arXiv subjects

Xueliang Li

Publications and source records attributed to Xueliang Li.

At least 91 records · Page 5Linked to original sources

arXiv2016

Degree sum conditions for graphs to have proper connection number 2

A path $P$ in an edge-colored graph $G$ is a \emph{proper path} if no two adjacent edges of $P$ are colored with the same color. The graph $G$ is \emph{proper connected} if, between every pair of vertices, there exists a proper path in $G$. The \emph{proper connection number} $pc(G)$ of a connected graph $G$ is defined as the minimum number of colors to make $G$ proper connected. In this paper, we study the degree sum condition for a general graph or a bipartite graph to have proper connection number 2. First, we show that if $G$ is a connected noncomplete graph of order $n\geq 5$ such that $d(x)+d(y)\geq \frac{n}{2}$ for every pair of nonadjacent vertices $x,y\in V(G)$, then $pc(G)=2$ except for three small graphs on 6, 7 and 8 vertices. In addition, we obtain that if $G$ is a connected bipartite graph of order $n\geq 4$ such that $d(x)+d(y)\geq \frac{n+6}{4}$ for every pair of nonadjacent vertices $x,y\in V(G)$, then $pc(G)=2$. Examples are given to show that the above conditions are best possible.

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arXiv2016

Solutions for two conjectures on kaleidoscopic edge-colorings

For an $r$-regular graph $G$, we define an edge-coloring $c$ with colors from $\{1,2,\cdots,$ $k\}$, in such a way that any vertex of $G$ is incident to at least one edge of each color. The multiset-color $c_m(v)$ of a vertex $v$ is defined as the ordered tuple $(a_1,a_2,\cdots ,a_k)$, where $a_i \ (1\leq i\leq k)$ denotes the number of edges with color $i$ which are incident with $v$ in $G$. Then this edge-coloring $c$ is called a {\it $k$-kaleidoscopic coloring} of $G$ if every two distinct vertices in $G$ have different multiset-colors and in this way the graph $G$ is defined as a {\it $k$-kaleidoscope}. In this paper, we determine the integer $k$ for a complete graph $K_n$ to be a $k$-kaleidoscope, and hence solve a conjecture in [P. Zhang, A Kaleidoscopic View of Graph Colorings, Springer, New York, 2016] that for any integers $n$ and $k$ with $n\geq k+3 \geq 6$, the complete graph $K_n$ is a $k$-kaleidoscope. Then, we construct an $r$-regular $3$-kaleidoscope of order $\binom{r-1}{2}-1$ for each integer $r\geq 7$, where $r\equiv 3\ (\text{mod}\ 4)$, which solves another conjecture in the same book on the maximum order for $r$-regular $3$-kaleidoscopes.

math.CO↗

arXiv2016

3-Rainbow index and forbidden subgraphs

A tree in an edge-colored connected graph $G$ is called \emph{a rainbow tree} if no two edges of it are assigned the same color. For a vertex subset $S\subseteq V(G)$, a tree is called an \emph{$S$-tree} if it connects $S$ in $G$. A \emph{$k$-rainbow coloring} of $G$ is an edge-coloring of $G$ having the property that for every set $S$ of $k$ vertices of $G$, there exists a rainbow $S$-tree in $G$. The minimum number of colors that are needed in a $k$-rainbow coloring of $G$ is the \emph{$k$-rainbow index} of $G$, denoted by $rx_k(G)$. The \emph{Steiner distance $d(S)$} of a set $S$ of vertices of $G$ is the minimum size of an $S$-tree $T$. The \emph{$k$-Steiner diameter $sdiam_k(G)$} of $G$ is defined as the maximum Steiner distance of $S$ among all sets $S$ with $k$ vertices of $G$. In this paper, we focus on the 3-rainbow index of graphs and find all finite families $\mathcal{F}$ of connected graphs, for which there is a constant $C_\mathcal{F}$ such that, for every connected $\mathcal{F}$-free graph $G$, $rx_3(G)\leq sdiam_3(G)+C_\mathcal{F}$.

math.CO↗

arXiv2016

A Source-level Energy Optimization Framework for Mobile Applications

Energy efficiency can have a significant influence on user experience of mobile devices such as smartphones and tablets. Although energy is consumed by hardware, software optimization plays an important role in saving energy, and thus software developers have to participate in the optimization process. The source code is the interface between the developer and hardware resources. In this paper, we propose an energy-optimization framework guided by a source code energy model that allows developers to be aware of energy usage induced by the code and to apply very targeted source-level refactoring strategies. The framework also lays a foundation for the code optimization by automatic tools. To the best of our knowledge, our work is the first that achieves this for a high-level language such as Java. In a case study, the experimental evaluation shows that our approach is able to save from 6.4% to 50.2% of the CPU energy consumption in various application scenarios.

