arXiv subjects

Weihua Yang

Publications and source records attributed to Weihua Yang.

70 records · Page 4Linked to original sources

arXiv2018

The Turán problem for a family of tight linear forests

Let $\mathcal{F}$ be a family of $r$-graphs. The Turán number $ex_r(n;\mathcal{F})$ is defined to be the maximum number of edges in an $r$-graph of order $n$ that is $\mathcal{F}$-free. The famous Erdős Matching Conjecture shows that \[ ex_r(n,M_{k+1}^{(r)})= \max\left\{\binom{rk+r-1}{r},\binom{n}{r}-\binom{n-k}{r}\right\}, \] where $M_{k+1}^{(r)}$ represents the $r$-graph consisting of $k+1$ disjoint edges. Motivated by this conjecture, we consider the Turán problem for tight linear forests. A tight linear forest is an $r$-graph whose connected components are all tight paths or isolated vertices. Let $\mathcal{L}_{n,k}^{(r)}$ be the family of all tight linear forests of order $n$ with $k$ edges in $r$-graphs. In this paper, we prove that for sufficiently large $n$, \[ ex_r(n;\mathcal{L}_{n,k}^{(r)})=\max\left\{\binom{k}{r}, \binom{n}{r}-\binom{n-\left\lfloor (k-1)/r\right \rfloor}{r}\right\}+d, \] where $d=o(n^r)$ and if $r=3$ and $k=cn$ with $0<c<1$, if $r\geq 4$ and $k=cn$ with $0<c<1/2$. The proof is based on the weak regularity lemma for hypergraphs. We also conjecture that for arbitrary $k$ satisfying $k \equiv 1\ (mod\ r)$, the error term $d$ in the above result equals 0. We prove that the proposed conjecture implies the Erdős Matching Conjecture directly.

math.CO↗

arXiv2018

Anti-$k$-labeling of graphs

It is well known that the labeling problems of graphs arise in many (but not limited to) networking and telecommunication contexts. In this paper we introduce the anti-$k$-labeling problem of graphs which we seek to minimize the similarity (or distance) of neighboring nodes. For example, in the fundamental frequency assignment problem in wireless networks where each node is assigned a frequency, it is usually desirable to limit or minimize the frequency gap between neighboring nodes so as to limit interference. Let $k\geq1$ be an integer and $ψ$ is a labeling function (anti-$k$-labeling) from $V(G)$ to $\{1,2,\cdots,k\}$ for a graph $G$. A {\em no-hole anti-$k$-labeling} is an anti-$k$-labeling using all labels between 1 and $k$. We define $w_ψ(e)=|ψ(u)-ψ(v)|$ for an edge $e=uv$ and $w_ψ(G)=\min\{w_ψ(e):e\in E(G)\}$ for an anti-$k$-labeling $ψ$ of the graph $G$. {\em The anti-$k$-labeling number} of a graph $G$, $mc_k(G)$ is $\max\{w_ψ(G): ψ\}$. In this paper, we first show that $mc_k(G)=\lfloor \frac{k-1}{χ-1}\rfloor$, and the problem that determines $mc_k(G)$ of graphs is NP-hard. We mainly obtain the lower bounds on no-hole anti-$n$-labeling number for trees, grids and $n$-cubes.

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arXiv2018

The Turán Number for Spanning Linear Forests

For a set of graphs $\mathcal{F}$, the extremal number $ex(n;\mathcal{F})$ is the maximum number of edges in a graph of order $n$ not containing any subgraph isomorphic to some graph in $\mathcal{F}$. If $\mathcal{F}$ contains a graph on $n$ vertices, then we often call the problem a spanning Turán problem. A linear forest is a graph whose connected components are all paths and isolated vertices. In this paper, we let $\mathcal{L}_n^k$ be the set of all linear forests of order $n$ with at least $n-k+1$ edges. We prove that when $n\geq 3k$ and $k\geq 2$, \[ ex(n;\mathcal{L}_n^k)=\binom{n-k+1}{2}+ O(k^2). \] Clearly, the result is interesting when $k=o(n)$.

