Oriented diameter and rainbow connection number of a graph
Xiaolong Huang, Hengzhe Li, Xueliang Li, Yuefang Sun
Abstract
The oriented diameter of a bridgeless graph G is
min { diam(H) | H is a strang orientation of
G}. A path in an edge-colored graph
G, where adjacent edges may have the same color, is
called rainbow if no two edges of the path are colored the same. The
rainbow connection number rc(G) of G is the
smallest integer number k for which there exists a
k-edge-coloring of G such that every two
distinct vertices of G are connected by a rainbow
path. In this paper, we obtain upper bounds for the oriented diameter
and the rainbow connection number of a graph in terms of
rad(G) and η(G), where
rad(G) is the radius of G and
η(G) is the smallest integer number such that every
edge of G is contained in a cycle of length at most
η(G). We also obtain constant bounds of the oriented
diameter and the rainbow connection number for a (bipartite) graph
G in terms of the minimum degree of G.
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