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arXiv2016

Distance proper connection of graphs and their complements

Let $G$ be an edge-colored connected graph. A path $P$ in $G$ is called a distance $\ell$-proper path if no two edges of the same color can appear with less than $\ell$ edges in between on $P$. The graph $G$ is called $(k,\ell)$-proper connected if there is an edge-coloring such that every pair of distinct vertices of $G$ are connected by $k$ pairwise internally vertex-disjoint distance $\ell$-proper paths in $G$. The minimum number of colors needed to make $G$ $(k,\ell)$-proper connected is called the $(k,\ell)$-proper connection number of $G$ and denoted by $pc_{k,\ell}(G)$. In this paper we first focus on the $(1,2)$-proper connection number of $G$ depending on some constraints of $\overline G$. Then, we characterize the graphs of order $n$ with $(1,2)$-proper connection number $n-1$ or $n-2$. Using this result, we investigate the Nordhaus-Gaddum-Type problem of $(1,2)$-proper connection number and prove that $pc_{1,2}(G)+pc_{1,2}(\overline{G})\leq n+2$ for connected graphs $G$ and $\overline{G}$. The equality holds if and only if $G$ or $\overline{G}$ is isomorphic to a double star.

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arXiv2016

Distance proper connection of graphs

Let $G$ be an edge-colored connected graph. A path $P$ in $G$ is called a distance $\ell$-proper path if no two edges of the same color appear with fewer than $\ell$ edges in between on $P$. The graph $G$ is called $(k,\ell)$-proper connected if every pair of distinct vertices of $G$ are connected by $k$ pairwise internally vertex-disjoint distance $\ell$-proper paths in $G$. For a $k$-connected graph $G$, the minimum number of colors needed to make $G$ $(k,\ell)$-proper connected is called the $(k,\ell)$-proper connection number of $G$ and denoted by $pc_{k,\ell}(G)$. In this paper, we prove that $pc_{1,2}(G)\leq 5$ for any $2$-connected graph $G$. Considering graph operations, we find that $3$ is a sharp upper bound for the $(1,2)$-proper connection number of the join and the Cartesian product of almost all graphs. In addition, we find some basic properties of the $(k,\ell)$-proper connection number and determine the values of $pc_{1,\ell}(G)$ where $G$ is a traceable graph, a tree, a complete bipartite graph, a complete multipartite graph, a wheel, a cube or a permutation graph of a nontrivial traceable graph.

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arXiv2016

ENTRA: Whole-Systems Energy Transparency

Promoting energy efficiency to a first class system design goal is an important research challenge. Although more energy-efficient hardware can be designed, it is software that controls the hardware; for a given system the potential for energy savings is likely to be much greater at the higher levels of abstraction in the system stack. Thus the greatest savings are expected from energy-aware software development, which is the vision of the EU ENTRA project. This article presents the concept of energy transparency as a foundation for energy-aware software development. We show how energy modelling of hardware is combined with static analysis to allow the programmer to understand the energy consumption of a program without executing it, thus enabling exploration of the design space taking energy into consideration. The paper concludes by summarising the current and future challenges identified in the ENTRA project.

