Showing papers by "Yuefang Sun published in 2010"

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

Abstract: 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 �-connected graph G and an integer k with 1 � k � �, the rainbow kconnectivity rck(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 Kr,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 Kr,r is

Rainbow connection numbers of complementary graphs

Abstract: 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 G. We first derive that for a connected graph G, if G does not belong to the following two cases: (i) diam(G) = 2,3, (ii) G contains exactly two connected components and one of them is trivial, then rc(G) � 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 G is triangle-free, then rc(G) � 6.

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

Abstract: 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 $\kappa$-connected graph $G$ and an integer $k$ with $1\leq k\leq \kappa$, 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, GL Johns, KA 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

Upper bounds for the rainbow connection numbers of line graphs

Abstract: 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.