arXiv subjects

Xueliang Li

Publications and source records attributed to Xueliang Li.

At least 37 records · Page 2Linked to original sources

arXiv2024

The maximum spectral radius of $θ_{1,3,3}$-free graphs with given size

A graph $G$ is said to be $F$-free if it does not contain $F$ as a subgraph. A theta graph, say $θ_{l_1,l_2,l_3}$, is the graph obtained by connecting two distinct vertices with three internally disjoint paths of length $l_1, l_2, l_3$, where $l_1\leq l_2\leq l_3$ and $l_2\geq2$. Recently, Li, Zhao and Zou [arXiv:2409.15918v1] characterized the $θ_{1,p,q}$-free graph of size $m$ having the largest spectral radius, where $q\geq p\geq3$ and $p+q\geq2k+1\geq7$, and proposed a problem on characterizing the graphs with the maximum spectral radius among $θ_{1,3,3}$-free graphs. In this paper, we consider this problem and determine the maximum spectral radius of $θ_{1,3,3}$-free graphs with size $m$ and characterize the extremal graph. Up to now, all the graphs in $\mathcal{G}(m,θ_{1,p,q})$ which have the largest spectral radius have been determined, where $q\geq p\geq 2$.

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arXiv2024

Rainbow and Gallai-Rado numbers involving binary function equations

Let $\mathcal{E}$, $\mathcal{E}_1$, and $\mathcal{E}_2$ be equations, $n$ and $k$ be positive integers. The rainbow number $\operatorname{rb}([n],\mathcal{E})$ is difined as the minimum number of colors such that for every exact $(\operatorname{rb}([n],\mathcal{E}))$-coloring of $[n]$, there exists a rainbow solution of $\mathcal{E}$. The Gallai-Rado number $\operatorname{GR}_k(\mathcal{E}_1:\mathcal{E}_2)$ is defined as the minimum positive integer $N$, if it exists, such that for all $n\ge N$, every $k$-colored $[n]$ contains either a rainbow solution of $\mathcal{E}_1$ or a monochromatic solution of $\mathcal{E}_2$. In this paper, we get some exact values of rainbow and Gallai-Rado numbers involving binary function equations. We also provide an algorithm to calculate the rainbow numbers of nonlinear binary function equations.

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arXiv2024

Gallai-Ramsey multiplicity for rainbow small trees

Let $G, H$ be two non-empty graphs and $k$ be a positive integer. The Gallai-Ramsey number $\operatorname{gr}_k(G:H)$ is defined as the minimum positive integer $N$ such that for all $n\geq N$, every $k$-edge-coloring of $K_n$ contains either a rainbow subgraph $G$ or a monochromatic subgraph $H$. The Gallai-Ramsey multiplicity $\operatorname{GM}_k(G:H)$ is defined as the minimum total number of rainbow subgraphs $G$ and monochromatic subgraphs $H$ for all $k$-edge-colored $K_{\operatorname{gr}_k(G:H)}$. In this paper, we get some exact values of the Gallai-Ramsey multiplicity for rainbow small trees versus general monochromatic graphs under a sufficiently large number of colors. We also study the bipartite Gallai-Ramsey multiplicity.

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arXiv2023

Sufficient Conditions for the Exact Relaxation of Complementarity Constraints for Storages in Multi-period OPF Problems

Storage-concerned Optimal Power Flow (OPF) with complementarity constraints is highly non-convex and intractable. In this paper, we propose two generalized sufficient conditions which guarantee no simultaneous charging and discharging (SCD) in the relaxed multi-period OPF excluding the complementarity constraints. Moreover, we prove that the regions on the locational marginal prices (LMPs) formed by the proposed two conditions both contain the other existing representative ones. We also generalize the application premise of sufficient conditions from the positive electricity price requirements to the negative electricity price scenarios. In contrast to participating solely in the energy dispatch, we highlight that offering reserve services can broaden the application range of relaxation conditions for storages. The case studies verify the exactness and advantages of the proposed conditions.

