arXiv subjects

Hongliang Lu

Publications and source records attributed to Hongliang Lu.

At least 91 records · Page 5Linked to original sources

arXiv2019

Binding Number, Toughness and General Matching Extendability in Graphs

A connected graph $G$ with at least $2m + 2n + 2$ vertices which contains a perfect matching is $E(m, n)$-{\it extendable}, if for any two sets of disjoint independent edges $M$ and $N$ with $|M| = m$ and $|N|= n$, there is a perfect matching $F$ in $G$ such that $M\subseteq F$ and $N\cap F=\emptyset$. Similarly, a connected graph with at least $n+2k+2$ vertices is called $(n,k)$-{\it extendable} if for any vertex set $S$ of size $n$ and any matching $M$ of size $k$ of $G-S$, $G-S-V(M)$ contains a perfect matching. Let $\varepsilon$ be a small positive constant, $b(G)$ and $t(G)$ be the binding number and toughness of a graph $G$. The two main theorems of this paper are: for every graph $G$ with sufficiently large order, 1) if $b(G)\geq 4/3+\varepsilon$, then $G$ is $E(m,n)$-extendable and also $(n,k)$-extendable; 2) if $t(G)\geq 1+\varepsilon$ and $G$ has a high connectivity, then $G$ is $E(m,n)$-extendable and also $(n,k)$-extendable. It is worth to point out that the binding number and toughness conditions for the existence of the general matching extension properties are almost same as that for the existence of perfect matchings.

math.CO↗

arXiv2018

The Tutte's condition in terms of graph factors

Let $G$ be a connected general graph of even order, with a function $f\colon V(G)\to\Z^+$. We obtain that $G$ satisfies the Tutte's condition \[ o(G-S)\le \sum_{v\in S}f(v)\qquad\text{for any nonempty set $S\subset V(G)$}, \] with respect to $f$ if and only if $G$ contains an $H$-factor for any function $H\colon V(G)\to 2^\N$ such that $H(v)\in \{J_f(v),\,J_f^+(v)\}$ for each $v\in V(G)$, where the set $J_f(v)$ consists of the integer $f(v)$ and all positive odd integers less than $f(v)$, and the set $J^+_f(v)$ consists of positive odd integers less than or equal to $f(v)+1$. We also obtain a characterization for graphs of odd order satisfying the Tutte's condition with respect to a function.

math.CO↗

arXiv2018

The Complexity of All $(g,f)$-Factor Problem

Let $G$ be a graph with vertex set $V$ and let $g, f : V\rightarrow \mathbb{Z}^+$ be two functions such that $g\le f$. We say that $G$ has all $(g, f )$-factors if $G$ has an $h$-factor for every $h: V\rightarrow \mathbb{Z}^+$ such that $g(v)\le h(v)\le f (v)$ for every $v\in V$ and $\sum_{v\in{V}}h(v)\equiv 0\pmod 2$. Two decades ago, Niessen derived from Tutte's $f$-factor theorem a similar characterization for the property of graphs having all $(g, f )$-factors and asked whether there is a polynomial time algorithm for testing whether a graph $G$ has all $(g, f )$-factors (A characterization of graphs having all $(g, f )$-Factors, \emph{J. Combin. Theory, Ser. B}, \textbf{72} (1998), 152--156). In this paper, we show that it is NP-hard to determine whether a graph $G$ has all $(g,f)$-factors, which gives a negative answer to the question of Niessen.

math.CO↗

arXiv2018

Theoretical investigation of novel electronic, optical, mechanical and thermal properties of metallic hydrogen at 495 GPa

Atomic metallic hydrogen has been produced in the laboratory at high pressure and low temperature, prompting further investigations of its different properties. However, purely experimental approaches are infeasible because of the extreme requirements in producing and preserving the metastable phase. Here we perform a systematic investigation of the electronic, optical, mechanical and thermal properties of $I4_1/amd$ hydrogen at 495 GPa using first-principles calculations. We calculate the electronic structure and dielectric function to verify the metallic behaviour of $I4_1/amd$ hydrogen. The calculated total plasma frequency from both intraband and interband transitions, 33.40 eV, agrees well with the experimental result. The mechanical properties including elastic stability and sound velocity are also investigated. The mechanical stability of $I4_1/amd$ hydrogen is limited by shear modulus other than bulk modulus, and the high Young's modulus indicates that $I4_1/amd$ hydrogen is a stiff material. After investigating the lattice vibrational properties, we study the thermodynamical properties and lattice anharmonicity to understand thermal behaviours in metallic hydrogen. Finally, the lattice thermal conductivity of $I4_1/amd$ hydrogen is calculated to be 194.72 W/mK and 172.96 W/mK along the $x$ and $z$ directions, respectively. Using metallic hydrogen as an example, we demonstrate that first-principles calculations can be a game-changing solution to understand a variety of material properties in extreme conditions.