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arXiv2016

The $(k,\ell)$-proper index of graphs

A tree $T$ in an edge-colored graph is called a {\it proper tree} if no two adjacent edges of $T$ receive the same color. Let $G$ be a connected graph of order $n$ and $k$ be an integer with $2\leq k \leq n$. For $S\subseteq V(G)$ and $|S| \ge 2$, an $S$-tree is a tree containing the vertices of $S$ in $G$. Suppose $\{T_1,T_2,\ldots,T_\ell\}$ is a set of $S$-trees, they are called \emph{internally disjoint} if $E(T_i)\cap E(T_j)=\emptyset$ and $V(T_i)\cap V(T_j)=S$ for $1\leq i\neq j\leq \ell$. For a set $S$ of $k$ vertices of $G$, the maximum number of internally disjoint $S$-trees in $G$ is denoted by $κ(S)$. The $κ$-connectivity $κ_k(G)$ of $G$ is defined by $κ_k(G)=\min\{κ(S)\mid S$ is a $k$-subset of $V(G)\}$. For a connected graph $G$ of order $n$ and for two integers $k$ and $\ell$ with $2\le k\le n$ and $1\leq \ell \leq κ_k(G)$, the \emph{$(k,\ell)$-proper index $px_{k,\ell}(G)$} of $G$ is the minimum number of colors that are needed in an edge-coloring of $G$ such that for every $k$-subset $S$ of $V(G)$, there exist $\ell$ internally disjoint proper $S$-trees connecting them. In this paper, we show that for every pair of positive integers $k$ and $\ell$ with $k \ge 3$, there exists a positive integer $N_1=N_1(k,\ell)$ such that $px_{k,\ell}(K_n) = 2$ for every integer $n \ge N_1$, and also there exists a positive integer $N_2=N_2(k,\ell)$ such that $px_{k,\ell}(K_{m,n}) = 2$ for every integer $n \ge N_2$ and $m=O(n^r) (r \ge 1)$. In addition, we show that for every $p \ge c\sqrt[k]{\frac{\log_a n}{n}}$ ($c \ge 5$), $px_{k,\ell}(G_{n,p})\le 2$ holds almost surely, where $G_{n,p}$ is the Erdös-Rényi random graph model.

math.CO↗

arXiv2016

Fine-Grained Energy Modeling for the Source Code of a Mobile Application

Energy efficiency has a significant influence on user experience of battery-driven devices such as smartphones and tablets. The goal of an energy model for source code is to lay a foundation for the application of energy-saving techniques during software development. The challenge is to relate hardware energy consumption to high-level application code, considering the complex run-time context and software stack. Traditional techniques build the energy model by mapping a hardware energy model onto software constructs; this approach faces obstacles when the software stack consists of a number of abstract layers. Another approach that has been followed is to utilize hardware or operating system features to estimate software energy information at a coarse level of granularity such as blocks, methods or even applications. In this paper, we explain how to construct a fine-grained energy model for the source code, which is based on "energy operations" identified directly from the source code and able to provide more valuable information for code optimization. We apply the approach to a class of applications based on a game-engine, and explain the wider applicability of the method.

cs.OH↗

arXiv2016

An Energy-Aware Programming Approach for Mobile Application Development Guided by a Fine-Grained Energy Model

Energy efficiency has a significant influence on user experience of battery-driven devices such as smartphones and tablets. It is shown that software optimization plays an important role in reducing energy consumption of system. However, in mobile devices, the conventional nature of compiler considers not only energy-efficiency but also limited memory usage and real-time response to user inputs, which largely limits the compiler's positive impact on energy-saving. As a result, the code optimization relies more on developers. In this paper, we propose an energy-aware programming approach, which is guided by an operation-based source-code-level energy model. And this approach is placed at the end of software engineering life cycle to avoid distracting developers from guaranteeing the correctness of system. The experimental result shows that our approach is able to save from 6.4% to 50.2% of the overall energy consumption depending on different scenarios.

cs.SE↗

arXiv2016

More on total monochromatic connection of graphs

A graph is said to be {\it total-colored} if all the edges and the vertices of the graph are colored. A total-coloring of a graph is a {\it total monochromatically-connecting coloring} ({\it TMC-coloring}, for short) if any two vertices of the graph are connected by a path whose edges and internal vertices on the path have the same color. For a connected graph $G$, the {\it total monochromatic connection number}, denoted by $tmc(G)$, is defined as the maximum number of colors used in a TMC-coloring of $G$. Note that a TMC-coloring does not exist if $G$ is not connected, in which case we simply let $tmc(G)=0$. In this paper, we first characterize all graphs of order $n$ and size $m$ with $tmc(G)=3,4,5,6,m+n-2,m+n-3$ and $m+n-4$, respectively. Then we determine the threshold function for a random graph to have $tmc(G)\geq f(n)$, where $f(n)$ is a function satisfying $1\leq f(n)<\frac{1}{2}n(n-1)+n$. Finally, we show that for a given connected graph $G$, and a positive integer $L$ with $L\leq m+n$, it is NP-complete to decide whether $tmc(G)\geq L$.

