arXiv subjects

Xueliang Li

Publications and source records attributed to Xueliang Li.

At least 55 records · Page 3Linked to original sources

arXiv2020

Bounds for the rainbow disconnection number of graphs

An edge-cut $R$ of an edge-colored connected graph is called a rainbow-cut if no two edges in the edge-cut are colored the same. An edge-colored graph is rainbow disconnected if for any two distinct vertices $u$ and $v$ of the graph, there exists a $u$-$v$-rainbow-cut separating them. For a connected graph $G$, the rainbow disconnection number of $G$, denoted by rd$(G)$, is defined as the smallest number of colors that are needed in order to make $G$ rainbow disconnected. In this paper, we first give some tight upper bounds for rd$(G)$, and moreover, we completely characterize the graphs which meet the upper bound of the Nordhaus-Gaddum type results obtained early by us. Secondly, we propose a conjecture that $λ^+(G)\leq \textnormal{rd}(G)\leq λ^+(G)+1$, where $λ^+(G)$ is the upper edge-connectivity, and prove the conjecture for many classes of graphs, to support it. Finally, we give the relationship between rd$(G)$ of a graph $G$ and the rainbow vertex-disconnection number rvd$(L(G))$ of the line graph $L(G)$ of $G$.

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arXiv2020

The asymptotic value of energy for matrices with degree-distance-based entries of random graphs

For a graph $G=(V, E)$ and $i, j\in V$, denote the distance between $i$ and $j$ in $G$ by $D(i, j)$ and the degrees of $i$, $j$ by $d_i$, $d_j$, respectively. Let $f(D(i, j), d_{i}, d_{j})$ be a function symmetric in $i$ and $j$. Define a matrix $W_f(G)$, called the weighted distance matrix, of $G$, with the $ij$-entry $W_f(G)(i, j)=f(D(i, j), d_{i}, d_{j})$ if $i\neq j$ and $W_f(G)(i, j)=0$ if $i=j$. In this paper, we prove that if the symmetric function $f$ satisfies that $f(D(i, j), (1+o(1))np, (1+o(1))np)=(1+o(1))f(D(i, j), np, np)$, then for almost all graphs $G_p$ in the $Erd\ddot{o}s$-$R\acute{e}nyi$ random graph model $\mathcal{G}_{n, p}$, the energy of $W_f(G_p)$ is $\{(\frac{8}{3π}\sqrt{p(1-p)}+o(1))\cdot|f(1, np, np)-f(2, np, np)|+o(|f(2, np, np)|)\}\cdot n^{3/2}$. As a consequence, we give the asymptotic values of energies of a variety of weighted distance matrices with function $f$ from distance-based only and mixed with degree-distance-based topological indices of chemical use. This generalizes our former result with only degree-based weights.

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arXiv2020

The asymptotic value of graph energy for random graphs with degree-based weights

In this paper, we investigate the energy of a weighted random graph $G_p(f)$ in $G_{n,p}(f)$, in which each edge $ij$ takes the weight $f(d_i,d_j)$, where $d_v$ is a random variable, the degree of vertex $v$ in the random graph $G_p$ of the Erdös--Rényi random graph model $G_{n,p}$, and $f$ is a symmetric real function on two variables. Suppose $|f(d_i,d_j)|\leq C n^m$ for some constants $C, m>0$, and $f((1+o(1))np,(1+o(1))np)=(1+o(1))f(np,np)$. Then, for almost all graphs $G_p(f)$ in $G_{n,p}(f)$, the energy of $G_p(f)$ is $(1+o(1))f(np,np)\frac{8}{3π}\sqrt{p(1-p)}\cdot n^{3/2},$ where $p\in(0,1)$ is any fixed and independent of $n$. Consequently, with this one basket we can get the asymptotic values of various kinds of graph energies of chemical use, such as Randić energy, ABC energy, and energies of random matrices obtained from various kinds of degree-based chemical indices.

