Anti-van der Waerden numbers on Graphs

TL;DR

This paper extends the concept of anti-van der Waerden numbers to graphs, providing bounds and exact values for 3-term arithmetic progressions, and exploring their relation to graph properties and Ramsey numbers.

Contribution

It introduces the anti-van der Waerden number for graphs and derives bounds and exact values for specific classes, connecting it to graph parameters and Ramsey theory.

Findings

Bounds for 3-term progressions based on graph radius and diameter

Exact values for certain classes of graphs

Connection established between Ramsey numbers and anti-van der Waerden numbers

Abstract

In this paper arithmetic progressions on the integers and the integers modulo n are extended to graphs. This allows for the definition of the anti-van der Waerden number of a graph. Much of the focus of this paper is on 3-term arithmetic progressions for which general bounds are obtained based on the radius and diameter of a graph. The general bounds are improved for trees and Cartesian products and exact values are determined for some classes of graphs. Larger k-term arithmetic progressions are considered and a connection between the Ramsey number of paths and the anti-van der Waerden number of graphs is established.