Anti-van der Waerden Numbers of Graph Products of Cycles

TL;DR

This paper determines the anti-van der Waerden number for specific graph products involving cycles, focusing on 3-term arithmetic progressions and providing exact values for various Cartesian products of paths and cycles.

Contribution

It precisely computes the anti-van der Waerden number for 3-APs in certain graph products involving cycles, extending understanding of rainbow arithmetic progressions in these graphs.

Findings

Exact anti-van der Waerden numbers for P_m d7 C_n, C_m d7 C_n, and G d7 C_{2n+1}

Characterization of rainbow 3-APs in these graph products

Extension of anti-van der Waerden theory to complex graph structures

Abstract

A $k$-term arithmetic progression ($k$-AP) in a graph $G$ is a list of vertices such that each consecutive pair of vertices is the same distance apart. If $c$ is a coloring function of the vertices of $G$ and a $k$-AP in $G$ has each vertex colored distinctly, then that $k$-AP is a rainbow $k$-AP. The anti-van der Waerden number of a graph $G$ with respect to $k$ is the least positive integer $r$ such that every surjective coloring with domain $V(G)$ and codomain $\{1,2,\dots,r\} = [r]$ is guaranteed to have a rainbow $k$-AP. This paper focuses on $3$-APs and graph products with cycles. Specifically, the anti-van der Waerden number with respect to $3$ is determined precisely for $P_m \square C_n$, $C_m\square C_n$ and $G\square C_{2n+1}$.