The rainbow vertex-index of complementary graphs

A vertex-colored graph $G$ is \emph{rainbow vertex-connected} if two vertices are connected by a path whose internal vertices have distinct colors. The \emph{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 vertex-rainbow $u-v$ geodesic, then $G$ is \emph{strongly rainbow vertex-connected}. The minimum $k$ for which there exists a $k$-coloring of $G$ that results in a strongly rainbow-vertex-connected graph is called the \emph{strong rainbow vertex number} $srvc(G)$ of $G$. Thus $rvc(G)\leq srvc(G)$ for every nontrivial connected graph $G$. A tree $T$ in $G$ is called a \emph{rainbow vertex tree} if the internal vertices of $T$ receive different colors. 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{rainbow vertex $S$-tree} if the internal vertices of $T$ receive distinct colors. The minimum number of colors that are needed in a vertex-coloring of $G$ such that there is a rainbow vertex $S$-tree for every $k$-set $S$ of $V(G)$ is called the {\it $k$-rainbow vertex-index} of $G$, denoted by $rvx_k(G)$. In this paper, we first investigate the strong rainbow vertex-connection of complementary graphs. The $k$-rainbow vertex-index of complementary graphs are also studied

Fractional matching preclusion for butterfly derived networks

The matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost perfect matchings. As a generalization, Liu and Liu [18] recently introduced the concept of fractional matching preclusion number. The fractional matching preclusion number (FMP number) of G, denoted by fmp(G), is the minimum number of edges whose deletion leaves the resulting graph without a fractional perfect matching. The fractional strong matching preclusion number (FSMP number) of G, denoted by fsmp(G), is the minimum number of vertices and edges whose deletion leaves the resulting graph without a fractional perfect matching. In this paper, we study the fractional matching preclusion number and the fractional strong matching preclusion number for butterfly network, augmented butterfly network and enhanced butterfly network

On Cyclic-Vertex Connectivity of n,k-Star Graphs

A vertex subset F ⊆ VG is a cyclic vertex-cut of a connected graph G if G−F is disconnected and at least two of its components contain cycles. The cyclic vertex-connectivity κcG is denoted as the cardinality of a minimum cyclic vertex-cut. In this paper, we show that the cyclic vertex-connectivity of the n,k-star network Sn,k is κcSn,k=n+2k−5 for any integer n≥4 and k≥2

The rainbow vertex-index of complementary graphs

A vertex-colored graph $G$ is \emph{rainbow vertex-connected} if two vertices are connected by a path whose internal vertices have distinct colors. The \emph{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 vertex-rainbow $u-v$ geodesic, then $G$ is \emph{strongly rainbow vertex-connected}. The minimum $k$ for which there exists a $k$-coloring of $G$ that results in a strongly rainbow-vertex-connected graph is called the \emph{strong rainbow vertex number} $srvc(G)$ of $G$. Thus $rvc(G)\leq srvc(G)$ for every nontrivial connected graph $G$. A tree $T$ in $G$ is called a \emph{rainbow vertex tree} if the internal vertices of $T$ receive different colors. 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{rainbow vertex $S$-tree} if the internal vertices of $T$ receive distinct colors. The minimum number of colors that are needed in a vertex-coloring of $G$ such that there is a rainbow vertex $S$-tree for every $k$-set $S$ of $V(G)$ is called the {\it $k$-rainbow vertex-index} of $G$, denoted by $rvx_k(G)$. In this paper, we first investigate the strong rainbow vertex-connection of complementary graphs. The $k$-rainbow vertex-index of complementary graphs are also studied

The chromatic equivalence class of graph Bn−6,1,2‾\overline{B_{n-6,1,2}}

By h(G,x) and P(G,λ) we denote the adjoint polynomial and the chromatic polynomial of graph G, respectively. A new invariant of graph G, which is the fourth character R₄(G), is given in this paper. Using the properties of the adjoint polynomials, the adjoint equivalence class of graph $B_{n-6,1,2}$ is determined, which can be regarded as the continuance of the paper written by Wang et al. [J. Wang, R. Liu, C. Ye and Q. Huang, A complete solution to the chromatic equivalence class of graph $\overline{B_{n-7,1,3}}$, Discrete Math. (2007), doi: 10.1016/j.disc.2007.07.030]. According to the relations between h(G,x) and P(G,λ), we also simultaneously determine the chromatic equivalence class of $\overline{B_{n-6,1,2}}$ that is the complement of $B_{n-6,1,2}$