Anti-van der Waerden Numbers of Graph Products

TL;DR

This paper studies the anti-van der Waerden numbers in Cartesian graph products, proving an upper bound of four and determining exact values for products of paths, advancing understanding of combinatorial colorings in graph theory.

Contribution

It proves a universal upper bound of four for the anti-van der Waerden number of Cartesian graph products and determines exact values for products of paths.

Findings

Anti-van der Waerden number of Cartesian product of two graphs is at most four.

Exact anti-van der Waerden numbers are found for Cartesian products of paths.

The results confirm a conjecture and extend previous bounds in graph coloring theory.

Abstract

In this paper, anti-van der Waerden numbers on Cartesian products of graphs are investigated and a conjecture made by Schulte, et al (see arXiv:1802.01509) is answered. In particular, the anti-van der Waerden number of the Cartesian product of two graphs has an upper bound of four. This result is then used to determine the anti-van der Waerden number for any Cartesian product of two paths.