Showing papers by "Yuefang Sun published in 2012"
Book•
23 Feb 2012
TL;DR: This chapter discusses the motivation and definitions for rainbow connection number, and some graph classes, for dense and sparse graphs, and graph operations, an upper bound for strong Rainbow connection number.
Abstract: 1. Introduction (Motivation and definitions, Terminology and notations).- 2. (Strong) Rainbow connection number(Basic results, Upper bounds for rainbow connection number, For some graph classes, For dense and sparse graphs, For graph operations, An upper bound for strong rainbow connection number).- 3. Rainbow k-connectivity.- 4. k-rainbow index.- 5. Rainbow vertex-connection number.- 6. Algorithms and computational complexity.- References.
215 citations
TL;DR: In this paper, it was shown that for any two connected graphs G and H with kappa(3) (G) >= kappa (3)(H) (H), if kappa[G] > kappa [3] (G), then kappa((G) = kappa |H) + kappa|H) ≤ 1.
Abstract: The generalized connectivity of a graph, which was introduced by Chartrand et al. in 1984, is a generalization of the concept of vertex connectivity. Let S be a nonempty set of vertices of G, a collection \T-1, T (2), ... , T-r\ of trees in G is said to be internally disjoint trees connecting S if E(T-i) boolean AND E(T-j) - empty set and V (T-i) boolean AND V(T-j) = S for any pair of distinct integers i, j, where 1 = kappa(G) + kappa(H) for any two connected graphs G and H. In this paper, we prove that for any two connected graphs G and H with kappa(3) (G) >= kappa(3) (H), if kappa(G) > kappa(3) (G), then kappa(3) (G square H) >= kappa(3) (G) + kappa(3) (H); if kappa(G) = kappa(3)(G), then kappa(3)(G square H) >= kappa(3)(G) + kappa(3) (H) - 1. Our result could be seen as an extension of Sabidussi's Theorem. Moreover, all the bounds are sharp.
48 citations
TL;DR: In this article, the authors studied the rainbow connection number of the line graph L(G) of a graph G which contains triangles and gave two sharp upper bounds for rc(L(G)), in terms of the number of edge disjoint triangles of G.
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 rc(G), is defined as the smallest number of colors such that G is rainbow connected. In this paper, we mainly study the rainbow connection number rc(L(G)) of the line graph L(G) of a graph G which contains triangles. We get two sharp upper bounds for rc(L(G)), in terms of the number of edge-disjoint triangles of G. We also give results on the iterated line graphs.
21 citations
01 Jan 2012
TL;DR: Krivelevich and Yuster as discussed by the authors proposed a vertex version of the Rainbow Connection Number (VPN) and proved a theorem analogous to the result of Theorem 3.10.1.
Abstract: All the above parameters on rainbow connections involved edge-colorings of graphs. A natural idea is to introduce a similar parameter involving vertex-colorings of graphs. It is, as mentioned above, a vertex version of the rainbow connection number. Krivelevich and Yuster (J. Graph Theory 63(3):185–191, 2009) is the first to propose this new concept and proved a theorem analogous to the result of Theorem 3.1.10.
4 citations
Posted Content•
TL;DR: This paper improves the lower bound of t(n, 2), and gets a lowerbound of t (n, d) for $$3 \leq d < \lceil\frac{n}{2}\rceil}$$ .
Abstract: An edge-colored graph $G$, where adjacent edges may have the same color, is {\it rainbow connected} if every two vertices of $G$ are connected by a path whose edge has distinct colors. A graph $G$ is {\it $k$-rainbow connected} if one can use $k$ colors to make $G$ rainbow connected. For integers $n$ and $d$ let $t(n,d)$ denote the minimum size (number of edges) in $k$-rainbow connected graphs of order $n$. Schiermeyer got some exact values and upper bounds for $t(n,d)$. However, he did not get a lower bound of $t(n,d)$ for $3\leq d<\lceil\frac{n}{2}\rceil $. In this paper, we improve his lower bound of $t(n,2)$, and get a lower bound of $t(n,d)$ for $3\leq d<\lceil\frac{n}{2}\rceil$.