A linear algorithm for the grundy number of a tree

TL;DR

This paper introduces a linear-time algorithm to determine the Grundy number of a tree, providing bounds and conditions for equality related to girth, advancing understanding of graph colorings.

Contribution

It presents a natural upper bound on the Grundy number and shows trees with large girth meet this bound, leading to an efficient algorithm for trees.

Findings

Derived a natural upper bound on the Grundy number.

Proved trees with large girth achieve equality in the bound.

Developed a linear time algorithm for trees' Grundy number.

Abstract

A coloring of a graph G = (V,E) is a partition {V1, V2, . . ., Vk} of V into independent sets or color classes. A vertex v Vi is a Grundy vertex if it is adjacent to at least one vertex in each color class Vj . A coloring is a Grundy coloring if every color class contains at least one Grundy vertex, and the Grundy number of a graph is the maximum number of colors in a Grundy coloring. We derive a natural upper bound on this parameter and show that graphs with sufficiently large girth achieve equality in the bound. In particular, this gives a linear time algorithm to determine the Grundy number of a tree.