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arXiv2020

Hardness results for three kinds of colored connections of graphs

The concept of rainbow connection number of a graph was introduced by Chartrand et al. in 2008. Inspired by this concept, other concepts on colored version of connectivity in graphs were introduced, such as the monochromatic connection number by Caro and Yuster in 2011, the proper connection number by Borozan et al. in 2012, and the conflict-free connection number by Czap et al. in 2018, as well as some other variants of connection numbers later on. Chakraborty et al. proved that to compute the rainbow connection number of a graph is NP-hard. For a long time, it has been tried to fix the computational complexity for the monochromatic connection number, the proper connection number and the conflict-free connection number of a graph. However, it has not been solved yet. Only the complexity results for the strong version, i.e., the strong proper connection number and the strong conflict-free connection number, of these connection numbers were determined to be NP-hard. In this paper, we prove that to compute each of the monochromatic connection number, the proper connection number and the conflict free connection number for a graph is NP-hard. This solves a long standing problem in this field, asked in many talks of workshops and papers.

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arXiv2020

Rainbow monochromatic $k$-edge-connection colorings of graphs

A path in an edge-colored graph is called a monochromatic path if all edges of the path have a same color. We call $k$ paths $P_1,\cdots,P_k$ rainbow monochromatic paths if every $P_i$ is monochromatic and for any two $i\neq j$, $P_i$ and $P_j$ have different colors. An edge-coloring of a graph $G$ is said to be a rainbow monochromatic $k$-edge-connection coloring (or $RMC_k$-coloring for short) if every two distinct vertices of $G$ are connected by at least $k$ rainbow monochromatic paths. We use $rmc_k(G)$ to denote the maximum number of colors that ensures $G$ has an $RMC_k$-coloring, and this number is called the rainbow monochromatic $k$-edge-connection number. We prove the existence of $RMC_k$-colorings of graphs, and then give some bounds of $rmc_k(G)$ and present some graphs whose $rmc_k(G)$ reaches the lower bound. We also obtain the threshold function for $rmc_k(G(n,p))\geq f(n)$, where $\lfloor\frac{n}{2}\rfloor> k\geq 1$.

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arXiv2019

Monochromatic disconnection of graphs

For an edge-colored graph $G$, we call an edge-cut $M$ of $G$ monochromatic if the edges of $M$ are colored with a same color. The graph $G$ is called monochromatically disconnected if any two distinct vertices of $G$ are separated by a monochromatic edge-cut. For a connected graph $G$, the monochromatic disconnection number, denoted by $md(G)$, of $G$ is the maximum number of colors that are needed in order to make $G$ monochromatically disconnected. We will show that almost all graphs have monochromatic disconnection numbers equal to 1. We also obtain the Nordhaus-Gaddum-type results for $md(G)$.

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arXiv2019

Spanning Triangle-trees and Flows of Graphs

In this paper we study the flow-property of graphs containing a spanning triangle-tree. Our main results provide a structure characterization of graphs with a spanning triangle-tree admitting a nowhere-zero $3$-flow. All these graphs without nowhere-zero $3$-flows are constructed from $K_4$ by a so-called bull-growing operation. This generalizes a result of Fan et al. in 2008 on triangularly-connected graphs and particularly shows that every $4$-edge-connected graph with a spanning triangle-tree has a nowhere-zero $3$-flow. A well-known classical theorem of Jaeger in 1979 shows that every graph with two edge-disjoint spanning trees admits a nowhere-zero $4$-flow. We prove that every graph with two edge-disjoint spanning triangle-trees has a flow strictly less than $3$.

