Anti-van der Waerden Numbers on Graphs

Abstract

In this paper arithmetic progressions on the integers and the integers modulo n are extended to graphs. A k-term arithmetic progression of a graph G (k-AP) is a list of k distinct vertices such that the distance between consecutive pairs is constant. A rainbow k-AP is a k-AP where each vertex is colored distinctly. This allows for the definition of the anti-van der Waerden number of a graph G, which is the least positive integer r such that every exact r-coloring of G contains a rainbow k-AP. 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. Longer 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.Please confirm if the inserted city and country name for all affiliations is correct. Amend if necessary.The cities and affiliations are correct.