On L(2,1)-labelings of some products of oriented cycles

TL;DR

This paper refines previous results on the $L(2,1)$-labeling number for Cartesian and strong products of oriented cycles, providing exact values or bounds for large cycle lengths.

Contribution

It computes exact $L(2,1)$-labeling numbers for Cartesian products of large oriented cycles and establishes bounds for strong products, improving understanding of these graph labelings.

Findings

Exact $L(2,1)$-labeling numbers for Cartesian products with cycles ≥40.

Bounds and gaps for strong product labelings with cycles ≥48.

Enhanced understanding of labelings in product graphs.

Abstract

We refine two results of Jiang, Shao and Vesel on the $L(2,1)$-labeling number $\lambda$ of the Cartesian and the strong product of two oriented cycles. For the Cartesian product, we compute the exact value of $\lambda(\overrightarrow{C_m} \square \overrightarrow{C_n})$ for $m$, $n \geq 40$; in the case of strong product, we either compute the exact value or establish a gap of size one for $\lambda(\overrightarrow{C_m} \boxtimes \overrightarrow{C_n})$ for $m$, $n \geq 48$.