math.CO↗

arXiv2016

Some upper bounds for the $3$-proper index of graphs

A tree $T$ in an edge-colored graph is a {\it proper tree} if no two adjacent edges of $T$ receive the same color. Let $G$ be a connected graph of order $n$ and $k$ be a fixed integer with $2\le k\le n$. For a vertex subset $S \subseteq V(G)$ with $\left|S\right| \ge 2$, a tree containing all the vertices of $S$ in $G$ is called an $S$-tree. An edge-coloring of $G$ is called a \emph{$k$-proper coloring} if for every $k$-subset $S$ of $V(G)$, there exists a proper $S$-tree in $G$. For a connected graph $G$, the \emph{$k$-proper index} of $G$, denoted by $px_k(G)$, is the smallest number of colors that are needed in a $k$-proper coloring of $G$. In this paper, we show that for every connected graph $G$ of order $n$ and minimum degree $δ\geq 3$, $px_{3}(G)\le n\frac{\ln(δ+1)}{δ+1}(1+o_δ(1))+2$. We also prove that for every connected graph $G$ with minimum degree at least $3$, $px_{3}(G) \le px_{3}(G[D])+3$ when $D$ is a connected $3$-way dominating set of $G$ and $px_{3}(G) \le px_{3}(G[D])+1$ when $D$ is a connected $3$-dominating set of $G$. In addition, we obtain tight upper bounds of the 3-proper index for two special graph classes: threshold graphs and chain graphs. Finally, we prove that $px_3(G) \le \lfloor\frac{n}{2}\rfloor$ for any 2-connected graphs with at least four vertices.

math.CO↗

arXiv2016

On two conjectures about the proper connection number of graphs

A path in an edge-colored graph is called proper if no two consecutive edges of the path receive the same color. For a connected graph $G$, the proper connection number $pc(G)$ of $G$ is defined as the minimum number of colors needed to color its edges so that every pair of distinct vertices of $G$ are connected by at least one proper path in $G$. In this paper, we consider two conjectures on the proper connection number of graphs. The first conjecture states that if $G$ is a noncomplete graph with connectivity $κ(G) = 2$ and minimum degree $δ(G)\ge 3$, then $pc(G) = 2$, posed by Borozan et al.~in [Discrete Math. 312(2012), 2550-2560]. We give a family of counterexamples to disprove this conjecture. However, from a result of Thomassen it follows that 3-edge-connected noncomplete graphs have proper connection number 2. Using this result, we can prove that if $G$ is a 2-connected noncomplete graph with $diam(G)=3$, then $pc(G) = 2$, which solves the second conjecture we want to mention, posed by Li and Magnant in [Theory \& Appl. Graphs 0(1)(2015), Art.2].

math.CO↗

arXiv2016

The (vertex-)monochromatic index of a graph

A tree $T$ in an edge-colored graph $H$ is called a \emph{monochromatic tree} if all the edges of $T$ have the same color. For $S\subseteq V(H)$, a \emph{monochromatic $S$-tree} in $H$ is a monochromatic tree of $H$ containing the vertices of $S$. For a connected graph $G$ and a given integer $k$ with $2\leq k\leq |V(G)|$, the \emph{$k$-monochromatic index $mx_k(G)$} of $G$ is the maximum number of colors needed such that for each subset $S\subseteq V(G)$ of $k$ vertices, there exists a monochromatic $S$-tree. In this paper, we prove that for any connected graph $G$, $mx_k(G)=|E(G)|-|V(G)|+2$ for each $k$ such that $3\leq k\leq |V(G)|$. A tree $T$ in a vertex-colored graph $H$ is called a \emph{vertex-monochromatic tree} if all the internal vertices of $T$ have the same color. For $S\subseteq V(H)$, a \emph{vertex-monochromatic $S$-tree} in $H$ is a vertex-monochromatic tree of $H$ containing the vertices of $S$. For a connected graph $G$ and a given integer $k$ with $2\leq k\leq |V(G)|$, the \emph{$k$-monochromatic vertex-index $mvx_k(G)$} of $G$ is the maximum number of colors needed such that for each subset $S\subseteq V(G)$ of $k$ vertices, there exists a vertex-monochromatic $S$-tree. We show that for a given a connected graph $G$, and a positive integer $L$ with $L\leq |V(G)|$, to decide whether $mvx_k(G)\geq L$ is NP-complete for each integer $k$ such that $2\leq k\leq |V(G)|$. We also obtain some Nordhaus-Gaddum-type results for the $k$-monochromatic vertex-index.

