arXiv subjects
Publications and source records attributed to Xueliang Li.
The generalized $k$-edge-connectivity $λ_k(G)$ of a graph $G$ is a generalization of the concept of edge-connectivity. The lexicographic product of two graphs $G$ and $H$, denoted by $G\circ H$, is an important graph product. In this paper, we mainly study the generalized 3-edge-connectivity of $G \circ H$, and get upper and lower bounds of $λ_3(G \circ H)$. Moreover, all bounds are sharp.
Let $G$ be a nontrivial connected graph with an edge-coloring $c:E(G)\rightarrow \{1,2,\ldots,q\},$ $q\in \mathbb{N}$, where adjacent edges may be colored the same. A tree $T$ in $G$ is called a $rainbow~tree$ if no two edges of $T$ receive the same color. For a vertex set $S\subseteq V(G)$, a tree that connects $S$ in $G$ is called an {\it $S$-tree}. The minimum number of colors that are needed in an edge-coloring of $G$ such that there is a rainbow $S$-tree for every $k$-set $S$ of $V(G)$ is called the {\it $k$-rainbow index} of $G$, denoted by $rx_k(G)$. Notice that an lower bound and an upper bound of the $k$-rainbow index of a graph with order $n$ is $k-1$ and $n-1$, respectively. Chartrand et al. got that the $k$-rainbow index of a tree with order $n$ is $n-1$ and the $k$-rainbow index of a unicyclic graph with order $n$ is $n-1$ or $n-2$. Li and Sun raised the open problem of characterizing the graphs of order $n$ with $rx_k(G)=n-1$ for $k\geq 3$. In early papers we characterized the graphs of order $n$ with 3-rainbow index 2 and $n-1$. In this paper, we focus on $k=4$, and characterize the graphs of order $n$ with 4-rainbow index 3 and $n-1$, respectively.
A path in an edge-colored graph, where adjacent edges may be colored the same, is a rainbow path if no two edges of it are colored the same. A nontrivial connected graph $G$ is rainbow connected if there is a rainbow path connecting any two vertices, and the rainbow connection number of $G$, denoted by $rc(G)$, is the minimum number of colors that are needed in order to make $G$ rainbow connected. Chartrand et al. obtained that $G$ is a tree if and only if $rc(G)=m$, and it is easy to see that $G$ is not a tree if and only if $rc(G)\leq m-2$, where $m$ is the number of edge of $G$. So there is an interesting problem: Characterize the graphs $G$ with $rc(G)=m-2$. In this paper, we settle down this problem. Furthermore, we also characterize the graphs $G$ with $rc(G)=m-3$.
A new turán-type problem on distances on graphs was introduced by Tyomkyn and Uzzell. In this paper, we focus on the case that the distance is two. We primely show that for any value of $n$, a graph on $n$ vertices without three vertices pairwise at distance $2$, if it has a vertex $v \in V(G)$, whose neighbours are covered by at most two cliques, then it has at most $(n^2 - 1)/4 + 1$ pairs of vertices at distance $2$. This partially answers a guess of Tyomkyn and Uzzell [Tyomkyn, M., Uzzell, A.J.: A new Turán-Type promble on distaces of graphs. Graphs Combin. {\bf 29}(6), 1927--1942 (2012)].
A remarkable connection between the order of a maximum clique and the Lagrangian of a graph was established by Motzkin and Straus in 1965. This connection and its extensions were applied in Turán problems of graphs and uniform hypergraphs. Very recently, the study of Turán densities of non-uniform hypergraphs has been motivated by extremal poset problems. In this paper, we give some Motzkin-Straus type results for non-uniform hypergraphs.
A tree in an edge-colored graph $G$ is said to be a rainbow tree if no two edges on the tree share the same color. Given two positive integers $k$, $\ell$ with $k\geq 3$, the \emph{$(k,\ell)$-rainbow index} $rx_{k,\ell}(G)$ of $G$ is the minimum number of colors needed in an edge-coloring of $G$ such that for any set $S$ of $k$ vertices of $G$, there exist $\ell$ internally disjoint rainbow trees connecting $S$. This concept was introduced by Chartrand et al., and there have been very few results about it. In this paper, we investigate the $(k,\ell)$-rainbow index for complete bipartite graphs and complete multipartite graphs. Some asymptotic values of their $(k,\ell)$-rainbow index are obtained.
A tree in an edge colored graph is said to be a rainbow tree if no two edges on the tree share the same color. Given two positive integers $k$, $\ell$ with $k\geq 3$, the \emph{$(k,\ell)$-rainbow index} $rx_{k,\ell}(G)$ of $G$ is the minimum number of colors needed in an edge-coloring of $G$ such that for any set $S$ of $k$ vertices of $G$, there exist $\ell$ internally disjoint rainbow trees connecting $S$. This concept was introduced by Chartrand et. al., and there have been very few related results about it. In this paper, We establish a sharp threshold function for $rx_{k,\ell}(G_{n,p})\leq k$ and $rx_{k,\ell}(G_{n,M})\leq k,$ respectively, where $G_{n,p}$ and $G_{n,M}$ are the usually defined random graphs.
