arXiv subjects

Yuefang Sun

Publications and source records attributed to Yuefang Sun.

42 records · Page 3Linked to original sources

arXiv2011

Rainbow connections of graphs -- A survey

The concept of rainbow connection was introduced by Chartrand et al. in 2008. It is fairly interesting and recently quite a lot papers have been published about it. In this survey we attempt to bring together most of the results and papers that dealt with it. We begin with an introduction, and then try to organize the work into five categories, including (strong) rainbow connection number, rainbow $k$-connectivity, $k$-rainbow index, rainbow vertex-connection number, algorithms and computational complexity. This survey also contains some conjectures, open problems or questions.

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arXiv2010

Rainbow connection numbers of complementary graphs

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. In this paper, we provide a new approach to investigate the rainbow connection number of a graph $G$ according to some constraints to its complement graph $\bar{G}$. We first derive that for a connected graph $G$, if $\bar{G}$ does not belong to the following two cases: $(i)$~$diam(\bar{G})=2,3$, $(ii)~\bar{G}$ contains exactly two connected components and one of them is trivial, then $rc(G)\leq 4$, where $diam(G)$ is the diameter of $G$. Examples are given to show that this bound is best possible. Next we derive that for a connected graph $G$, if $\bar{G}$ is triangle-free, then $rc(G)\leq 6$.

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arXiv2010

On strong rainbow connection number

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. For any two vertices $u$ and $v$ of $G$, a rainbow $u-v$ geodesic in $G$ is a rainbow $u-v$ path of length $d(u,v)$, where $d(u,v)$ is the distance between $u$ and $v$. The graph $G$ is strongly rainbow connected if there exists a rainbow $u-v$ geodesic for any two vertices $u$ and $v$ in $G$. The strong rainbow connection number of $G$, denoted $src(G)$, is the minimum number of colors that are needed in order to make $G$ strong rainbow connected. In this paper, we first investigate the graphs with large strong rainbow connection numbers. Chartrand et al. obtained that $G$ is a tree if and only if $src(G)=m$, we will show that $src(G)\neq m-1$, so $G$ is not a tree if and only if $src(G)\leq m-2$, where $m$ is the number of edge of $G$. Furthermore, we characterize the graphs $G$ with $src(G)=m-2$. We next give a sharp upper bound for $src(G)$ according to the number of edge-disjoint triangles in graph $G$, and give a necessary and sufficient condition for the equality.

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arXiv2010

Note on the Rainbow $k$-Connectivity of Regular Complete Bipartite Graphs

A path in an edge-colored graph $G$, where adjacent edges may be colored the same, is called a rainbow path if no two edges of the path are colored the same. For a $κ$-connected graph $G$ and an integer $k$ with $1\leq k\leq κ$, the rainbow $k$-connectivity $rc_k(G)$ of $G$ is defined as the minimum integer $j$ for which there exists a $j$-edge-coloring of $G$ such that any two distinct vertices of $G$ are connected by $k$ internally disjoint rainbow paths. Denote by $K_{r,r}$ an $r$-regular complete bipartite graph. Chartrand et al. in "G. Chartrand, G.L. Johns, K.A. McKeon, P. Zhang, The rainbow connectivity of a graph, Networks 54(2009), 75-81" left an open question of determining an integer $g(k)$ for which the rainbow $k$-connectivity of $K_{r,r}$ is 3 for every integer $r\geq g(k)$. This short note is to solve this question by showing that $rc_k(K_{r,r})=3$ for every integer $r\geq 2k\lceil\frac{k}{2}\rceil$, where $k\geq 2$ is a positive integer.

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arXiv2010

Upper bounds for the rainbow connection numbers of line graphs

A path in an edge-colored graph $G$, where adjacent edges may be colored the same, is called a rainbow path if no two edges of it are colored the same. A nontrivial connected graph $G$ is rainbow connected if for any two vertices of $G$ there is a rainbow path connecting them. The rainbow connection number of $G$, denoted by $rc(G)$, is defined as the smallest number of colors by using which there is a coloring such that $G$ is rainbow connected. In this paper, we mainly study the rainbow connection number of the line graph of a graph which contains triangles and get two sharp upper bounds for $rc(L(G))$, in terms of the number of edge-disjoint triangles of $G$ where $L(G)$ is the line graph of $G$. We also give results on the iterated line graphs.

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arXiv2009

The rainbow $k$-connectivity of two classes of graphs

A path in an edge-colored graph $G$, where adjacent edges may be colored the same, is called a rainbow path if no two edges of $G$ are colored the same. For a $κ$-connected graph $G$ and an integer $k$ with $1\leq k\leq κ$, the rainbow $k$-connectivity $rc_k(G)$ of $G$ is defined as the minimum integer $j$ for which there exists a $j$-edge-coloring of $G$ such that every two distinct vertices of $G$ are connected by $k$ internally disjoint rainbow paths. Let $G$ be a complete $(\ell+1)$-partite graph with $\ell$ parts of size $r$ and one part of size $p$ where $0\leq p <r$ (in the case $p=0$, $G$ is a complete $\ell$-partite graph with each part of size $r$). This paper is to investigate the rainbow $k$-connectivity of $G$. We show that for every pair of integers $k\geq 2$ and $r\geq 1$, there is an integer $f(k,r)$ such that if $\ell\geq f(k,r)$, then $rc_k(G)=2$. As a consequence, we improve the upper bound of $f(k)$ from $(k+1)^2$ to $ck^{3/2}+C$, where $0<c<1$, $C=o(k^{3/2})$, and $f(k)$ is the integer such that if $n \geq f(k)$ then $rc_k(K_n)=2$.

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