math.CO↗

arXiv2016

Solution to a conjecture on the proper connection number of graphs

A path in an edge-colored graph is called a proper path if no two adjacent edges of the path receive the same color. For a connected graph $G$, the proper connection number $pc(G)$ of $G$ is defined as the minimum number of colors needed to color its edges, so that every pair of distinct vertices of $G$ is connected by at least one proper path in $G$. Recently, Li and Magnant in [Theory Appl. Graphs 0(1)(2015), Art.2] posed the following conjecture: If $G$ is a connected noncomplete graph of order $n \geq 5$ and minimum degree $δ(G) \geq n/4$, then $pc(G)=2$. In this paper, we show that this conjecture is true except for two small graphs on 7 and 8 vertices, respectively. As a byproduct we obtain that if $G$ is a connected bipartite graph of order $n\geq 4$ with $δ(G)\geq \frac{n+6}{8}$, then $pc(G)=2$.

math.CO↗

arXiv2016

Rainbow vertex-connection and forbidden subgraphs

A path in a vertex-colored graph is called \emph{vertex-rainbow} if its internal vertices have pairwise distinct colors. A graph $G$ is \emph{rainbow vertex-connected} if for any two distinct vertices of $G$, there is a vertex-rainbow path connecting them. For a connected graph $G$, the \emph{rainbow vertex-connection number} of $G$, denoted by $rvc(G)$, is defined as the minimum number of colors that are required to make $G$ rainbow vertex-connected. In this paper, we find all the families $\mathcal{F}$ of connected graphs with $|\mathcal{F}|\in\{1,2\}$, for which there is a constant $k_\mathcal{F}$ such that, for every connected $\mathcal{F}$-free graph $G$, $rvc(G)\leq diam(G)+k_\mathcal{F}$, where $diam(G)$ is the diameter of $G$.

math.CO↗

arXiv2016

The $k$-proper index of graphs

A tree $T$ in an edge-colored graph is a \emph{proper tree} if any two adjacent edges of $T$ are colored with different colors. Let $G$ be a graph of order $n$ and $k$ be a fixed integer with $2\leq k\leq n$. For a vertex set $S\subseteq V(G)$, a tree containing the vertices of $S$ in $G$ is called an \emph{$S$-tree}. An edge-coloring of $G$ is called a \emph{$k$-proper coloring} if for every set $S$ of $k$ vertices in $G$, there exists a proper $S$-tree in $G$. The \emph{$k$-proper index} of a nontrivial connected graph $G$, denoted by $px_k(G)$, is the smallest number of colors needed in a $k$-proper coloring of $G$. In this paper, some simple observations about $px_k(G)$ for a nontrivial connected graph $G$ are stated. Meanwhile, the $k$-proper indices of some special graphs are determined, and for every pair of positive integers $a$, $b$ with $2\leq a\leq b$, a connected graph $G$ with $px_k(G)=a$ and $rx_k(G)=b$ is constructed for each integer $k$ with $3\leq k\leq n$. Also, the graphs with $k$-proper index $n-1$ and $n-2$ are respectively characterized.

math.CO↗

arXiv2016

Total monochromatic connection of graphs

A graph is said to be {\it total-colored} if all the edges and the vertices of the graph are colored. A path in a total-colored graph is a {\it total monochromatic path} if all the edges and internal vertices on the path have the same color. A total-coloring of a graph is a {\it total monochromatically-connecting coloring} ({\it TMC-coloring}, for short) if any two vertices of the graph are connected by a total monochromatic path of the graph. For a connected graph $G$, the {\it total monochromatic connection number}, denoted by $tmc(G)$, is defined as the maximum number of colors used in a TMC-coloring of $G$. These concepts are inspired by the concepts of monochromatic connection number $mc(G)$, monochromatic vertex connection number $mvc(G)$ and total rainbow connection number $trc(G)$ of a connected graph $G$. Let $l(T)$ denote the number of leaves of a tree $T$, and let $l(G)=\max\{ l(T) | $ $T$ is a spanning tree of $G$ $\}$ for a connected graph $G$. In this paper, we show that there are many graphs $G$ such that $tmc(G)=m-n+2+l(G)$, and moreover, we prove that for almost all graphs $G$, $tmc(G)=m-n+2+l(G)$ holds. Furthermore, we compare $tmc(G)$ with $mvc(G)$ and $mc(G)$, respectively, and obtain that there exist graphs $G$ such that $tmc(G)$ is not less than $mvc(G)$ and vice versa, and that $tmc(G)=mc(G)+l(G)$ holds for almost all graphs. Finally, we prove that $tmc(G)\leq mc(G)+mvc(G)$, and the equality holds if and only if $G$ is a complete graph.

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