Let $G$ be a nontrivial connected graph with an edge-coloring $c:E(G)\rightarrow \{1,2,\ldots,q\},$ $q\in \mathbb{N}$, where adjacent edges may be colored the same. A tree $T$ in $G$ is a $rainbow tree$ if no two edges of $T$ receive the same color. For a vertex set $S\subseteq V(G)$, the tree connecting $S$ in $G$ is called an $S$-tree. The minimum number of colors that are needed in an edge-coloring of $G$ such that there is a rainbow $S$-tree for each $k$-set $S$ of $V(G)$ is called the $k$-rainbow index of $G$, denoted by $rx_k(G)$. In \cite{Zhang}, they got that the $k$-rainbow index of a tree is $n-1$ and the $k$-rainbow index of a unicyclic graph is $n-1$ or $n-2$. So there is an intriguing problem: Characterize graphs with the $k$-rainbow index $n-1$ and $n-2$. In this paper, we focus on $k=3$, and characterize the graphs whose 3-rainbow index is $n-1$ and $n-2$, respectively.
The generalized $k$-connectivity $κ_k(G)$ of a graph $G$, introduced by Chartrand et al., is a natural and nice generalization of the concept of (vertex-)connectivity. In this paper, we prove that for any two connected graphs $G$ and $H$, $κ_3(G\circ H)\geq κ_3(G)|V(H)|$. We also give upper bounds for $κ_3(G\Box H)$ and $κ_3(G\circ H)$. Moreover, all the bounds are sharp.
For $S\subseteq V(G)$ and $|S|\geq 2$, $λ(S)$ is the maximum number of edge-disjoint trees connecting $S$ in $G$. For an integer $k$ with $2\leq k\leq n$, the \emph{generalized $k$-edge-connectivity} $λ_k(G)$ of $G$ is then defined as $λ_k(G)= min\{λ(S) : S\subseteq V(G) \ and \ |S|=k\}$. It is also clear that when $|S|=2$, $λ_2(G)$ is nothing new but the standard edge-connectivity $λ(G)$ of $G$. In this paper, graphs of order $n$ such that $λ_3(G)=n-3$ is characterized. Furthermore, we determine the minimal number of edges of a graph of order $n$ with $λ_3=1,n-3,n-2$ and give a sharp lower bound for $2\leq λ_3\leq n-4$.
For any graph $G$ of order $n$, the spanning tree packing number \emph{$STP(G)$}, is the maximum number of edge-disjoint spanning trees contained in $G$. In this paper, we obtain some sharp lower bounds for the spanning tree numbers of Cartesian product graphs and Lexicographic product graphs.
Let $G$ be a nontrivial connected graph with an edge-coloring $c: E(G)\rightarrow \{1,2,...,q\},$ $q \in \mathbb{N}$, where adjacent edges may be colored the same. A tree $T$ in $G$ is a $rainbow tree$ if no two edges of $T$ receive the same color. For a vertex subset $S\subseteq V(G)$, a tree that connects $S$ in $G$ is called an $S$-tree. The minimum number of colors that are needed in an edge-coloring of $G$ such that there is a rainbow $S$-tree for each $k$-subset $S$ of $V(G)$ is called $k$-rainbow index, denoted by $rx_k(G)$. In this paper, we first determine the graphs whose 3-rainbow index equals 2, $m,$ $m-1$, $m-2$, respectively. We also obtain the exact values of $rx_3(G)$ for regular complete bipartite and multipartite graphs and wheel graphs. Finally, we give a sharp upper bound for $rx_3(G)$ of 2-connected graphs and 2-edge connected graphs, and graphs whose $rx_3(G)$ attains the upper bound are characterized.
The revised Szeged index of a graph $G$ is defined as $Sz^*(G)=\sum_{e=uv \in E}(n_u(e)+ n_0(e)/2)(n_v(e)+ n_0(e)/2),$ where $n_u(e)$ and $n_v(e)$ are, respectively, the number of vertices of $G$ lying closer to vertex $u$ than to vertex $v$ and the number of vertices of $G$ lying closer to vertex $v$ than to vertex $u$, and $n_0(e)$ is the number of vertices equidistant to $u$ and $v$. In this paper, we give an upper bound of the revised Szeged index for a connected tricyclic graph, and also characterize those graphs that achieve the upper bound.