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arXiv2018

Minimum degree and size conditions for the proper connection number of graphs

An edge-coloured graph $G$ is called $properly$ $connected$ if every two vertices are connected by a proper path. The $proper$ $connection$ $number$ of a connected graph $G$, denoted by $pc(G)$, is the smallest number of colours that are needed in order to make $G$ properly connected. Susan A. van Aardt et al. gave a sufficient condition for the proper connection number to be at most $k$ in terms of the size of graphs. In this note, %optimizes the boundary of the number of edges %we study the $proper$ $connection$ $number$ is under the conditions of adding the minimum degree and optimizing the number of edges. our main result is the following, by adding a minimum degree condition: Let $G$ be a connected graph of order $n$, $k\geq3$. If $|E(G)|\geq \binom{n-m-(k+1-m)(δ+1)}{2} +(k+1-m)\binom{δ+1}{2}+k+2$, then $pc(G)\leq k$, where $m$ takes the value $t$ if $δ=1$ and $\lfloor \frac{k}{δ-1} \rfloor$ if $δ\geq2$. Furthermore, if $k=2$ and $δ=2$, %(i.e., $|E(G)|\geq \binom{n-5}{2} +7$) $pc(G)\leq 2$, except $G\in \{G_{1}, G_{n}\}$ ($n\geq8$), where $G_{1}=K_{1}\vee 3K_{2}$ and $G_{n}$ is obtained by taking a complete graph $K_{n-5}$ and $K_{1}\vee (2K_{2}$) with an arbitrary vertex of $K_{n-5}$ and a vertex with $d(v)=4$ in $K_{1}\vee (2K_{2}$) being joined. If $k=2$, $δ\geq 3$, we conjecture $pc(G)\leq 2$, where $m$ takes the value $1$ if $δ=3$ and $0$ if $δ\geq4$ in the assumption.

math.CO↗

arXiv2018

Vertex-disjoint cycles in tournaments

The Bermond-Thomassen conjecture states that, for any positive integer $r$, a digraph of minimum out-degree at least $2r-1$ contains at least $r$ vertex-disjoint directed cycles. Bessy, Sereni and Lichiardopol proved that a regular tournament $T$ of minimum degree $2r-1$ contains at least $r$ vertex-disjoint directed cycles, which shows that the above conjecture is true for tournaments. After that, Lichiardopol improved this result by showing that a $2r-1$-regular tournament contains at least $\frac{7}{6}r-\frac{7}{3}$ vertex-disjoint directed cycles. In this paper, we will extend the result to tournaments with minimum out-degree at least $2r-1$ by proving a more general result.

math.CO↗

arXiv2018

Vertex-disjoint directed cycles of prescribed length in tournaments with given minimum out-degree

The Bermond-Thomassen conjecture states that, for any positive integer $r$, a digraph of minimum out-degree at least $2r-1$ contains at least $r$ vertex-disjoint directed cycles. In 2014, Bang-Jensen, Bessy and Thomass\' e proved the conjecture for tournaments. In 2010, Lichiardopol conjectured that a tournament $T$ with minimum out-degree at least $(q-1)r-1$ contains at least $r$ vertex-disjoint $q$-cycles, where integer $q\geq3$ and $r\geq1$. In this paper, we address Lichiardopol's conjecture affirmatively. In particular, the case $q=3$ implies Bermond-Thomassen conjecture for tournaments.

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arXiv2018

Reliability evaluation of folded hypercubes in terms of component connectivity

The component connectivity is the generalization of connectivity which is an parameter for the reliability evaluation of interconnection networks. The $g$-component connectivity $cκ_{g}(G)$ of a non-complete connected graph $G$ is the minimum number of vertices whose deletion results in a graph with at least $g$ components. The results in [Component connectivity of the hypercubes, International Journal of Computer Mathematics 89 (2012) 137-145] by Hsu et al. determines the component connectivity of the hypercubes. As an invariant of the hypercube, we determine the $(g+1)$-component connectivity of the folded hypercube $cκ_{g}(FQ_{n})=g(n+1)-\frac{1}{2}g(g+1)+1$ for $1\leq g \leq n+1, n\geq 8$ in this paper.

