Results for grundy number of the complement of bipartite graphs

TL;DR

This paper investigates the Grundy chromatic number of the complement of bipartite graphs, providing an interpretation via total graphs and proving the problem's NP-Completeness.

Contribution

It offers a new interpretation of the Grundy number in terms of total graphs for complements of bipartite graphs and establishes the computational complexity as NP-Complete.

Findings

Interpretation of Grundy number via total graphs

Proved NP-Completeness of computing the Grundy number

Highlights computational difficulty in graph coloring problems

Abstract

A Grundy k-coloring of a graph G, is a vertex k-coloring of G such that for each two colors i and j with i < j, every vertex of G colored by j has a neighbor with color i. The Grundy chromatic number (G), is the largest integer k for which there exists a Grundy k-coloring for G. In this note we first give an interpretation of (G) in terms of the total graph of G, when G is the complement of a bipartite graph. Then we prove that determining the Grundy number of the complement of bipartite graphs is an NP-Complete problem