Full classification of anti-van der Waerden numbers of graph products of forests

TL;DR

The paper resolves the full classification of the anti-van der Waerden numbers for graph products of forests by determining $aw(T\square T',3)$ for trees $T,T'$. Building on the known bound $3\le aw(G\square H,3)\le 4$, it introduces and leverages concepts such as 3-peripheral, strongly non-$3$-peripheral, and weakly non-$3$-peripheral trees, along with the diameter-modifying operations $T_{v^-}$ and $T_{v^+}$. It shows that when $\operatorname{diam}(T\square T')$ is even, $aw(T\square T',3)=3$ if either tree is weakly non-$3$-peripheral or is $P_2$, and $=4$ if both are strongly non-$3$-peripheral and not $P_2$, with the argument extended to forest products. The results yield a complete classification for graph products of forests and provide structural insights into rainbow-free colorings in product graphs, including corollaries for disconnected components.

Abstract

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 $\operatorname{aw}(G,3)$. It is known that the anti-van der Waerden number of graph products is $3 \le \operatorname{aw}(G\square H,3)\le 4$. Previous work has been done on classifying families of graph products into $\operatorname{aw}(G\square H,3) = 3$ and $\operatorname{aw}(G\square H,3) = 4$. 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 $\operatorname{aw}(T\square T',3)$ where $T$ and $T'$ are trees. Moreover, this result extends to a full classification of products of forests.

Figures (2)

• Figure 1: The product of $G = P_4$ and broom graph $H$. The subgraph $G_1$ is bolded and the subgraph $H_3$ is dashed. Graph $H$ is an example of a weakly non-$3$-peripheral tree of odd diameter (see Definition \ref{['defn:weak/strong non-3-per']}) while the path $G$ is an example of a strongly non-$3$-peripheral graph of odd diameter.
• Figure 2: Relationship between peripheral vertices $u,x,y$ and $v$ when $d(u,s) \le d(u,t)$, as in Lemma \ref{['general tree peripheral lem']}.

Theorems & Definitions (41)

• Theorem 1.1: RSW
• Theorem 1.2: RSW
• Theorem 1.3
• Corollary 1.4
• Proposition 2.1: MW
• Corollary 2.2: MW
• Lemma 2.3: RSW
• Lemma 2.4: MW
• Corollary 2.5: MW
• Proposition 2.6: MW