math.CO↗

arXiv2018

Component edge connectivity of the folded hypercube

The $g$-component edge connectivity $cλ_g(G)$ of a non-complete graph $G$ is the minimum number of edges whose deletion results in a graph with at least $g$ components. In this paper, we determine the component edge connectivity of the folded hypercube $cλ_{g+1}(FQ_{n})=(n+1)g-(\sum\limits_{i=0}^{s}t_i2^{t_i-1}+\sum\limits_{i=0}^{s} i\cdot 2^{t_i})$ for $g\leq 2^{[\frac{n+1}2]}$ and $n\geq 5$, where $g$ be a positive integer and $g=\sum\limits_{i=0}^{s}2^{t_i}$ be the decomposition of $g$ such that $t_0=[\log_{2}{g}],$ and $t_i=[\log_2({g-\sum\limits_{r=0}^{i-1}2^{t_r}})]$ for $i\geq 1$.

math.CO↗

arXiv2018

Embedding 5-planar graphs in three pages

A \emph{book-embedding} of a graph $G$ is an embedding of vertices of $G$ along the spine of a book, and edges of $G$ on the pages so that no two edges on the same page intersect. the minimum number of pages in which a graph can be embedded is called the \emph{page number}. The book-embedding of graphs may be important in several technical applications, e.g., sorting with parallel stacks, fault-tolerant processor arrays design, and layout problems with application to very large scale integration (VLSI). Bernhart and Kainen firstly considered the book-embedding of the planar graph and conjectured that its page number can be made arbitrarily large [JCT, 1979, 320-331]. Heath [FOCS84] found that planar graphs admit a seven-page book embedding. Later, Yannakakis proved that four pages are necessary and sufficient for planar graphs in [STOC86]. Recently, Bekos et al. [STACS14] described an $O(n^{2})$ time algorithm of two-page book embedding for 4-planar graphs. In this paper, we embed 5-planar graphs into a book of three pages by an $O(n^{2})$ time algorithm.

math.CO↗

arXiv2017

On the number of proper paths between vertices in edge-colored hypercubes

Given an integer $1\leq j <n$, define the $(j)$-coloring of a $n$-dimensional hypercube $H_{n}$ to be the $2$-coloring of the edges of $H_{n}$ in which all edges in dimension $i$, $1\leq i \leq j$, have color $1$ and all other edges have color $2$. Cheng et al. [Proper distance in edge-colored hypercubes, Applied Mathematics and Computation 313 (2017) 384-391.] determined the number of distinct shortest properly colored paths between a pair of vertices for the $(1)$-colored hypercubes. It is natural to consider the number for $(j)$-coloring, $j\geq 2$. In this note, we determine the number of different shortest proper paths in $(j)$-colored hypercubes for arbitrary $j$.

math.CO↗

arXiv2017

A kind of conditional connectivity of transposition networks generated by $k$-trees

For a graph $G = (V, E)$, a subset $F\subset V(G)$ is called an $R_k$-vertex-cut of $G$ if $G -F$ is disconnected and each vertex $u \in V(G)- F$ has at least $k$ neighbors in $G -F$. The $R_k$-vertex-connectivity of $G$, denoted by $κ^k(G)$, is the cardinality of the minimum $R_k$-vertex-cut of $G$, which is a refined measure for the fault tolerance of network $G$. In this paper, we study $κ^2$ for Cayley graphs generated by $k$-trees. Let $Sym(n)$ be the symmetric group on $\{1, 2, \cdots ,n\}$ and $\mathcal{T}$ be a set of transpositions of $Sym(n)$. Let $G(\mathcal{T})$ be the graph on $n$ vertices $\{1, 2, . . . ,n\}$ such that there is an edge $ij$ in $G(\mathcal{T})$ if and only if the transposition $ij\in \mathcal{T}$. The graph $G(\mathcal{T})$ is called the transposition generating graph of $\mathcal{T}$. We denote by $Cay(Sym(n),\mathcal{T})$ the Cayley graph generated by $G(\mathcal{T})$. The Cayley graph $Cay(Sym(n),\mathcal{T})$ is denoted by $T_kG_n$ if $G(\mathcal{T})$ is a $k$-tree. We determine $κ^2(T_kG_n)$ in this work. The trees are $1$-trees, and the complete graph on $n$ vertices is a $n-1$-tree. Thus, in this sense, this work is a generalization of the such results on Cayley graphs generated by transposition generating trees and the complete-transposition graphs.

