

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arXiv2026

Trees with Prescribed Maximum Degree and Spectral Radius

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 $α\in \mathbb{P}$ 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 any positive integer $r$ and positive real number $α$, define ${\mathscr W}_r(α)$ recursively as follows. Initially, $α\in {\mathscr W}_r(α)$. Next, for any multiset $\left \{q_i: 1\le i\le s \right \}$ of positive elements of ${\mathscr W}_r(α)$ with $q:=α-\sum\limits_{i=1}^sq_i^{-1}\ge 0$, if either $s<r$ and $q\ge 0$, or $s=r$ and $q=0$, then $q\in {\mathscr W}_r(α)$. We prove that $0\in {\mathscr W}_r(α)$ if and only if there exists a tree $T$ with $Δ(T)\le r$ and $ρ(T)=α$. Consequently, $\mathbb{P}$ is exactly the set of positive numbers $α$ such that $0\in {\mathscr W}_{\lfloorα^2\rfloor}(α)$. 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 $\frac 14 k+1<r\le k$.

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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 $F(G,λ)$ 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 $\{1,2,3\}$.

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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

Characterization of Equimatchable Even-Regular Graphs

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.

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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 $τ(G)$, 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.

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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 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 $n \ge 10$.

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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. 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 $13$.

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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

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

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

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 Barát and Tóth 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 $ψ$ with a vertex orbit $\{ψ^i(u): i\ge 0\}$ 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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