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arXiv2021

Counting rainbow triangles in edge-colored graphs

Let $G$ be an edge-colored graph on $n$ vertices. The minimum color degree of $G$, denoted by $δ^c(G)$, is defined as the minimum number of colors assigned to the edges incident to a vertex in $G$. In 2013, H. Li proved that an edge-colored graph $G$ on $n$ vertices contains a rainbow triangle if $δ^c(G)\geq \frac{n+1}{2}$. In this paper, we obtain several estimates on the number of rainbow triangles through one given vertex in $G$. As consequences, we prove counting results for rainbow triangles in edge-colored graphs. One main theorem states that the number of rainbow triangles in $G$ is at least $\frac{1}{6}δ^c(G)(2δ^c(G)-n)n$, which is best possible by considering the rainbow $k$-partite Turán graph, where its order is divisible by $k$. This means that there are $Ω(n^2)$ rainbow triangles in $G$ if $δ^c(G)\geq \frac{n+1}{2}$, and $Ω(n^3)$ rainbow triangles in $G$ if $δ^c(G)\geq cn$ when $c>\frac{1}{2}$. Both results are tight in sense of the order of the magnitude. We also prove a counting version of a previous theorem on rainbow triangles under a color neighborhood union condition due to Broersma et al., and an asymptotically tight color degree condition forcing a colored friendship subgraph $F_k$ (i.e., $k$ rainbow triangles sharing a common vertex).

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arXiv2020

Rainbow triangles in edge-colored complete graphs

Let $G$ be a graph of order $n$ with an edge-coloring $c$, and let $δ^c(G)$ denote the minimum color-degree of $G$. A subgraph $F$ of $G$ is called rainbow if any two edges of $F$ have distinct colors. There have been a lot results in the existing literature on rainbow triangles in edge-colored complete graphs. Fujita and Magnant showed that for an edge-colored complete graph $G$ of order $n$, if $δ^c(G)\geq \frac{n+1}{2}$, then every vertex of $G$ is contained in a rainbow triangle. In this paper, we show that if $δ^c(G)\geq \frac{n+k}{2}$, then every vertex of $G$ is contained in at least $k$ rainbow triangles, which can be seen as a generalization of their result. Li showed that for an edge-colored graph $G$ of order $n$, if $δ^c(G)\geq \frac{n+1}{2}$, then $G$ contains a rainbow triangle. We show that if $G$ is complete and $δ^c(G)\geq \frac{n}{2}$, then $G$ contains a rainbow triangle and the bound is sharp. Hu et al. showed that for an edge-colored graph $G$ of order $n\geq 20$, if $δ^c(G)\geq \frac{n+2}{2}$, then $G$ contains two vertex-disjoint rainbow triangles. We show that if $G$ is complete with order $n\geq 8$ and $δ^c(G)\geq \frac{n+1}{2}$, then $G$ contains two vertex-disjoint rainbow triangles. Moreover, we improve the result of Hu et al. from $n\geq 20$ to $n\geq 7$, the best possible.

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arXiv2020

Note on rainbow cycles in edge-colored graphs

Let $G$ be a graph of order $n$ with an edge-coloring $c$, and let $δ^c(G)$ denote the minimum color degree of $G$. A subgraph $F$ of $G$ is called rainbow if all edges of $F$ have pairwise distinct colors. There have been a lot results on rainbow cycles of edge-colored graphs. In this paper, we show that (i) if $δ^c(G)>\frac{3n-3}{4}$, then every vertex of $G$ is contained in a rainbow triangle; (ii) $δ^c(G)>\frac{3n}{4}$, then every vertex of $G$ is contained in a rainbow $C_4$; and (iii) if $G$ is complete, $n\geq 8k-18$ and $δ^c(G)>\frac{n-1}{2}+k$, then $G$ contains a rainbow cycle of length at least $k$. Some gaps in previous publications are also found and corrected.

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arXiv2020

Extremal graphs and classification of planar graphs by MC-numbers

An edge-coloring of a connected graph $G$ is called a {\em monochromatic connection coloring} (MC-coloring for short) if any two vertices of $G$ are connected by a monochromatic path in $G$. For a connected graph $G$, the {\em monochromatic connection number} (MC-number for short) of $G$, denoted by $mc(G)$, is the maximum number of colors that ensure $G$ has a monochromatic connection coloring by using this number of colors. This concept was introduced by Caro and Yuster in 2011. They proved that $mc(G)\leq m-n+k$ if $G$ is not a $k$-connected graph. In this paper we depict all graphs with $mc(G)=m-n+k+1$ and $mc(G)= m-n+k$ if $G$ is a $k$-connected but not $(k+1)$-connected graph. We also prove that $mc(G)\leq m-n+4$ if $G$ is a planar graph, and classify all planar graphs by their monochromatic connectivity numbers.