cond-mat.mtrl-sci↗

arXiv2018

Electronic, optical and transport properties of van der Waals Transition-metal Dichalcogenides Heterostructures: A First-principle Study

Two-dimensional (2D) transition-metal dichalcogenide (TMD) MX$_2$ (M = Mo, W; X= S, Se, Te) possess unique properties and novel applications. In this work, we perform first-principles calculations on the van der Waals (vdW) stacked MX$_2$ heterostructures to investigate their electronic, optical and transport properties systematically. We perform the so-called Anderson's rule to classify the heterostructures by providing the scheme of the construction of energy band diagrams for the heterostructure consisting of two semiconductor materials. For most of the MX$_2$ heterostructures, the conduction band maximum (CBM) and valence band minimum (VBM) reside in two separate semiconductors, forming type II band structure, thus the electron-holes pairs are spatially separated. We also find strong interlayer coupling at $Γ$ point after forming MX$_2$ heterostructures, even leading to the indirect band gap. While the band structure near $K$ point remain as the independent monolayer. The carrier mobilities of MX$_2$ heterostructures depend on three decisive factors, elastic modulus, effective mass and deformation potential constant, which are discussed and contrasted with those of monolayer MX$_2$, respectively.

cond-mat.mtrl-sci↗

arXiv2017

Minimum co-degree condition for perfect matchings in k-partite k-graphs

Let $H$ be a $k$-partite $k$-graph with $n$ vertices in each partition class, and let $δ_{k-1}(H)$ denote the minimum co-degree of $H$. We characterize those $H$ with $δ_{k-1}(H) \geq n/2$ and with no perfect matching. As a consequence we give an affirmative answer to the following question of Rödl and Ruciński: If $k$ is even or $n \not\equiv 2 \pmod 4$, does $δ_{k-1}(H) \geq n/2$ imply that $H$ has a perfect matching? We also give an example indicating that it is not sufficient to impose this degree bound on only two types of $(k-1)$-sets.

math.CO↗

arXiv2017

$H$-Decomposition of $r$-graphs when $H$ is an $r$-graph with exactly $k$ independent edges

Let $ϕ_H^r(n)$ be the smallest integer such that, for all $r$-graphs $G$ on $n$ vertices, the edge set $E(G)$ can be partitioned into at most $ϕ_H^r(n)$ parts, of which every part either is a single edge or forms an $r$-graph isomorphic to $H$. The function $ϕ^2_H(n)$ has been well studied in literature, but for the case $r\ge 3$, the problem that determining the value of $ϕ_H^r(n)$ is widely open. Sousa (2010) gave an asymptotic value of $ϕ_H^r(n)$ when $H$ is an $r$-graph with exactly 2 edges, and determined the exact value of $ϕ_H^r(n)$ in some special cases. In this paper, we first give the exact value of $ϕ_H^r(n)$ when $H$ is an $r$-graph with exactly 2 edges, which improves Sousa's result. Second we determine the exact value of $ϕ_H^r(n)$ when $H$ is an $r$-graph consisting of exactly $k$ independent edges.

math.CO↗

arXiv2017

Almost perfect matchings in $k$-partite $k$-graphs

The minimum co-degree threshold for a perfect matching in a $k$-graph with $n$ vertices was determined by Rödl, Ruciński and Szemerédi for the case when $n\equiv 0\pmod k$. Recently, Han resolved the remaining cases when $n \not\equiv 0\pmod k$, establishing a conjecture of Rödl, Ruciński and Szemerédi. In this paper, we determine the minimum co-degree threshold for almost perfect matchings in $k$-partite $k$-graphs, answering a question of Rödl and Ruciński.

math.CO↗

arXiv2017

Electronic, transport and optical properties of monolayer $α$ and $β-$GeSe: A first-principles study

The extraordinary properties and the novel applications of black phosphorene induce the research interest on the monolayer group-IV monochalcogenides. Here using the first-principles calculations, we systematically investigate the electronic, transport and optical properties of monolayer $α-$ and $β-$GeSe, the latter of which was recently experimentally realized. We found that, monolayer $α-$GeSe is a semiconductor with direct band gap of 1.6 eV, and $β-$GeSe displays indirect semiconductor with the gap of 2.47 eV, respectively. For monolayer $β-$GeSe, the electronic/hole transport is anisotropic with an extremely high electron mobility of 7.84 $\times10^4$$cm^2/V\cdot {s}$ along the zigzag direction, comparable to that of black phosphorene. Furthermore, for $β-$GeSe, robust band gaps nearly disregarding the applied tensile strain along the zigzag direction is observed. Both monolayer $α-$ and $β-$GeSe exhibit anisotropic optical absorption in the visible spectrum.

