Pendant-tree connectivity of line graphs
The concept of pendant-tree connectivity, introduced by Hager in 1985, is a generalization of classical vertex-connectivity. In this paper, we study pendant-tree connectivity of line graphs.
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arXiv subjects
Yaping Mao
Publications and source records attributed to Yaping Mao.
Line k-Arboricity in Product Networks
A \emph{linear $k$-forest} is a forest whose components are paths of length at most $k$. The \emph{linear $k$-arboricity} of a graph $G$, denoted by ${\rm la}_k(G)$, is the least number of linear $k$-forests needed to decompose $G$. Recently, Zuo, He and Xue studied the exact values of the linear $(n-1)$-arboricity of Cartesian products of various combinations of complete graphs, cycles, complete multipartite graphs. In this paper, for general $k$ we show that $\max\{{\rm la}_{k}(G),{\rm la}_{\ell}(H)\}\leq {\rm la}_{\max\{k,\ell\}}(G\Box H)\leq {\rm la}_{k}(G)+{\rm la}_{\ell}(H)$ for any two graphs $G$ and $H$. Denote by $G\circ H$, $G\times H$ and $G\boxtimes H$ the lexicographic product, direct product and strong product of two graphs $G$ and $H$, respectively. We also derive upper and lower bounds of ${\rm la}_{k}(G\circ H)$, ${\rm la}_{k}(G\times H)$ and ${\rm la}_{k}(G\boxtimes H)$ in this paper. The linear $k$-arboricity of a $2$-dimensional grid graph, a $r$-dimensional mesh, a $r$-dimensional torus, a $r$-dimensional generalized hypercube and a $2$-dimensional hyper Petersen network are also studied.
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Path connectivity of line graphs
Dirac showed that in a $(k-1)$-connected graph there is a path through each $k$ vertices. The path $k$-connectivity $π_k(G)$ of a graph $G$, which is a generalization of Dirac's notion, was introduced by Hager in 1986. In this paper, we study path connectivity of line graphs.
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On the Equitable Vertex Arboricity of Graphs
The equitable coloring problem, introduced by Meyer in 1973, has received considerable attention and research. Recently, Wu, Zhang and Li introduced the concept of equitable $(t,k)$-tree-coloring, which can be regarded as a generalization of proper equitable $t$-coloring. The \emph{strong equitable vertex $k$-arboricity} of $G$, denoted by ${va_k}^\equiv(G)$, is the smallest integer $t$ such that $G$ has an equitable $(t', k)$-tree-coloring for every $t'\geq t$. The exact value of strong equitable vertex $k$-arboricity of complete equipartition bipartite graph $K_{n,n}$ was studied by Wu, Zhang and Li. In this paper, we first get a sharp upper bound of strong equitable vertex arboricity of complete bipartite graph$K_{n,n+\ell} \ (1\leq \ell\leq n)$, that is, ${va_2}^\equiv(K_{n,n+\ell})\leq2\left\lfloor{\frac{n+\ell+1}{3}}\right\rfloor$. Next, we obtain a sufficient and necessary condition on an equitable $(q,\infty)$-tree coloring of a complete equipartition tripartite graph, and study the strong equitable vertex arboricity of forests. For a simple graph $G$ of order $n$, we show that $1\leq {va_k}^\equiv(G)\leq \lceil n/2 \rceil$. Furthermore, graphs with ${va_k}^\equiv(G)=1,\lceil\frac{n}{2}\rceil,\lceil\frac{n}{2}\rceil-1$ are characterized, respectively. In the end, we obtain the Nordhaus-Gaddum type results of strong equitable vertex $k$-arboricity for general $k$.