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arXiv2020

Graph colorings under global structural conditions

More than ten years ago in 2008, a new kind of graph coloring appeared in graph theory, which is the {\it rainbow connection coloring} of graphs, and then followed by some other new concepts of graph colorings, such as {\it proper connection coloring, monochromatic connection coloring, and conflict-free connection coloring} of graphs. In about ten years of our consistent study, we found that these new concepts of graph colorings are actually quite different from the classic graph colorings. These {\it colored connection colorings} of graphs are brand-new colorings and they need to take care of global structural properties (for example, connectivity) of a graph under the colorings; while the traditional colorings of graphs are colorings under which only local structural properties (adjacent vertices or edges) of a graph are taken care of. Both classic colorings and the new colored connection colorings can produce the so-called chromatic numbers. We call the colored connection numbers the {\it global chromatic numbers}, and the classic or traditional chromatic numbers the {\it local chromatic numbers}. This paper intends to clarify the difference between the colored connection colorings and the traditional colorings, and finally to propose the new concepts of global colorings under which global structural properties of the colored graph are kept, and the global chromatic numbers.

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arXiv2020

Group Connectivity under $3$-Edge-Connectivity

Let $S,T$ be two distinct finite Abelian groups with $|S|=|T|$. A fundamental theorem of Tutte shows that a graph admits a nowhere-zero $S$-flow if and only if it admits a nowhere-zero $T$-flow. Jaeger, Linial, Payan and Tarsi in 1992 introduced group connectivity as an extension of flow theory, and they asked whether such a relation holds for group connectivity analogy. It was negatively answered by Hušek, Mohelníková and Šámal in 2017 for graphs with edge-connectivity 2 for the groups $S=\mathbb{Z}_4$ and $T=\mathbb{Z}_2^2$. In this paper, we extend their results to $3$-edge-connected graphs (including both cubic and general graphs), which answers open problems proposed by Hušek, Mohelníková and Šámal(2017) and Lai, Li, Shao and Zhan(2011). Combining some previous results, this characterizes all the equivalence of group connectivity under $3$-edge-connectivity, showing that every $3$-edge-connected $S$-connected graph is $T$-connected if and only if $\{S,T\}\neq \{\mathbb{Z}_4,\mathbb{Z}_2^2\}$.

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arXiv2020

Asymptotic values of four Laplacian-type energies for matrices with degree-distance-based entries of random graphs

Let $f(D(i, j), d_i, d_j)$ be a real function symmetric in $i$ and $j$ with the property that $f(d, (1+o(1))np, (1+o(1))np)=(1+o(1))f(d, np, np)$ for $d=1,2$. Let $G$ be a graph, $d_i$ denote the degree of a vertex $i$ of $G$ and $D(i, j)$ denote the distance between vertices $i$ and $j$ in $G$. In this paper, we define the $f$-weighted Laplacian matrix for random graphs in the Erd$\ddot{o}$s-R$\acute{e}$nyi random graph model $\mathcal{G}_{n, p}$, where $p\in (0, 1)$ is fixed. Four weighted Laplacian type energies: the weighted Laplacian energy $\mathscr{LE}_f(G)$, weighted signless Laplacian energy $\mathscr{LE}^{+}_f(G)$, weighted incidence energy $\mathscr{IE}_f(G)$ and the weighted Laplacian-energy like invariant $\mathscr{LEL}_f(G)$ are introduced and studied. We obtain the asymptotic values of $\mathscr{IE}_f(G)$ and $\mathscr{LEL}_f(G)$, and the values of $\mathscr{LE}_f(G)$ and $\mathscr{LE}_f^{+}(G)$ under the condition that $f(D(i, j), d_i, d_j)$ is a function dependent only on $D(i, j)$. As a consequence, we get that for almost all graphs $G_p\in \mathcal{G}_{n, p}$, the energy for the matrix with degree-distance-based entries of $G_p$, $\mathscr{E}(W_f(G_p)) < \mathscr{LE}_f(G_p),$ the Laplacian energy of the matrix, which is a generalization of a conjecture by Gutman et al.