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arXiv2019

Proper disconnection of graphs

For an edge-colored graph $G$, a set $F$ of edges of $G$ is called a \emph{proper cut} if $F$ is an edge-cut of $G$ and any pair of adjacent edges in $F$ are assigned by different colors. An edge-colored graph is \emph{proper disconnected} if for each pair of distinct vertices of $G$ there exists a proper edge-cut separating them. For a connected graph $G$, the \emph{proper disconnection number} of $G$, denoted by $pd(G)$, is the minimum number of colors that are needed in order to make $G$ proper disconnected. In this paper, we first give the exact values of the proper disconnection numbers for some special families of graphs. Next, we obtain a sharp upper bound of $pd(G)$ for a connected graph $G$ of order $n$, i.e, $pd(G)\leq \min\{ χ'(G)-1, \left \lceil \frac{n}{2} \right \rceil\}$. Finally, we show that for given integers $k$ and $n$, the minimum size of a connected graph $G$ of order $n$ with $pd(G)=k$ is $n-1$ for $k=1$ and $n+2k-4$ for $2\leq k\leq \lceil\frac{n}{2}\rceil$.

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arXiv2019

Steiner (revised) Szeged index of graphs

The Steiner distance in a graph, introduced by Chartrand et al. in 1989, is a natural generalization of the concept of classical graph distance. For a connected graph $G$ of order at least 2 and $S\subseteq V(G)$, the Steiner distance $d_G(S)$ of the set $S$ of vertices in $G$ is the minimum size of a connected subgraph whose vertex set contains or connects $S$. In this paper, we introduce the concept of the Steiner (revised) Szeged index ($rSz_k(G)$) $Sz_k(G)$ of a graph $G$, which is a natural generalization of the well-known (revised) Szeged index of chemical use. We determine the $Sz_k(G)$ for trees in general. Then we give a formula for computing the Steiner Szeged index of a graph in terms of orbits of automorphism group action on the edge set of the graph. Finally, we give sharp upper and lower bounds of ($rSz_k(G)$) $Sz_k(G)$ of a connected graph $G$, and establish some of its properties. Formulas of ($rSz_k(G)$) $Sz_k(G)$ for small and large $k$ are also given in this paper.

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arXiv2019

Conflict-free (vertex)-connection numbers of graphs with small diameters

A path in an(a) edge(vertex)-colored graph is called a conflict-free path if there exists a color used on only one of its edges(vertices). An(A) edge(vertex)-colored graph is called conflict-free (vertex-)connected if for each pair of distinct vertices, there is a conflict-free path connecting them. For a connected graph $G$, the conflict-free (vertex-)connection number of $G$, denoted by $cfc(G)(\text{or}~vcfc(G))$, is defined as the smallest number of colors that are required to make $G$ conflict-free (vertex-)connected. In this paper, we first give the exact value $cfc(T)$ for any tree $T$ with diameters $2,3$ and $4$. Based on this result, the conflict-free connection number is determined for any graph $G$ with $diam(G)\leq 4$ except for those graphs $G$ with diameter $4$ and $h(G)=2$. In this case, we give some graphs with conflict-free connection number $2$ and $3$, respectively. For the conflict-free vertex-connection number, the exact value $vcfc(G)$ is determined for any graph $G$ with $diam(G)\leq 4$.

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arXiv2019

Detecting and Diagnosing Energy Issues for Mobile Applications

Energy efficiency is an important criterion to judge the quality of mobile apps, but one third of our randomly sampled apps suffer from energy issues that can quickly drain battery power. To understand these issues, we conducted an empirical study on 27 well-maintained apps such as Chrome and Firefox, whose issue tracking systems are publicly accessible. Our study revealed that the main root causes of energy issues include unnecessary workload and excessively frequent operations. Surprisingly, these issues are beyond the application of present technology on energy issue detection. We also found that 20.6% of energy issues can only manifest themselves under specific contexts such as poor network performance, but such contexts are again neglected by present technology. Therefore, we proposed a novel testing framework for detecting energy issues in real-world apps. Our framework examines apps with well-designed input sequences and runtime contexts. To identify the root causes mentioned above, we employed a machine learning algorithm to cluster the workloads and further evaluate their necessity. For the issues concealed by the specific contexts, we carefully set up several execution contexts to pinpoint them. More importantly, we developed leading edge technology, e.g. pre-designing input sequences with potential energy overuse and tuning tests on-the-fly, to achieve high efficacy in detecting energy issues. A large-scale evaluation shows that 91.6% issues detected in our test were previously unknown to developers. On average, these issues double the energy costs of the apps. Furthermore, our test achieves a low number of false positives. Finally, we show how our test reports can help developers fix the issues.