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The Steiner diameter of a graph
The Steiner distance of a graph, introduced by Chartrand, Oellermann, Tian and Zou 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 \emph{Steiner distance} $d(S)$ among the vertices of $S$ is the minimum size among all connected subgraphs whose vertex sets contain $S$. Let $n,k$ be two integers with $2\leq k\leq n$. Then the \emph{Steiner $k$-eccentricity $e_k(v)$} of a vertex $v$ of $G$ is defined by $e_k(v)=\max \{d(S)\,|\,S\subseteq V(G), \ |S|=k, \ and \ v\in S \}$. Furthermore, the \emph{Steiner $k$-diameter} of $G$ is $sdiam_k(G)=\max \{e_k(v)\,|\, v\in V(G)\}$. In 2011, Chartrand, Okamoto and Zhang showed that $k-1\leq sdiam_k(G)\leq n-1$. In this paper, graphs with $sdiam_3(G)=2,3,n-1$ are characterized, respectively. We also consider the Nordhaus-Gaddum-type results for the parameter $sdiam_k(G)$. We determine sharp upper and lower bounds of $sdiam_k(G)+sdiam_k(\overline{G})$ and $sdiam_k(G)\cdot sdiam_k(\overline{G})$ for a graph $G$ of order $n$. Some graph classes attaining these bounds are also given.
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Interval minors of complete multipartite graphs
Interval minors of bipartite graphs were introduced by Jacob Fox in the study of Stanley-Wilf limits. Recently, Mohar, Rafiey, Tayfeh-Rezaie and Wu investigated the maximum number of edges in $K_{k,\ell}$-interval minor free bipartite graphs when $k=2$ and $k=3$. In this paper, we investigate the maximum number of edges in $K_{k,\ell}$-interval minor free bipartite graphs for general $k$ and $\ell$. We also study the maximum number of edges in $K_{\ell_1,\ell_2,\cdots,\ell_t}$-interval minor free multipartite graphs.
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A survey on the generalized connectivity of graphs
The generalized $k$-connectivity $κ_k(G)$ of a graph $G$ was introduced by Hager before 1985. As its a natural counterpart, we introduced the concept of generalized edge-connectivity $λ_k(G)$, recently. In this paper we summarize the known results on the generalized connectivity and generalized edge-connectivity. After an introductory section, the paper is then divided into nine sections: the generalized (edge-)connectivity of some graph classes, algorithms and computational complexity, sharp bounds of $κ_k(G)$ and $λ_k(G)$, graphs with large generalized (edge-)connectivity, Nordhaus-Gaddum-type results, graph operations, extremal problems, and some results for random graphs and multigraphs. It also contains some conjectures and open problems for further studies.
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Graphs with large generalized (edge-)connectivity
The generalized $k$-connectivity $κ_k(G)$ of a graph $G$, introduced by Hager in 1985, is a nice generalization of the classical connectivity. Recently, as a natural counterpart, we proposed the concept of generalized $k$-edge-connectivity $λ_k(G)$. In this paper, graphs of order $n$ such that $κ_k(G)=n-\frac{k}{2}-1$ and $λ_k(G)=n-\frac{k}{2}-1$ for even $k$ are characterized.
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On the regular k-independence number of graphs
The \emph{regular independence number}, introduced by Albertson and Boutin in 1990, is the maximum cardinality of an independent set of $G$ in which all vertices have equal degree in $G$. Recently, Caro, Hansberg and Pepper introduced the concept of regular $k$-independence number, which is a natural generalization of the regular independence number. A \emph{$k$-independent set} is a set of vertices whose induced subgraph has maximum degree at most $k$. The \emph{regular $k$-independence number} of $G$, denoted by $α_{k-reg}(G)$, is defined as the maximum cardinality of a $k$-independent set of $G$ in which all vertices have equal degree in $G$. In this paper, the exact values of the regular $k$-independence numbers of some special graphs are obtained. We also get some lower and upper bounds for the regular $k$-independence number of trees with given diameter, and the lower bounds for the regular $k$-independence number of line graphs. For a simple graph $G$ of order $n$, we show that $1\leqα_{k-reg}(G)\leq n$ and characterize the extremal graphs. The Nordhaus-Gaddum-type results for the regular $k$-independence number of graphs are also obtained.
