arXiv subjects

Fengming Dong

Publications and source records attributed to Fengming Dong.

At least 37 records · Page 2Linked to original sources

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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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 $ι(G)$, 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

Comparing list-color functions of uniform hypergraphs with their chromatic polynomials

In [J. Combin. Theory Ser. B 161 (2023), 109--119], the authors showed that the list-color function $P_l(G,k)$ of any simple graph $G$ of size $m$ coincides with its chromatic polynomial $P(G,k)$ for all integers $k\ge m-1$. In this article, we extend this conclusion to any uniform hypergraph. Furthermore, we show that for any $r$-uniform hypergraph ${\cal H}=(V,E)$, where $r\ge 2$, $P({\cal H}, L)-P({\cal H},k)\ge (k-|E|+1)k^{|V|-r-1}\sum\limits_{e\in E}\left (k-\left|\bigcap\limits_{v\in e}L(v)\right|\right )$ holds for all integers $k$ with $k\ge |E|-1\ge 4$ and all $k$-assignments $L$ of ${\cal H}$, where $P({\cal H}, L)$ is the number of $L$-colorings of ${\cal H}$.

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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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arXiv2024

Comparing list-color functions of uniform hypergraphs with their chromatic polynomials (III)

For a hypergraph ${\cal H}$, let $P({\cal H},k)$ and $P_l({\cal H},k)$ be its chromatic polynomial and list-color function respectively, and let $τ'({\cal H})$ be the least non-negative integer $q$ such that $P({\cal H},k)=P_l({\cal H},k)$ holds for all integers $k\ge q$. In this article, we show that for any $r$-uniform hypergraph ${\cal H}$ of order $n$ and size $m$ and any $k$-assignment $L$ of ${\cal H}$, where $r\ge 3$, $P({\cal H},L)-P({\cal H},k)\ge \min \{0.02k, k-(m-1)\} k^{n-r-1}\sum_{e\in E({\cal H})} \left ( k-\left |\bigcap_{v\in e}L(v)\right | \right )$ holds for $k\ge m-1\ge 4$. It follows that $τ'({\cal H})\le m-1$, improving the current best result on $τ'({\cal H})$.

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arXiv2024

4-connected 1-planar chordal graphs are Hamiltonian-connected

Tutte proved that 4-connected planar graphs are Hamiltonian. It is unknown if there is an analogous result on 1-planar graphs. In this paper, we characterize 4-connected 1-planar chordal graphs, and show that all such graphs are Hamiltonian-connected. A crucial tool used in our proof is a characteristic of 1-planar 4-trees.

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arXiv2024

Partial-dual polynomial as a framed weight system

Recently, Chmutov proved that the partial-dual polynomial considered as a function on chord diagrams satisfies the four-term relations. In this paper, we show that this function on framed chord diagrams also satisfies the four-term relations, i.e., is a framed weight system.

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arXiv2024

A new infinite family of 4-regular crossing-critical graphs

A graph $G$ is said to be crossing-critical if $cr(G-e)< cr(G)$ for every edge $e$ of $G$, where $cr(G)$ is the crossing number of $G$. Richter and Thomassen [Journal of Combinatorial Theory, Series B 58 (1993), 217-224] constructed an infinite family of 4-regular crossing-critical graphs with crossing number $3$. In this article, we present a new infinite family of 4-regular crossing-critical graphs.

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arXiv2023

On the colorability of bi-hypergraphs

A {\it mixed hypergraph} ${\cal H}=({\cal V},{\cal C},{\cal D})$ consists of the vertex set ${\cal V}$ and two families of subsets of $2^{\cal V}$: the family ${\cal C}$ of co-edges and the family ${\cal D}$ of edges. ${\cal H}$ is said to be colorable if there is a mapping $f$ from ${\cal V}$ to the set of positive integers such that $|\{f(v):v\in e\}|<|e|$ for each $e\in {\cal C}$ and $|\{f(v):v\in e\}|>1$ for each $e\in {\cal D}$. There exist mixed hypergraphs which are uncolorable, and quite little about these mixed hypergraphs is known. A mixed hypergraph is called a bi-hypergraph if its co-edge set and edge set are the same. In this article, we first apply Lovász local lemma to show that any $r$-uniform bi-hypergraph with $r\ge 4$ is colorable if every edge is incident to less than $(r-1)^{r-1}e^{-1}-1$ other edges, where $e$ is the base of natural logarithms. Then, we show that among all the uncolorable $3$-uniform bi-hypergraphs, the smallest size of a minimal one is ten, which answers a question raised by Tuza and Voloshin in 2000. As an extension, we provide a minimal uncolorable $3$-uniform bi-hypergraph of order $n$ and size at most $\frac{7n}3-4$ for every $n\ge 6$.