math.CO↗

arXiv2017

A note on edge degree and spanning trail containing given edges

Let $G$ be a simple graph with $n\geq4$ vertices and $d(x)+d(y)\geq n+k$ for each edge $xy\in E(G)$. In this work we prove that $G$ either contains a spanning closed trail containing any given edge set $X$ if $|X|\leq k$, or $G$ is a well characterized graph. As a corollary, we show that line graphs of such graphs are $k$-hamiltonian.

math.CO↗

arXiv2017

On the existence of specified cycles in bipartite tournaments

For two integers $n\geq 3$ and $2\leq p\leq n$, we denote $D(n,p)$ the digraph obtained from a directed $n$-cycle by changing the orientations of $p-1$ consecutive arcs. In this paper, we show that a family of $k$-regular $(k\geq 3)$ bipartite tournament $BT_{4k}$ contains $D(4k,p)$ for all $2\leq p\leq 4k$ unless $BT_{4k}$ is isomorphic to a digraph $D$ such that $(1,2,3,...,4k,1)$ is a Hamilton cycle and $(4m+i-1,i)\in A(D)$ and $(i,4m+i+1)\in A(D)$, where $1\leq m\leq k-1$.

math.CO↗

arXiv2017

On the spanning connectivity of tournaments

Let $D$ be a digraph. A $k$-container of $D$ between $u$ and $v$, $C(u,v)$, is a set of $k$ internally disjoint paths between $u$ and $v$. A $k$-container $C(u,v)$ of $D$ is a strong (resp. weak) $k^{*}$-container if there is a set of $k$ internally disjoint paths with the same direction (resp. with different directions allowed) between $u$ and $v$ and it contains all vertices of $D$. A digraph $D$ is $k^{*}$-strongly (resp. $k^{*}$-weakly) connected if there exists a strong (resp. weak) $k^{*}$-container between any two distinct vertices. We define the strong (resp. weak) spanning connectivity of a digraph $D$, $κ_{s}^{*}(D)$ (resp. $κ_{w}^{*}(D)$ ), to be the largest integer $k$ such that $D$ is $ω^{*}$-strongly (resp. $ω^{*}$-weakly) connected for all $1\leq ω\leq k$ if $D$ is a $1^{*}$-strongly (resp. $1^{*}$-weakly) connected. In this paper, we show that a tournament with $n$ vertices and irregularity $i(T)\leq k$, if $n\geq6t+5k$ $(t\geq2)$, then $κ_{s}^{*}(T)\geq t$ and $κ_{w}^{*}(T)\geq t+1$ if $n\geq6t+5k-3$ $(t\geq2)$.

math.CO↗

arXiv2016

Rainbow connection of bridgeless outerplanar graphs with small diameters

In this paper, we investigate rainbow connection number $rc(G)$ of bridgeless outerplanar graphs $G$ with diameter 2 or 3. We proved the following results: If $G$ has diameter $2,$ then $rc(G)=3$ for fan graphs $F_{n}$ with $n\geq 7$ or $C_5,$ otherwise $rc(G)=2;$ if $G$ has diameter $3,$ then $rc(G)\leq 4$ and the bound is sharp.

math.CO↗

arXiv2012

Continuous Dependence of Cauchy Problem For Nonlinear Schrödinger Equation in $H^{s}$

We consider the Cauchy problem for the nonlinear Schrödinger equation $i \partial_{t}u+ Δu=λ_{0}u+λ_{1}|u|^αu$ in $\mathbb{R}^{N}$, where $λ_{0},λ_{1}\in\mathbb{C}$, in $H^s$ subcritical and critical case: $0<α\leq\frac{4}{N-2s}$ when $1<s<\frac{N}{2}$ and $0<α<+\infty$ when $s\geq\frac{N}{2}$. We show that the solution depends continuously on the initial value in the standard sense in $H^{s}(\mathbb{R}^{N})$ if $α$ satisfies certain assumptions.

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