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On the pedant tree-connectivity of graphs
The concept of pedant tree-connectivity was introduced by Hager in 1985. For a graph $G=(V,E)$ and a set $S\subseteq V(G)$ of at least two vertices, \emph{an $S$-Steiner tree} or \emph{a Steiner tree connecting $S$} (or simply, \emph{an $S$-tree}) is a such subgraph $T=(V',E')$ of $G$ that is a tree with $S\subseteq V'$. For an $S$-Steiner tree, if the degree of each vertex in $S$ is equal to one, then this tree is called a \emph{pedant $S$-Steiner tree}. Two pedant $S$-Steiner trees $T$ and $T'$ are said to be \emph{internally disjoint} if $E(T)\cap E(T')=\varnothing$ and $V(T)\cap V(T')=S$. For $S\subseteq V(G)$ and $|S|\geq 2$, the \emph{local pedant-tree connectivity} $τ_G(S)$ is the maximum number of internally disjoint pedant $S$-Steiner trees in $G$. For an integer $k$ with $2\leq k\leq n$, \emph{$k$-pedant tree-connectivity} is defined as $τ_k(G)=\min\{τ_G(S)\,|\,S\subseteq V(G),|S|=k\}$. In this paper, we first study the sharp bounds of pedant tree-connectivity. Next, we obtain the exact value of a threshold graph, and give an upper bound of the pedant-tree $k$-connectivity of a complete multipartite graph. For a connected graph $G$, we show that $0\leq τ_k(G)\leq n-k$, and graphs with $τ_k(G)=n-k,n-k-1,n-k-2,0$ are characterized in this paper. In the end, we obtain the Nordhaus-Guddum type results for pedant tree-connectivity.
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Constructing Internally Disjoint Pendant Steiner Trees in Cartesian Product Networks
The concept of pedant tree-connectivity was introduced by Hager in 1985. For a graph $G=(V,E)$ and a set $S\subseteq V(G)$ of at least two vertices, \emph{an $S$-Steiner tree} or \emph{a Steiner tree connecting $S$} (or simply, \emph{an $S$-tree}) is a such subgraph $T=(V',E')$ of $G$ that is a tree with $S\subseteq V'$. For an $S$-Steiner tree, if the degree of each vertex in $S$ is equal to one, then this tree is called a \emph{pedant $S$-Steiner tree}. Two pedant $S$-Steiner trees $T$ and $T'$ are said to be \emph{internally disjoint} if $E(T)\cap E(T')=\varnothing$ and $V(T)\cap V(T')=S$. For $S\subseteq V(G)$ and $|S|\geq 2$, the \emph{local pedant tree-connectivity} $τ_G(S)$ is the maximum number of internally disjoint pedant $S$-Steiner trees in $G$. For an integer $k$ with $2\leq k\leq n$, \emph{pedant tree $k$-connectivity} is defined as $τ_k(G)=\min\{τ_G(S)\,|\,S\subseteq V(G),|S|=k\}$. In this paper, we prove that for any two connected graphs $G$ and $H$, $τ_3(G\Box H)\geq \min\{3\lfloor\frac{τ_3(G)}{2}\rfloor,3\lfloor\frac{τ_3(H)}{2}\rfloor\}$. Moreover, the bound is sharp.
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On the equitable vertex arboricity of complete tripartite graphs
The equitable coloring problem, introduced by Meyer in 1973, has received considerable attention and research. Recently, Wu et al. introduced the concept of equitable (t,k)-tree-coloring, which can be viewed as a generalization of proper equitable t-coloring. The strong equitable vertex k-arboricity of complete bipartite equipartition graphs was investigated in 2013. In this paper, we study the exact value of the strong equitable vertex 3-arboricity of complete equipartition tripartite graphs.
