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 uβv geodesic in G is a rainbow uβv
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 uβv
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)ξ =mβ1, so G is not a tree if and only if src(G)β€mβ2, where
m is the number of edge of G. Furthermore, we characterize the graphs G
with src(G)=mβ2. 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.Comment: 16 page