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arXiv2019

Complementary Graphs with Flows Less Than Three

X. Hou, H.-J. Lai, P. Li and C.-Q. Zhang [J. Graph Theory 69 (2012) 464-470] showed that for a simple graph $G$ with $|V(G)|\ge 44$, if $\min\{δ(G),δ(G^c)\}\ge 4$, then either $G$ or its complementary graph $G^c$ has a nowhere-zero $3$-flow. In this paper, we improve this result by showing that if $|V(G)|\ge 32$ and $\min\{δ(G),δ(G^c)\}\ge 4$, then either $G$ or $G^c$ has flow index strictly less than $3$. Our result is proved by a newly developed closure operation and contraction method.

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arXiv2019

Strong conflict-free connection of graphs

A path $P$ in an edge-colored graph is called \emph{a conflict-free path} if there exists a color used on only one of the edges of $P$. An edge-colored graph $G$ is called \emph{conflict-free connected} if for each pair of distinct vertices of $G$ there is a conflict-free path in $G$ connecting them. The graph $G$ is called \emph{strongly conflict-free connected }if for every pair of vertices $u$ and $v$ of $G$ there exists a conflict-free path of length $d_G(u,v)$ in $G$ connecting them. For a connected graph $G$, the \emph{strong conflict-free connection number} of $G$, denoted by $\mathit{scfc}(G)$, is defined as the smallest number of colors that are required in order to make $G$ strongly conflict-free connected. In this paper, we first show that if $G_t$ is a connected graph with $m$ $(m\geq 2)$ edges and $t$ edge-disjoint triangles, then $\mathit{scfc}(G_t)\leq m-2t$, and the equality holds if and only if $G_t\cong S_{m,t}$. Then we characterize the graphs $G$ with $scfc(G)=k$ for $k\in \{1,m-3,m-2,m-1,m\}$. In the end, we present a complete characterization for the cubic graphs $G$ with $scfc(G)=2$.

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arXiv2019

Erdős-Gallai-type results for the rainbow disconnection number of graphs

Let $G$ be a nontrivial connected and edge-colored graph. An edge-cut $R$ of $G$ is called a rainbow cut if no two edges of it are colored with a same color. An edge-colored graph $G$ is called rainbow disconnected if for every two distinct vertices $u$ and $v$ of $G$, there exists a $u-v$ rainbow cut separating them. For a connected graph $G$, the rainbow disconnection number of $G$, denoted by $rd(G)$, is defined as the smallest number of colors that are needed in order to make $G$ rainbow disconnected. In this paper, we will study the Erdős-Gallai-type results for $rd(G)$, and completely solve them.

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arXiv2018

Erdös-Gallai-type results for conflict-free connection of graphs

A path in an edge-colored graph is called \emph{a conflict-free path} if there exists a color used on only one of its edges. An edge-colored graph is called \emph{conflict-free connected} if there is a conflict-free path between each pair of distinct vertices. The \emph{conflict-free connection number} of a connected graph $G$, denoted by $\mathit{cfc}(G)$, is defined as the smallest number of colors that are required to make $G$ conflict-free connected. In this paper, we obtain Erdös-Gallai-type results for the conflict-free connection numbers of graphs.

