Yindi Weng | Searcharxiv

arXiv2023

Some results on the rainbow vertex-disconnection colorings of graphs

Let $G$ be a nontrivial connected and vertex-colored graph. A vertex subset $X$ 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 $(G-xy)-S$. For a connected graph $G$, the \emph{rainbow vertex-disconnection number} of $G$, $rvd(G)$, is the minimum number of colors that are needed to make $G$ rainbow vertex-disconnected. In this paper, we prove for any $K_4$-minor free graph, $rvd(G)\leq Δ(G)$ and the bound is sharp. We show it is $NP$-complete to determine the rainbow vertex-disconnection number for bipartite graphs and split graphs. Moreover, we show for every $ε>0$, it is impossible to efficiently approximate the rainbow vertex-disconnection number of any bipartite graph and split graph within a factor of $n^{\frac{1}{3}-ε}$ unless $ZPP=NP$.

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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

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

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 $S$ of $G$ 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 $(G-xy)-S$. 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 characterize all graphs of order $n$ with rainbow vertex-disconnection number $k$ for $k\in\{1,2,n\}$, and determine the rainbow vertex-disconnection numbers of some special graphs. Moreover, we study the extremal problems on the number of edges of a connected graph $G$ with order $n$ and $rvd(G)=k$ for given integers $k$ and $n$ with $1\leq k\leq n$.

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arXiv2019

Proper disconnection of graphs

For an edge-colored graph $G$, a set $F$ of edges of $G$ is called a \emph{proper cut} if $F$ is an edge-cut of $G$ and any pair of adjacent edges in $F$ are assigned by different colors. An edge-colored graph is \emph{proper disconnected} if for each pair of distinct vertices of $G$ there exists a proper edge-cut separating them. For a connected graph $G$, the \emph{proper disconnection number} of $G$, denoted by $pd(G)$, is the minimum number of colors that are needed in order to make $G$ proper disconnected. In this paper, we first give the exact values of the proper disconnection numbers for some special families of graphs. Next, we obtain a sharp upper bound of $pd(G)$ for a connected graph $G$ of order $n$, i.e, $pd(G)\leq \min\{ χ'(G)-1, \left \lceil \frac{n}{2} \right \rceil\}$. Finally, we show that for given integers $k$ and $n$, the minimum size of a connected graph $G$ of order $n$ with $pd(G)=k$ is $n-1$ for $k=1$ and $n+2k-4$ for $2\leq k\leq \lceil\frac{n}{2}\rceil$.

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