2025/03/31 by Zhanar Berikkyzy, Berikkyzy, Zhanar, Miller, Joe +2
Computer Science · Mathematics · #05C05 #05C12 #05C15 #05C35 #05C76 #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Graph Labeling and Dimension Problems #Limits and Structures in Graph Theory
paper · pdf · doi:10.48550/arxiv.2504.00288
openalex publication_date 2025/03/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The anti-van der Waerden number of a graph G is the fewest number of colors needed to guarantee a rainbow 3-term arithmetic progression in G, denoted aw(G,3). It is known that the anti-van der Waerden number of graph products is 3 ≤ aw(G\square H,3)≤ 4. Previous work has been done on classifying families of graph products into aw(G\square H,3) = 3 and aw(G\square H,3) = 4. Some of these families include the product of two paths, the product of paths and cycles, the product of two cycles, and the product of odd cycles with any graph. Recently, a partial characterization of the product of two trees was established. This paper completes the characterization for aw(T\square T',3) where T and T' are trees. Moreover, this result extends to a full classification of products of forests.