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arXiv2020

Monochromatic disconnection: Erdős-Gallai-type problems and product 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. The monochromatic disconnection number, denoted by $md(G)$, of a connected graph $G$ is the maximum number of colors that are allowed to make $G$ monochromatically disconnected. In this paper, we solve the Erdős-Gallai-type problems for the monochromatic disconnection, and give the monochromatic disconnection numbers for four graph products, i.e., Cartesian, strong, lexicographic, and tensor products.

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arXiv2020

Strong rainbow disconnection in graphs

Let $G$ be a nontrivial edge-colored connected graph. An edge-cut $R$ of $G$ is called a {\it rainbow edge-cut} if no two edges of $R$ are colored with the same color. For two distinct vertices $u$ and $v$ of $G$, if an edge-cut separates them, then the edge-cut is called a {\it $u$-$v$-edge-cut}. An edge-colored graph $G$ is called \emph{strong rainbow disconnected} if for every two distinct vertices $u$ and $v$ of $G$, there exists a both rainbow and minimum $u$-$v$-edge-cut ({\it rainbow minimum $u$-$v$-edge-cut} for short) in $G$, separating them, and this edge-coloring is called a {\it strong rainbow disconnection coloring} (srd-{\it coloring} for short) of $G$. For a connected graph $G$, the \emph{strong rainbow disconnection number} (srd-{\it number} for short) of $G$, denoted by $\textnormal{srd}(G)$, is the smallest number of colors that are needed in order to make $G$ strong rainbow disconnected. In this paper, we first characterize the graphs with $m$ edges such that $\textnormal{srd}(G)=k$ for each $k \in \{1,2,m\}$, respectively, and we also show that the srd-number of a nontrivial connected graph $G$ equals the maximum srd-number among the blocks of $G$. Secondly, we study the srd-numbers for the complete $k$-partite graphs, $k$-edge-connected $k$-regular graphs and grid graphs. Finally, we show that for a connected graph $G$, to compute $\textnormal{srd}(G)$ is NP-hard. In particular, we show that it is already NP-complete to decide if $\textnormal{srd}(G)=3$ for a connected cubic graph. Moreover, we show that for a given edge-colored (with an unbounded number of colors) connected graph $G$ it is NP-complete to decide whether $G$ is strong rainbow disconnected.

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arXiv2020

Upper bounds for the $MD$-numbers and characterization of extremal graphs

For an edge-colored graph $G$, we call an edge-cut $M$ of $G$ monochromatic if the edges of $M$ are colored with the same color. The graph $G$ is called monochromatic disconnected if any two distinct vertices of $G$ are separated by a monochromatic edge-cut. For a connected graph $G$, the monochromatic disconnection number (or $MD$-number for short) of $G$, denoted by $md(G)$, is the maximum number of colors that are allowed in order to make $G$ monochromatic disconnected. For graphs with diameter one, they are complete graphs and so their $MD$-numbers are $1$. For graphs with diameter at least 3, we can construct $2$-connected graphs such that their $MD$-numbers can be arbitrarily large; whereas for graphs $G$ with diameter two, we show that if $G$ is a $2$-connected graph then $md(G)\leq 2$, and if $G$ has a cut-vertex then $md(G)$ is equal to the number of blocks of $G$. So, we will focus on studying $2$-connected graphs with diameter two, and give two upper bounds of their $MD$-numbers depending on their connectivity and independent numbers, respectively. We also characterize the $\left\lfloor\frac{n}{2}\right\rfloor$-connected graphs (with large connectivity) whose $MD$-numbers are $2$ and the $2$-connected graphs (with small connectivity) whose $MD$-numbers archive the upper bound $\left\lfloor\frac{n}{2}\right\rfloor.$ For graphs with connectivity less than $\frac n 2$, we show that if the connectivity of a graph is in linear with its order $n$, then its $MD$-number is upper bounded by a constant, and this suggests us to leave a conjecture that for a $k$-connected graph $G$, $md(G)\leq \left\lfloor\frac{n}{k}\right\rfloor$.

