# On the Rainbow Mean Indexes of Caterpillars

## Discrete and Computational Geometry, Graphs, and Games

## Keywords:

rainbow mean coloring, rainbow mean index, caterpillars, brooms, double stars, double brooms## Abstract

Let $G$ be a simple connected graph and $c$ an edge coloring with colors that are positive integers. Given a vertex $v$ of $G,$ we define its chromatic mean, denoted by cm$\left( v\right) $, as the average of the colors of the incident edges. If cm$\left( v\right) $ is an integer for each $v\in V\left( G\right) $ and distinct vertices have distinct chromatic means, then $c$ is called a rainbow mean coloring. The maximum chromatic mean of a vertex in the coloring $c$ is called the rainbow mean index of $c$ and is denoted by rm$\left( c\right) $. On the other hand, the rainbow mean index of $G$, denoted by $\operatorname{rm}(G)$, is the minimum value of rm$\left( c\right) $ among all rainbow mean colorings $c$ of $G$. In this paper, we determine the rainbow mean indexes of families of caterpillars, including brooms, and double brooms.

## Downloads

## Published

## How to Cite

*Thai Journal of Mathematics*,

*21*(4), 821–834. Retrieved from https://thaijmath2.in.cmu.ac.th/index.php/thaijmath/article/view/1548