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arXiv2023

On the maximum local mean order of sub-k-trees of a k-tree

For a k-tree T, a generalization of a tree, the local mean order of sub-k-trees of T is the average order of sub-k-trees of T containing a given k-clique. The problem whether the largest local mean order of a tree (i.e., a 1-tree) at a vertex always takes on at a leaf was asked by Jamison in 1984 and was answered by Wagner and Wang in 2016. In 2018, Stephens and Oellermann asked a similar problem: for any k-tree T, does the maximum local mean order of sub-k-trees containing a given k-clique occur at a k-clique that is not a major k-clique of T? In this paper, we give it an affirmative answer.

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arXiv2023

The absolute values of the perfect matching derangement graph's eigenvalues almost follow the lexicographic order of partitions

In 2013, Ku and Wong showed that for any partitions $μ$ and $μ'$ of a positive integer $n$ with the same first part $u$ and the lexicographic order $μ\triangleleft μ'$, the eigenvalues $ξ_μ$ and $ξ_{μ'}$ of the derangement graph $Γ_n$ have the property $|ξ_μ|\le |ξ_{μ'}|$, where the equality holds if and only if $u=3$ and all other parts are less than $3$. In this article, we obtain an analogous conclusion on the eigenvalues of the perfect matching derangement graph $\mathcal{M}_{2n}$ of $K_{2n}$ by finding a new recurrence formula for the eigenvalues of $\mathcal{M}_{2n}$.

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arXiv2023

An improved lower bound of $P(G, L)-P(G,k)$ for $k$-assignments $L$

Let $G=(V,E)$ be a simple graph with $n$ vertices and $m$ edges, $P(G,k)$ be the chromatic polynomial of $G$, and $P(G,L)$ be the number of $L$-colorings of $G$ for any $k$-assignment $L$. In this article, we show that when $k\ge m-1\ge 3$, $P(G,L)-P(G,k)$ is bounded below by $\left ( (k-m+1)k^{n-3}+\frac{(k-m+3)c}{3} k^{n-5}\right )\sum\limits_{uv\in E}|L(u)\setminus L(v)|$, where $c\ge \frac{(m-1)(m-3)}{8}$, and in particular, if $G$ is $K_3$-free, then $c\ge {m-2\choose 2}+2\sqrt m-3$. Consequently, $P(G,L)\ge P(G,k)$ whenever $k\ge m-1$.

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arXiv2023

Compare list-color functions of uniform hypergraphs with their chromatic polynomials (II)

For any $r$-uniform hypergraph $\mathcal{H}$ with $m$ ($\geq 2$) edges, let $P(\mathcal{H},k)$ and $P_l(\mathcal{H},k)$ be the chromatic polynomial and the list-color function of $\mathcal{H}$ respectively, and let $ρ(\mathcal{H})$ denote the minimum value of $|e\setminus e'|$ among all pairs of distinct edges $e,e'$ in $\mathcal{H}$. We will show that if $r\ge3$, $ρ(\mathcal{H})\ge 2$ and $m\ge \frac{ρ(\mathcal{H})^3}2+1$, then $P_l(\mathcal{H},k)=P(\mathcal{H},k)$ holds for all integers $k\geq \frac{2.4(m-1)}{ρ(\mathcal{H})\log(m-1)}$.

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arXiv2022

Proving identities on weight polynomials of tiered trees via Tutte polynomials

A {\it tiered graph} $G=(V,E)$ with $m $ tiers is a simple graph with $V\subseteq \brk{n}$, where $\brk{n}=\{1,2,\cdots,n\}$, and with a surjective map $t$ from $V$ to $\brk{m}$ such that if $v$ is a vertex adjacent to $v'$ in $G$ with $v>v'$, then $t(v) >t(v')$. For any ordered partition $p=(p_1,p_2,\cdots,p_m)$ of $n$, let $\sett_p$ denote the set of tiered trees with vertex set $\brk{n}$ and with a map $t: \brk{n}\rightarrow \brk{m}$ such that $|t^{-1}(i)|=p_i$ for all $i=1,2,\ldots,m$. For any $T\in \sett_p$, let $K_T$ denote the complete tiered graph whose vertex set and tiering map are the same as those of $T$. If the edges of $K_T$ are ordered lexicographically by their endpoints, then the weight $w(T)$ of $T$ is the external activity of $T$ in $K_T$, i.e., the number of edges $e\in E(K_{T})\setminus E(T)$ such that $e$ is the least element in the unique cycle determined by $T\cup e$. Let $P_p(q)=\sum_{T\in \sett_{p}}q^{w(T)}$. Dugan, Glennon, Gunnells and Steingrímsson [J. Combin. Theory, Ser. A 164 (2019) pp. 24-49] asked for an elementary proof of the identity $P_p(q)=P_{π(p)}(q)$ for any permutation $π$ of $1,2,\cdots,m$, where $π(p)=p_{π(1)},p_{π(2)},\cdots,p_{π(m)})$. In this article, we will prove an extension of this identity by applying Tutte polynomials. Furthermore, we also provide a proof of the identity $P_{(1,p_1,p_2)}(q)=P_{(p_1+1,p_2+1)}(q)$ via Tutte polynomials.

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