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arXiv2020

Complexity results for two kinds of colored disconnections of graphs

The concept of rainbow disconnection number of graphs was introduced by Chartrand et al. in 2018. Inspired by this concept, we put forward the concepts of rainbow vertex-disconnection and proper disconnection in graphs. In this paper, we first show that it is $NP$-complete to decide whether a given edge-colored graph $G$ with maximum degree $Δ(G)=4$ is proper disconnected. Then, for a graph $G$ with $Δ(G)\leq 3$ we show that $pd(G)\leq 2$ and determine the graphs with $pd(G)=1$ and $2$, respectively. Furthermore, we show that for a general graph $G$, deciding whether $pd(G)=1$ is $NP$-complete, even if $G$ is bipartite. We also show that it is $NP$-complete to decide whether a given vertex-colored graph $G$ is rainbow vertex-disconnected, even though the graph $G$ has $Δ(G)=3$ or is bipartite.

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arXiv2020

Proper vertex-pancyclicity of edge-colored complete graphs without joint monochromatic triangles

In an edge-colored graph $(G,c)$, let $d^c(v)$ denote the number of colors on the edges incident with a vertex $v$ of $G$ and $δ^c(G)$ denote the minimum value of $d^c(v)$ over all vertices $v\in V(G)$. A cycle of $(G,c)$ is called proper if any two adjacent edges of the cycle have distinct colors. An edge-colored graph $(G,c)$ on $n\geq 3$ vertices is called properly vertex-pancyclic if each vertex of $(G,c)$ is contained in a proper cycle of length $\ell$ for every $\ell$ with $3 \le \ell \le n$. Fujita and Magnant conjectured that every edge-colored complete graph on $n\geq 3$ vertices with $δ^c(G)\geq \frac{n+1}{2}$ is properly vertex-pancyclic. Chen, Huang and Yuan partially solve this conjecture by adding an extra condition that $(G,c)$ does not contain any monochromatic triangle. In this paper, we show that this conjecture is true if the edge-colored complete graph contain no joint monochromatic triangles.

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arXiv2020

Further results on the rainbow vertex-disconnection of graphs

Let $G$ be a nontrivial connected and vertex-colored graph. A subset $X$ of the vertex set of $G$ is called rainbow if any two vertices in $X$ have distinct colors. The graph $G$ is called \emph{rainbow vertex-disconnected} if for any two vertices $x$ and $y$ of $G$, there exists a vertex subset $S$ such that when $x$ and $y$ are nonadjacent, $S$ is rainbow and $x$ and $y$ belong to different components of $G-S$; whereas when $x$ and $y$ are adjacent, $S+x$ or $S+y$ is rainbow and $x$ and $y$ belong to different components of $(G-xy)-S$. Such a vertex subset $S$ is called a \emph{rainbow vertex-cut} of $G$. For a connected graph $G$, the \emph{rainbow vertex-disconnection number} of $G$, denoted by $rvd(G)$, is the minimum number of colors that are needed to make $G$ rainbow vertex-disconnected. In this paper, we obtain bounds of the rainbow vertex-disconnection number of a graph in terms of the minimum degree and maximum degree of the graph. We give a tighter upper bound for the maximum size of a graph $G$ with $rvd(G)=k$ for $k\geq\frac{n}{2}$. We then characterize the graphs of order $n$ with rainbow vertex-disconnection number $n-1$ and obtain the maximum size of a graph $G$ with $rvd(G)=n-1$. Moreover, we get a sharp threshold function for the property $rvd(G(n,p))=n$ and prove that almost all graphs $G$ have $rvd(G)=rvd(\overline{G})=n$. Finally, we obtain some Nordhaus-Gaddum-type results: $n-5\leq rvd(G)+rvd(\overline{G})\leq 2n$ and $n-1\leq rvd(G)\cdot rvd(\overline{G})\leq n^2$ for the rainbow vertex-disconnection numbers of nontrivial connected graphs $G$ and $\overline{G}$ with order $n\geq 24$.

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arXiv2020

The rainbow vertex-disconnection in graphs

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