The (strong) rainbow connection numbers of Cayley graphs of Abelian groups

TL;DR

This paper investigates the rainbow connection and strong rainbow connection numbers of Cayley graphs of Abelian groups, providing bounds and exact values for specific cases.

Contribution

It establishes bounds and exact values for the (strong) rainbow connection numbers of Cayley graphs of Abelian groups, advancing understanding in graph coloring and algebraic graph theory.

Findings

Derived upper and lower bounds for rainbow connection numbers.

Determined exact (strong) rainbow connection numbers for certain special Cayley graphs.

Enhanced understanding of coloring properties in algebraic graph structures.

Abstract

A path in an edge-colored graph $G$, where adjacent edges may have the same color, is called a rainbow path if no two edges of the path are colored the same. The rainbow connection number $rc(G)$ of $G$ is the minimum integer $i$ for which there exists an $i$-edge-coloring of $G$ such that every two distinct vertices of $G$ are connected by a rainbow path. The strong rainbow connection number $src(G)$ of $G$ is the minimum integer $i$ for which there exists an $i$-edge-coloring of $G$ such that every two distinct vertices $u$ and $v$ of $G$ are connected by a rainbow path of length $d(u,v)$. In this paper, we give upper and lower bounds of the (strong) rainbow connection Cayley graphs of Abelian groups. Moreover, we determine the (strong) rainbow connection numbers of some special cases.