Maximizing and minimizing the number of generalized colorings of trees

TL;DR

This paper classifies trees with the maximum and minimum counts of various generalized colorings, extending previous results on homomorphisms to looped graphs, providing a comprehensive understanding of coloring extremities in trees.

Contribution

It provides a complete classification of trees with extremal numbers of several generalized colorings and extends existing results on homomorphisms to looped graphs.

Findings

Identified trees with maximum and minimum generalized colorings

Extended results on existence homomorphisms to looped graphs

Provided a unified framework for coloring extremities in trees

Abstract

We classify the trees on $n$ vertices with the maximum and the minimum number of certain generalized colorings, including conflict-free, odd, non-monochromatic, star, and star rainbow vertex colorings. We also extend a result of Cutler and Radcliffe on the maximum and minimum number of existence homomorphisms from a tree to a completely looped graph on $q$ vertices.