Given a graph $G$, let $G^σ$ be an oriented graph of $G$ with the orientation $σ$ and skew-adjacency matrix $S(G^σ)$. Then the spectrum of $S(G^σ)$ consisting of all the eigenvalues of $S(G^σ)$ is called the skew-spectrum of $G^σ$, denoted by $Sp(G^σ)$. The skew energy of the oriented graph $G^σ$, denoted by $\mathcal{E}_S(G^σ)$, is defined as the sum of the norms of all the eigenvalues of $S(G^σ)$. In this paper, we give orientations of the Kronecker product $H\otimes G$ and the strong product $H\ast G$ of $H$ and $G$ where $H$ is a bipartite graph and $G$ is an arbitrary graph. Then we determine the skew-spectra of the resultant oriented graphs. As applications, we construct new families of oriented graphs with maximum skew energy. Moreover, we consider the skew energy of the orientation of the lexicographic product $H[G]$ of a bipartite graph $H$ and a graph $G$.
The Turán number of a $k$-uniform hypergraph $H$, denoted by $e{x_k}\left({n;H} \right)$, is the maximum number of edges in any $k$-uniform hypergraph $F$ on $n$ vertices which does not contain $H$ as a subgraph. Let $\mathcal{C}_{\ell}^{\left(k \right)}$ denote the family of all $k$-uniform minimal cycles of length $\ell$, $\mathcal{S}(\ell_1,\ldots,\ell_r)$ denote the family of hypergraphs consisting of unions of $r$ vertex disjoint minimal cycles of length $\ell_1,\ldots,\ell_r$, respectively, and $\mathbb{C}_{\ell}^{\left(k \right)}$ denote a $k$-uniform linear cycle of length $\ell$. We determine precisely $e{x_k}\left({n;\mathcal{S}(\ell_1,\ldots,\ell_r)} \right)$ and $e{x_k}\left({n;\mathbb{C}_{\ell_1}^{\left(k \right)}, \ldots, \mathbb{C}_{\ell_r}^{\left(k \right)}} \right)$ for sufficiently large $n$. The results extend recent results of Füredi and Jiang who determined the Turán numbers for single $k$-uniform minimal cycles and linear cycles.
Let $\mathcal{T}_n$ be the set of trees with $n$ vertices. Suppose that each tree in $\mathcal{T}_n$ is equally likely. We show that the number of different rooted trees of a tree equals $(μ_r+o(1))n$ for almost every tree of $\mathcal{T}_n$, where $μ_r$ is a constant. As an application, we show that the number of any given pattern in $\mathcal{T}_n$ is also asymptotically normally distributed with mean $\sim μ_M n$ and variance $\sim σ_M n$, where $μ_M, σ_M$ are some constants related to the given pattern. This solves an open question claimed in Kok's thesis.
Given a graph $G$, let $G^σ$ be an oriented graph of $G$ with the orientation $σ$ and skew-adjacency matrix $S(G^σ)$. Then the spectrum of $S(G^σ)$ is called the skew-spectrum of $G^σ$, denoted by $Sp_S(G^σ)$. It is known that a graph $G$ is bipartite if and only if there is an orientation $σ$ of $G$ such that $Sp_S(G^σ)=iSp(G)$. In [D. Cui, Y. Hou, On the skew spectra of Cartesian products of graphs, Electron. J. Combin. 20(2013), #P19], Cui and Hou conjectured that such orientation of a bipartite graph is unique under switching-equivalence. In this paper, we prove that the conjecture is true. Moreover, we give an orientation of the Cartesian product of a bipartite graph and a graph, and then determine the skew-spectrum of the resulting oriented product graph, which generalizes Cui and Hou's result, and can be used to construct more oriented graphs with maximum skew energy.
The generalized $k$-connectivity $κ_k(G)$ of a graph $G$ was introduced by Chartrand et al. in 1984. It is natural to introduce the concept of generalized $k$-edge-connectivity $λ_k(G)$. For general $k$, the generalized $k$-edge-connectivity of a complete graph is obtained. For $k\geq 3$, tight upper and lower bounds of $κ_k(G)$ and $λ_k(G)$ are given for a connected graph $G$ of order $n$, that is, $1\leq κ_k(G)\leq n-\lceil\frac{k}{2}\rceil$ and $1\leq λ_k(G)\leq n-\lceil\frac{k}{2}\rceil$. Graphs of order $n$ such that $κ_k(G)=n-\lceil\frac{k}{2}\rceil$ and $λ_k(G)=n-\lceil\frac{k}{2}\rceil$ are characterized, respectively. Nordhaus-Gaddum-type results for the generalized $k$-connectivity are also obtained. For $k=3$, we study the relation between the edge-connectivity and the generalized 3-edge-connectivity of a graph. Upper and lower bounds of $λ_3(G)$ for a graph $G$ in terms of the edge-connectivity $λ$ of $G$ are obtained, that is, $\frac{3λ-2}{4}\leq λ_3(G)\leq λ$, and two graph classes are given showing that the upper and lower bounds are tight. From these bounds, we obtain that $λ(G)-1\leq λ_3(G)\leq λ(G)$ if $G$ is a connected planar graph, and the relation between the generalized 3-connectivity and generalized 3-edge-connectivity of a graph and its line graph.