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Proper connection number and graph products
A path $P$ in an edge-colored graph $G$ is called \emph{a proper path} if no two adjacent edges of $P$ are colored the same, and $G$ is \emph{proper connected} if every two vertices of $G$ are connected by a proper path in $G$. The \emph{proper connection number} of a connected graph $G$, denoted by $pc(G)$, is the minimum number of colors that are needed to make $G$ proper connected. In this paper, we study the proper connection number on the lexicographical, strong, Cartesian, and direct product and present several upper bounds for these products of graphs.
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Monochromatic connectivity and graph products
The concept of monochromatic connectivity was introduced by Caro and Yuster. A path in an edge-colored graph is called a \emph{monochromatic path} if all the edges on the path are colored the same. An edge-coloring of $G$ is a \emph{monochromatic connection coloring} ($MC$-coloring, for short) if there is a monochromatic path joining any two vertices in $G$. The \emph{monochromatic connection number}, denoted by $mc(G)$, is defined to be the maximum number of colors used in an $MC$-coloring of a graph $G$. In this paper, we study the monochromatic connection number on the lexicographical, strong, Cartesian and direct product and present several upper and lower bounds for these products of graphs.
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The vertex-rainbow index of a graph
The $k$-rainbow index $rx_k(G)$ of a connected graph $G$ was introduced by Chartrand, Okamoto and Zhang in 2010. As a natural counterpart of the $k$-rainbow index, we introduced the concept of $k$-vertex-rainbow index $rvx_k(G)$ in this paper. For a graph $G=(V,E)$ and a set $S\subseteq V$ of at least two vertices, \emph{an $S$-Steiner tree} or \emph{a Steiner tree connecting $S$} (or simply, \emph{an $S$-tree}) is a such subgraph $T=(V',E')$ of $G$ that is a tree with $S\subseteq V'$. For $S\subseteq V(G)$ and $|S|\geq 2$, an $S$-Steiner tree $T$ is said to be a \emph{vertex-rainbow $S$-tree} if the vertices of $V(T)\setminus S$ have distinct colors. For a fixed integer $k$ with $2\leq k\leq n$, the vertex-coloring $c$ of $G$ is called a \emph{$k$-vertex-rainbow coloring} if for every $k$-subset $S$ of $V(G)$ there exists a vertex-rainbow $S$-tree. In this case, $G$ is called \emph{vertex-rainbow $k$-tree-connected}. The minimum number of colors that are needed in a $k$-vertex-rainbow coloring of $G$ is called the \emph{$k$-vertex-rainbow index} of $G$, denoted by $rvx_k(G)$. When $k=2$, $rvx_2(G)$ is nothing new but the vertex-rainbow connection number $rvc(G)$ of $G$. In this paper, sharp upper and lower bounds of $srvx_k(G)$ are given for a connected graph $G$ of order $n$,\ that is, $0\leq srvx_k(G)\leq n-2$. We obtain the Nordhaus-Guddum results for $3$-vertex-rainbow index, and show that $rvx_3(G)+rvx_3(\overline{G})=4$ for $n=4$ and $2\leq rvx_3(G)+rvx_3(\overline{G})\leq n-1$ for $n\geq 5$. Let $t(n,k,\ell)$ denote the minimal size of a connected graph $G$ of order $n$ with $rvx_k(G)\leq \ell$, where $2\leq \ell\leq n-2$ and $2\leq k\leq n$. The upper and lower bounds for $t(n,k,\ell)$ are also obtained.
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Formally self-dual linear binary codes from circulant graphs
In 2002, Tonchev first constructed some linear binary codes defined by the adjacency matrices of undirected graphs. So, graph is an important tool for searching optimum codes. In this paper, we introduce a new method of searching (proposed) optimum formally self-dual linear binary codes from circulant graphs.
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Additive codes over $GF(4)$ from circulant graphs
In $2006$, Danielsen and Parker \cite{DP} proved that every self-dual additive code over $GF(4)$ is equivalent to a graph code. So, graph is an important tool for searching (proposed) optimum codes. In this paper, we introduce a new method of searching (proposed) optimum additive codes from circulant graphs.
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The generalized 3-connectivity of Lexicographic product graphs
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.
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