Oriented diameter and rainbow connection number of a graph

TL;DR

This paper establishes upper bounds for the oriented diameter and rainbow connection number of graphs based on radius, cycle length parameters, and minimum degree, providing new insights into graph connectivity measures.

Contribution

It introduces new upper bounds for oriented diameter and rainbow connection number using radius, cycle length, and minimum degree parameters.

Findings

Upper bounds in terms of radius and cycle length

Constant bounds for bipartite graphs based on minimum degree

Connections between graph parameters and connectivity measures

Abstract

The oriented diameter of a bridgeless graph $G$ is $\min\{diam(H)\ | H\ is\ an orientation\ of\ G\}$. 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 $k$ 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 $\eta(G)$, where $rad(G)$ is the radius of $G$ and $\eta(G)$ is the smallest integer number such that every edge of $G$ is contained in a cycle of length at most $\eta(G)$. 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 $G$.