arXiv subjects

Yaping Mao

Publications and source records attributed to Yaping Mao.

99 records · Page 6Linked to original sources

arXiv2013

The minimal size of a graph with given generalized 3-edge-connectivity

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 $2\leq λ_3\leq n-4$.

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arXiv2013

Lower bounds for the spanning tree numbers of two graph products

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.

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arXiv2013

On the generalized (edge-)connectivity of graphs

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.

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arXiv2013

Further hardness results on the generalized connectivity of graphs

The generalized $k$-connectivity $κ_k(G)$ of a graph $G$ was introduced by Chartrand et al. in 1984, which is a nice generalization of the classical connectivity. Recently, as a natural counterpart, Li et al. proposed the concept of generalized edge-connectivity for a graph. In this paper, we determine the computational complexity of the generalized connectivity and generalized edge-connectivity of a graph. Two conjectures are also proved to be true.

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arXiv2013

On extremal graphs with at most $\ell$ internally disjoint Steiner trees connecting any n-1 vertices

The concept of maximum local connectivity $\bar κ$ of a graph was introduced by Bollobás. One of the problems about it is to determine the largest number of edges $f(n;\barκ\leq \ell)$ for graphs of order $n$ that have local connectivity at most $\ell$. We consider a generalization of the above concept and problem. For $S\subseteq V(G)$ and $|S|\geq 2$, the \emph{generalized local connectivity} $κ(S)$ is the maximum number of internally disjoint trees connecting $S$ in $G$. The parameter $\barκ_k(G)=max\{κ(S)|S\subseteq V(G),|S|=k\}$ is called the \emph{maximum generalized local connectivity} of $G$. This paper it to consider the problem of determining the largest number $f(n;\barκ_k\leq \ell)$ of edges for graphs of order $n$ that have maximum generalized local connectivity at most $\ell$. The exact value of $f(n;\barκ_k\leq \ell)$ for $k=n,n-1$ is determined. For a general $k$, we construct a graph to obtain a sharp lower bound.

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arXiv2013

On extremal graphs with at most two internally disjoint Steiner trees connecting any three vertices

The problem of determining the smallest number of edges, $h(n;\barκ\geq r)$, which guarantees that any graph with $n$ vertices and $h(n;\barκ\geq r)$ edges will contain a pair of vertices joined by $r$ internally disjoint paths was posed by Erdös and Gallai. Bollobás considered the problem of determining the largest number of edges $f(n;\barκ\leq \ell)$ for graphs with $n$ vertices and local connectivity at most $\ell$. One can see that $f(n;\barκ\leq \ell)= h(n;\barκ\geq \ell+1)-1$. These two problems had received a wide attention of many researchers in the last few decades. In the above problems, only pairs of vertices connected by internally disjoint paths are considered. In this paper, we study the number of internally disjoint Steiner trees connecting sets of vertices with cardinality at least 3.

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arXiv2012

Nordhaus-Gaddum-type results for the generalized edge-connectivity of graphs

Let $G$ be a graph, $S$ be a set of vertices of $G$, and $λ(S)$ be the maximum number $\ell$ of pairwise edge-disjoint trees $T_1, T_2,..., T_{\ell}$ in $G$ such that $S\subseteq V(T_i)$ for every $1\leq i\leq \ell$. The generalized $k$-edge-connectivity $λ_k(G)$ of $G$ is defined as $λ_k(G)= min\{λ(S) | S\subseteq V(G) \ and \ |S|=k\}$. Thus $λ_2(G)=λ(G)$. In this paper, we consider the Nordhaus-Gaddum-type results for the parameter $λ_k(G)$. We determine sharp upper and lower bounds of $λ_k(G)+λ_k(\bar{G})$ and $λ_k(G)... λ_k(\bar{G})$ for a graph $G$ of order $n$, as well as for a graph of order $n$ and size $m$. Some graph classes attaining these bounds are also given.

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arXiv2012

Graphs with large generalized 3-connectivity

Let $S$ be a nonempty set of vertices of a connected graph $G$. A collection $T_1,..., T_\ell$ of trees in $G$ is said to be internally disjoint trees connecting $S$ if $E(T_i)\cap E(T_j)= \emptyset$ and $V(T_i)\cap V(T_j)=S$ for any pair of distinct integers $i, j$, where $1 \leq i, j \leq r$. For an integer $k$ with $2 \leq k \leq n$, the generalized $k$-connectivity $κ_k(G)$ of $G$ is the greatest positive integer $r$ such that $G$ contains at least $r$ internally disjoint trees connecting $S$ for any set $S$ of $k$ vertices of $G$. Obviously, $κ_2(G)$ is the connectivity of $G$. In this paper, sharp upper and lower bounds of $κ_3(G)$ are given for a connected graph $G$ of order $n$, that is, $1 \leq κ_3(G) \leq n - 2$. Graphs of order $n$ such that $κ_3(G) = n - 2, n - 3$ are characterized, respectively.

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arXiv2012

The strong rainbow vertex-connection of graphs

A vertex-colored graph $G$ is said to be rainbow vertex-connected if every two vertices of $G$ are connected by a path whose internal vertices have distinct colors, such a path is called a rainbow path. The rainbow vertex-connection number of a connected graph $G$, denoted by $rvc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow vertex-connected. If for every pair $u, v$ of distinct vertices, $G$ contains a rainbow $u-v$ geodesic, then $G$ is strong rainbow vertex-connected. The minimum number $k$ for which there exists a $k$-vertex-coloring of $G$ that results in a strongly rainbow vertex-connected graph is called the strong rainbow vertex-connection number of $G$, denoted by $srvc(G)$. Observe that $rvc(G)\leq srvc(G)$ for any nontrivial connected graph $G$. In this paper, sharp upper and lower bounds of $srvc(G)$ are given for a connected graph $G$ of order $n$, that is, $0\leq srvc(G)\leq n-2$. Graphs of order $n$ such that $srvc(G)= 1, 2, n-2$ are characterized, respectively. It is also shown that, for each pair $a, b$ of integers with $a\geq 5$ and $b\geq (7a-8)/5$, there exists a connected graph $G$ such that $rvc(G)=a$ and $srvc(G)=b$.

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