cond-mat.mtrl-sci↗

arXiv2017

Atomically sharp 1D SbSeI, SbSI and SbSBr with high stability and novel properties for microelectronic, optoelectronic, and thermoelectric applications

In scaling of transistor dimensions with low source-to-drain currents, 1D semiconductors with certain electronic properties are highly desired. We discover three new 1D materials, SbSeI, SbSI and SbSBr with high stability and novel electronic properties based on first principles calculations. Both dynamical and thermal stability of these 1D materials are examined. The bulk-to-1D transition results in dramatic changes in band gap, effective mass and static dielectric constant due to quantum confinement, making 1D SbSeI a highly promising channel material for transistors with gate length shorter than 1 nm. Under small uniaxial strain, these materials are transformed from indirect into direct band gap semiconductors, paving the way for optoelectronic devices and mechanical sensors. Moreover, the thermoelectric performance of these materials is significantly improved over their bulk counterparts. Finally, we demonstrate the experimental feasibility of synthesizing such atomically sharp V-VI-VII compounds. These highly desirable properties render SbSeI, SbSI and SbSBr promising 1D materials for applications in future microelectronics, optoelectronics, mechanical sensors, and thermoelectrics.

cond-mat.mtrl-sci↗

arXiv2016

On rainbow matchings for hypergraphs

For any posotive integer $m$, let $[m]:=\{1,\ldots,m\}$. Let $n,k,t$ be positive integers. Aharoni and Howard conjectured that if, for $i\in [t]$, $\mathcal{F}_i\subset[n]^k:= \{(a_1,\ldots,a_k): a_j\in [n] \mbox{ for } j\in [k]\}$ and $|\mathcal{F}_i|>(t-1)n^{k-1}$, then there exist $M\subseteq [n]^k$ such that $|M|=t$ and $|M\cap \mathcal{F}_i|=1$ for $i\in [t]$ We show that this conjecture holds when $n\geq 3(k-1)(t-1)$. Let $n, t, k_1\ge k_2\geq \ldots\geq k_t $ be positive integers. Huang, Loh and Sudakov asked for the maximum $Π_{i=1}^t |{\cal R}_i|$ over all ${\cal R}=\{{\cal R}_1, \ldots ,{\cal R}_t\}$ such that each ${\cal R}_i$ is a collection of $k_i$-subsets of $[n]$ for which there does not exist a collection $M$ of subsets of $[n]$ such that $|M|=t$ and $|M\cap \mathcal{R}_i|=1$ for $i\in [t]$ %and ${\cal R}$ does not admit a rainbow matching. We show that for sufficiently large $n$ with $\sum_{i=1}^t k_i\leq n(1-(4k\ln n/n)^{1/k}) $, $\prod_{i=1}^t |\mathcal{R}_i|\leq {n-1\choose k_1-1}{n-1\choose k_2-1}\prod_{i=3}^{t}{n\choose k_i}$. This bound is tight.

math.CO↗

arXiv2016

Stability and Strength of Atomically Thin Borophene from First Principles Calculations

A new two-dimensional (2D) material, borophene (2D boron sheet), has been grown successfully recently on single crystal Ag substrates by two parallel experiments [Mannix \textit{et al., Science}, 2015, \textbf{350}, 1513] [Feng \textit{et al., Nature Chemistry}, 2016, \textbf{advance online publication}]. Three main structures have been proposed ($β_{12}$, $χ_3$ and striped borophene). However, the stability of three structures is still in debate. Using first principles calculations, we examine the dynamical, thermodynamical and mechanical stability of $β_{12}$, $χ_3$ and striped borophene. Free-standing $β_{12}$ and $χ_3$ borophene is dynamically, thermodynamically, and mechanically stable, while striped borophene is dynamically and thermodynamically unstable due to high stiffness along $a$ direction. The origin of high stiffness and high instability in striped borophene along $a$ direction can both be attributed to strong directional bonding. This work provides a benchmark for examining the relative stability of different structures of borophene.

cond-mat.mtrl-sci↗

arXiv2016

A Degree Condition for a Graph to have $(a,b)$-Parity Factors

Let $a,b,n$ be three positive integers such that $a\equiv b\pmod 2$ and $n\geq b(a+b)(a+b+2)/(2a)$. Let $G$ be a graph of order $n$ with minimum degree at least $a+b/a-1$. We show that $G$ has an $(a,b)$-parity factor, if $max\{d_G(u),d_G(v)\}\geq \frac{an}{a+b}$ for any two nonadjacent vertices $u,v$ of $G$. It is an extension of Nishimura's results for the existence of $k$-factors (\emph{J. Graph Theory}, \textbf{16} (1992), 141--151) and generalizes Li and Cai's result in some senses (\emph{J. Graph Theory}, \textbf{27} (1998), 1--6). These conditions are tight.