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arXiv2018

Hardness results for rainbow disconnection of graphs

Let $G$ be a nontrivial connected, edge-colored graph. An edge-cut $S$ of $G$ is called a rainbow cut if no two edges in $S$ are colored with a same color. An edge-coloring of $G$ is a rainbow disconnection coloring if for every two distinct vertices $s$ and $t$ of $G$, there exists a rainbow cut $S$ in $G$ such that $s$ and $t$ belong to different components of $G\setminus S$. For a connected graph $G$, the {\it rainbow disconnection number} of $G$, denoted by $rd(G)$, is defined as the smallest number of colors such that $G$ has a rainbow disconnection coloring by using this number of colors. In this paper, we show that for a connected graph $G$, computing $rd(G)$ is NP-hard. In particular, it is already NP-complete to decide if $rd(G)=3$ for a connected cubic graph. Moreover, we prove that for a given edge-colored (with an unbounded number of colors) connected graph $G$ it is NP-complete to decide whether $G$ is rainbow disconnected.

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arXiv2018

Monochromatic $k$-edge-connection colorings of graphs

A path in an edge-colored graph $G$ is called monochromatic if any two edges on the path have the same color. For $k\geq 2$, an edge-colored graph $G$ is said to be monochromatic $k$-edge-connected if every two distinct vertices of $G$ are connected by at least $k$ edge-disjoint monochromatic paths, and $G$ is said to be uniformly monochromatic $k$-edge-connected if every two distinct vertices are connected by at least $k$ edge-disjoint monochromatic paths such that all edges of these $k$ paths colored with a same color. We use $mc_k(G)$ and $umc_k(G)$ to denote the maximum number of colors that ensures $G$ to be monochromatic $k$-edge-connected and, respectively, $G$ to be uniformly monochromatic $k$-edge-connected. In this paper, we first conjecture that for any $k$-edge-connected graph $G$, $mc_k(G)=e(G)-e(H)+\lfloor\frac{k}{2}\rfloor$, where $H$ is a minimum $k$-edge-connected spanning subgraph of $G$. We verify the conjecture for $k=2$. We also prove the conjecture for $G=K_{k+1}$ when $k\geq4$ is even, and for $G=K_{k,n}$ when $k\geq4$ is even, or when $k=3$ and $n\geq k$. When $G$ is a minimal $k$-edge-connected graph, we give an upper bound of $mc_k(G)$, i.e., $mc_k(G)\leq k-1$, and $mc_k(G)\leq \lfloor\frac{k}{2}\rfloor$ when $G=K_{k,n}$. For the uniformly monochromatic $k$-edge-connectivity, we prove that for all $k$, $umc_k(G)=e(G)-e(H)+1$, where $H$ is a minimum $k$-edge-connected spanning subgraph of $G$.

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arXiv2018

More on rainbow disconnection in graphs

Let $G$ be a nontrivial edge-colored connected graph. An edge-cut $R$ of $G$ is called a rainbow cut if no two edges of it are colored the same. An edge-colored graph $G$ is rainbow disconnected if for every two vertices $u$ and $v$, there exists a $u-v$ rainbow cut. For a connected graph $G$, the rainbow disconnection number of $G$, denoted by $rd(G)$, is defined as the smallest number of colors that are needed in order to make $G$ rainbow disconnected. In this paper, we first solve a conjecture that determines the maximum size of a connected graph $G$ of order $n$ with $rd(G) = k$ for given integers $k$ and $n$ with $1\leq k\leq n-1$, where $n$ is odd, posed by Chartrand et al. in \cite{CDHHZ}. Secondly, we discuss bounds of the rainbow disconnection numbers for complete multipartite graphs, critical graphs, minimal graphs with respect to chromatic index and regular graphs, and give the rainbow disconnection numbers for several special graphs. Finally, we get the Nordhaus-Gaddum-type theorem for the rainbow disconnection number of graphs. We prove that if $G$ and $\overline{G}$ are both connected, then $n-2 \leq rd(G)+rd(\overline{G})\leq 2n-5$ and $n-3\leq rd(G)\cdot rd(\overline{G})\leq (n-2)(n-3)$. Furthermore, examples are given to show that the upper bounds are sharp for $n\geq 6$, and the lower bounds are sharp when $G=\overline{G}=P_4$.

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