math.CO↗

arXiv2016

First-principle calculations of phononic, electronic and optical properties of monolayer arsenene and antimonene allotropes

Recently a stable monolayer of antimony in buckled honeycomb structure called antimonene was successfully grown on 3D topological insulator Bi$_2$Te$_3$ and Sb$_2$Te$_3$, which displays semiconducting properties. By first principle calculations, we systematically investigate the phononic, electronic and optical properties of $α-$ and $β-$ allotropes of monolayer arsenene/antimonene. We investigate the dynamical stabilities of these four materials by considering the phonon dispersions. The obtained electronic structures reveal the direct band gap of monolayer $α-$As/Sb and indirect band gap of $β-$As/Sb. Significant absorption is observed in $α-$Sb, which can be used as a broad saturable absorber.

cond-mat.mtrl-sci↗

arXiv2015

A Tutte-type characterization for graph factors

Let $G$ be a connected general graph. Let $f\colon V(G)\to \Z^+$ be a function. We show that $G$ satisfies the Tutte-type condition \[ o(G-S)\le f(S)\qquad\text{for all vertex subsets $S$}, \] if and only if it contains a colored $J_f^*$-factor for any $2$-end-coloring, where $J_f^*(v)$ is the union of all odd integers smaller than $f(v)$ and the integer $f(v)$ itself. This is a generalization of the $(1,f)$-odd factor characterization theorem, and answers a problem of Cui and Kano. We also derive an analogous characterization for graphs of odd orders, which addresses a problem of Akiyama and Kano.

math.CO↗

arXiv2015

The maximum number of perfect matchings of semi-regular graphs

Let $n\ge 34$ be an even integer, and $D_n=2\lceil n/4 \rceil-1$. In this paper, we prove that every $\{D_n,\,D_n+1\}$-graph of order $n$ contains $\lceil n/4 \rceil$ disjoint perfect matchings. This result is sharp in the sense that (i) there exists a $\{D_n,\,D_n+1\}$-graph containing exactly $\lceil n/4 \rceil$ disjoint perfect matchings, and that (ii) there exists a $\{D_n-1,\,D_n\}$-graph without perfect matchings for each $n$. As a consequence, for any integer $D\ge D_n$, every $\{D,\,D+1\}$-graph of order $n$ contains $\lceil (D+1)/2 \rceil$ disjoint perfect matchings. This extends Csaba et~al.'s breathe-taking result that every $D$-regular graph of sufficiently large order is $1$-factorizable, generalizes Zhang and Zhu's result that every $D_n$-regular graph of order $n$ contains $\lceil n/4 \rceil$ disjoint perfect matchings, and improves Hou's result that for all $k\ge n/2$, every $\{k,\,k+1\}$-graph of order $n$ contains $(\lfloor n/3\rfloor+1+k-n/2)$ disjoint perfect matchings.

math.CO↗

arXiv2015

Surface embedding of non-bipartite $k$-extendable graphs

We find the minimum number $k=μ'(Σ)$ for any surface $Σ$, such that every $Σ$-embeddable non-bipartite graph is not $k$-extendable. In particular, we construct the so-called bow-tie graphs $C_6\bowtie P_n$, and show that they are $3$-extendable. This confirms the existence of an infinite number of $3$-extendable non-bipartite graphs which can be embedded in the Klein bottle.

math.CO↗

arXiv2014

Surface embedding of $(n,k)$-extendable graphs

This paper is concerned with the surface embedding of matching extendable graphs. There are two directions extending the theory of perfect matchings, that is, matching extendability and factor-criticality. In solving a problem posed by Plummer, Dean (The matching extendability of surfaces, J. Combin. Theory Ser. B 54 (1992), 133--141) established the fascinating formula for the minimum number $k= μ(Σ) $ such that every $Σ$-embeddable graph is not $k$-extendable. Su and Zhang, Plummer and Zha found the minimum number $n=ρ(Σ)$ such that every $Σ$-embeddable graph is not $n$-factor-critical. Based on the notion of $(n,k)$-graphs which associates these two parameters, we found the formula for the minimum number $k=μ(n,Σ)$ such that every $Σ$-embeddable graph is not an $(n,k)$-graph. To access this two-parameter-problem, we consider its dual problem and find out $μ(n,Σ)$ conversely. The same approach works for rediscovering the formula of the number $ρ(Σ)$.

math.CO↗